Generalized John Polynomials

1. Introduction

Pell sequence ${\left\{P_n\right\}}_{n\ge 0}$ (OEIS: A000129, [1]) and Pell-Lucas sequence ${\left\{Q_n\right\}}_{n\ge 0}$ (OEIS: A002203, [1]) are defined by the second-order recurrence relations

$$P_n=2P_{n-1}+P_{n-2},P_0=0,P_1=1$$

(1)

and

$$Q_n=2Q_{n-1}+Q_{n-2},Q_0=2,Q_1=2.$$

(2)

The sequences ${\left\{P_n\right\}}_{n\ge 0}$ and ${\left\{Q_n\right\}}_{n\ge 0}$ can be extended to negative subscripts by defining

$$P_{-n}=-2P_{-(n-1)}+P_{-(n-2)}$$

and

$$Q_{-n}=-2Q_{-(n-1)}+Q_{-(n-2)}$$

for $n=1,2,3,...$ respectively. Therefore, recurrences (1) and (2) hold for all integer $n.$

Pell sequence has been studied by many authors and more detail can be found in the extensive literature dedicated to these sequences, see for example, [2,3,4,5,6,7,8,9]. For higher order Pell sequences, see [10,11].

Now, Soykan [12] define two sequences related to Pell and Pell-Lucas numbers. John and John-Lucasnumbers are defined as

$$J_n=2J_{n-1}+J_{n-2}+1,\mathrm{with}{J}_0=0,J_1=1,n\ge 2,$$

and

$$H_n=2H_{n-1}+H_{n-2}-2,\mathrm{with}{H}_0=3,H_1=3,n\ge 2,$$

respectively. The first few values of John and John-Lucas numbers are

$$0,1,3,8,20,49,119,288,696,1681,4059,\dots$$

and

$$3,3,7,15,35,83,199,479,1155,2787,6727,\dots$$

respectively. The sequences $\left\{J_n\right\}$ and $\left\{H_n\right\}$ satisfy the following third order linear recurences:

$$\begin{array}{cc}\hfill J_n& =3J_{n-1}-J_{n-2}-J_{n-3},J_0=0,J_1=1,J_2=3,\hfill \\ \hfill H_n& =3H_{n-1}-H_{n-2}-H_{n-3},H_0=3,H_1=3,H_2=7.\hfill \end{array}$$

There are close relations between Johnand John-Lucas and Pell, Pell-Lucas numbers. For example, they satisfy the following interrelations:

$$\begin{array}{cc}\hfill J_n& ={\displaystyle \frac{1}{2}}(P_{n+2}-P_{n+1}-1),\hfill \\ \hfill H_n& =Q_n+1,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill J_n& ={\displaystyle \frac{1}{4}}(Q_{n+1}-2),\hfill \\ \hfill H_n& =2P_{n+1}-2P_n+1.\hfill \end{array}$$

In the study of number theory [13], MinMax numbers $\left\{M_n\right\}$ and their associated subsidiary numbers $\left\{N_n\right\}$ for Pell numbers $\left\{P_n\right\}$ represent a specialized class of integers where minimal and maximal representations coincide. A. F. Horadam [14] extended these concepts into the domain of algebraic polynomials, defining sequences such as $\left\{M_n\left(x\right)\right\}$, $\left\{N_n\left(x\right)\right\}$, $\left\{{\mathfrak{L}}_n\left(x\right)\right\}$, and $\left\{{\mathfrak{R}}_n\left(x\right)\right\}$ see (23)-(26).

The foundation of these systems lies in the Pell polynomials [16] $\left\{P_n\left(x\right)\right\}$, which are defined by the following second-order recurrence relation:

$$P_n\left(x\right)=2x{P}_{n-1}\left(x\right)+P_{n-2}\left(x\right),n\ge 2$$

(3)

with the initial condition $P_0\left(x\right)=0,$ $P_1\left(x\right)=1$ and the Pell-Lucas polynomials $\left\{Q_n\left(x\right)\right\}$, which satisfy:

$$Q_{n+2}\left(x\right)=2x{Q}_{n+1}\left(x\right)+Q_n\left(x\right),n\ge 2$$

(4)

with the initial condition $Q_0\left(x\right)=2,Q_1\left(x\right)=2x.$

Horadam introduce some key results related to $P_n\left(x\right)$ and $\left\{Q_n\left(x\right)\right\}$ in [14]. Some of these result is given as follows:

• The generating functions for the polynomials $\left\{P_n\left(x\right)\right\}$ and $\left\{Q_n\left(x\right)\right\}$ are

$${\displaystyle \sum _{r=0}^{\infty }}{P}_{r+1}\left(x\right)y^r={\displaystyle \frac{1}{1-2xy-y^2}}$$

(5)

and

$${\displaystyle \sum _{r=0}^{\infty }}{Q}_{r+1}\left(x\right)y^r={\displaystyle \frac{2x+2y}{1-2xy-y^2}}.$$

(6)

• Important elementary relationships involving polynomials $\left\{P_n\left(x\right)\right\}$ and $\left\{Q_n\left(x\right)\right\}$ can be given as:

  $$\begin{array}{cc}\hfill P_{n+1}\left(x\right)+P_{n-1}\left(x\right)& =Q_n\left(x\right)=2x{P}_n\left(x\right)+2P_{n-1}\left(x\right),\hfill \end{array}$$

  (7)

  $$\begin{array}{cc}\hfill Q_{n+1}\left(x\right)+Q_{n-1}\left(x\right)& =4(x^2+1)P_n\left(x\right),\hfill \end{array}$$

  (8)

  $$\begin{array}{cc}\hfill P_n\left(x\right)Q_n\left(x\right)& =P_{2n}\left(x\right),\hfill \end{array}$$

  (9)

  $$\begin{array}{cc}\hfill Q_{2n}\left(x\right)& ={\displaystyle \frac{1}{2}}\left(Q_n^2\left(x\right)+4(x^2+1)P_n^2\left(x\right)\right),\hfill \end{array}$$

  (10)

  $$\begin{array}{cc}\hfill P_{n+1}\left(x\right)P_{n-1}\left(x\right)-P_n^2\left(x\right)& ={\left(-1\right)}^n,\hfill \end{array}$$

  (11)

  $$\begin{array}{cc}\hfill Q_{n+1}\left(x\right)Q_{n-1}\left(x\right)-Q_n^2\left(x\right)& ={\left(-1\right)}^{n-1}4(x^2+1),\hfill \end{array}$$

  (12)

  $$\begin{array}{cc}\hfill P_{n+1}^2\left(x\right)-P_{n-1}^2\left(x\right)& =2x{P}_{2n}\left(x\right),\hfill \end{array}$$

  (13)

  $$\begin{array}{cc}\hfill 4(x^2+1)P_n^2\left(x\right)-Q_n^2\left(x\right)& =4{(-1)}^{n-1}.\hfill \end{array}$$

  (14)
• Summations of polynomials $\left\{P_n\left(x\right)\right\}$ and $\left\{Q_n\left(x\right)\right\}$ can be given as:

  $$\begin{array}{cc}\hfill {\displaystyle \sum _{r=1}^n}{P}_{2r}\left(x\right)& ={\displaystyle \frac{(P_{2n+1}\left(x\right)-1)}{2x}},\hfill \end{array}$$

  (15)

  $$\begin{array}{cc}\hfill {\displaystyle \sum _{r=1}^n}{P}_{2r-1}\left(x\right)& ={\displaystyle \frac{P_{2n}\left(x\right)}{2x}},\hfill \end{array}$$

  (16)

  $$\begin{array}{cc}\hfill {\displaystyle \sum _{r=1}^n}{P}_r\left(x\right)& ={\displaystyle \frac{(P_{n+1}\left(x\right)+P_n\left(x\right)-1)}{2x}},\hfill \end{array}$$

  (17)

  $$\begin{array}{cc}\hfill {\displaystyle \sum _{r=1}^n}{Q}_{2r}\left(x\right)& ={\displaystyle \frac{(Q_{2n+1}\left(x\right)-2x)}{2x}},\hfill \end{array}$$

  (18)

  $$\begin{array}{cc}\hfill {\displaystyle \sum _{r=1}^n}{Q}_{2r-1}\left(x\right)& ={\displaystyle \frac{(Q_{2n}\left(x\right)-2)}{2x}},\hfill \end{array}$$

  (19)

  $$\begin{array}{cc}\hfill {\displaystyle \sum _{r=1}^n}{Q}_r\left(x\right)& ={\displaystyle \frac{(Q_{n+1}\left(x\right)+Q_n\left(x\right)-2-2x)}{2x}}.\hfill \end{array}$$

  (20)
• The explicit expressions of Pell and Pell-Lucas polynomials are

  $$P_n\left(x\right)={\displaystyle \sum _{m=0}^{\left[{\displaystyle \frac{n-1}{2}}\right]}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-m-1}{m}}\right){\left(2x\right)}^{n-2m-1}$$

  (21)

and

$$Q_n\left(x\right)={\displaystyle \sum _{m=0}^{\left[{\displaystyle \frac{n}{2}}\right]}}{\displaystyle \frac{n}{n-m}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-m}{m}}\right){\left(2x\right)}^{n-2m},n\ne 0,$$

(22)

Building upon these structures, Horadam introduced the polynomials $\left\{M_n\left(x\right)\right\},$ $\left\{N_n\left(x\right)\right\},$ $\left\{{\mathfrak{L}}_n\left(x\right)\right\}$ and $\left\{{\mathfrak{R}}_n\left(x\right)\right\},$ see [14], with second order non-homogeneous recurence relations as follows.

• The MinMax polynomials $\left\{M_n\left(x\right)\right\},$ characterized by a non-homogeneous recurrence relation:

  $$M_n\left(x\right)=2x{M}_{n-1}\left(x\right)+M_{n-2}\left(x\right)+1,n\ge 2$$

  (23)

  with the initial condition $M_0\left(x\right)=0,$ $M_1\left(x\right)=1$.
• The corresponding subsidiary MinMax polynomials $\left\{N_n\left(x\right)\right\}$, related to the former by the identity

  $$N_n\left(x\right)=M_{n+1}\left(x\right)+M_{n-1}\left(x\right),N_0\left(x\right)=1$$

  were shown to satisfy:

  $$N_n\left(x\right)=2x{N}_{n-1}\left(x\right)+N_{n-2}\left(x\right)+2,n\ge 2$$

  (24)

  with the initial condition $N_0\left(x\right)=1,$ $N_1\left(x\right)=2x+1.$
• Furthermore, for the modified Pell sequence, the MinMax polynomials $\left\{{\mathfrak{L}}_n\left(x\right)\right\}$ and their subsidiary counterparts $\left\{{\mathfrak{R}}_n\left(x\right)\right\}$ are defined as follows:

  $${\mathfrak{L}}_n\left(x\right)=2x{\mathfrak{L}}_{n-1}\left(x\right)+{\mathfrak{L}}_{n-2}\left(x\right)+2,n\ge 2$$

  (25)

  with the initial condition ${\mathfrak{L}}_0\left(x\right)=0,{\mathfrak{L}}_1\left(x\right)=1.$

  $${\mathfrak{R}}_n\left(x\right)={\mathfrak{L}}_{n+1}\left(x\right)+{\mathfrak{L}}_{n-1}\left(x\right),{\mathfrak{R}}_0\left(x\right)=0$$

  $${\mathfrak{R}}_n\left(x\right)=2x{\mathfrak{R}}_{n-1}\left(x\right)+{\mathfrak{R}}_{n-2}\left(x\right)+4,n\ge 2$$

  (26)

  with the initial condition ${\mathfrak{R}}_0\left(x\right)=0,$ ${\mathfrak{R}}_0\left(x\right)=2x+2$.

2. Generalized John Polynomials

In this section, we provide a comprehensive analysis of the generalized Leonardo type Pell polynomials which is called generalized John polynomials in the rest of the study, exploring its fundamental algebraic and structural characteristics. We begin by establishing its Binet-type formula and the corresponding generating function, which serve as the primary tools for our subsequent derivations. Our investigation then extends to several classical identities well-documented in the literature, including those of Catalan, Cassini, Vajda, and d’Ocagne, demonstrating how these properties are integrated into this broader non-homogeneous framework. Furthermore, we present an explicit representation of these polynomials, typically expressed through nested summations or related sequences. In addition to these identities, we establish various theorems regarding matrix representations and determinantal values, which highlight unique properties that remain independent of the variable x. To initiate our formal treatment, we define the generalized $p\left(x\right)-$John polynomials through a second-order linear non-homogeneous recurrence relation as follows.

Definition 1.

Let n be an integer with $n\ge 2$, then the generalized $p\left(x\right)-$John polynomials is defined, with second order linear non homogeneous recurrence relation:

$$L_{n,p\left(x\right)}\left(x\right)=2x{L}_{\left(n-1\right),p\left(x\right)}\left(x\right)+L_{\left(n-2\right),p\left(x\right)}\left(x\right)+p\left(x\right)$$

(27)

with the initial condition $L_{0,p\left(x\right)}\left(x\right)$ and $L_{1,p\left(x\right)}\left(x\right)$ where $p\left(x\right)$ is an arbitrary polynomial in x.

If we take $p\left(x\right)=0,$ we obtain generalized Pell polynomial but in this study we assume that $p\left(x\right)\ne 0$ unless otherwise stated.

Next, we define four distinct cases of generalized $p\left(x\right)-$John polynomials, which satisfy a second-order linear non-homogeneous recurrence relation genereted by (27), as follows.

Definition 2.

For $n\ge 2$,

(a) 

Generalized $p_1\left(x\right)-$John polynomial is defined as

$$M_{n,p_1\left(x\right)}\left(x\right)=2x{M}_{\left(n-1\right),p_1\left(x\right)}\left(x\right)+M_{\left(n-2\right),p_1\left(x\right)}\left(x\right)+p_1\left(x\right)$$

(28)

with the initial condition $M_{0,p_1\left(x\right)}\left(x\right)=0$ and $M_{1,p_1\left(x\right)}\left(x\right)=1$ where $p_1\left(x\right)$ is a polynomial in $x.$

(b) 

Generalized $p_2\left(x\right)-$John polynomial is defined as

$$N_{n,p_2\left(x\right)}\left(x\right)=2x{N}_{\left(n-1\right),p_2\left(x\right)}\left(x\right)+N_{\left(n-2\right),p_2\left(x\right)}\left(x\right)+p_2\left(x\right)$$

(29)

with the initial condition $N_{0,p_2\left(x\right)}\left(x\right)=1$ and $N_{1,p_2\left(x\right)}\left(x\right)=2x+1$ where $p_2\left(x\right)$ is a polynomial in $x.$

(c) 

Generalized $p_3\left(x\right)-$John polynomial is defined as

$$R_{n,p_3\left(x\right)}\left(x\right)=2x{R}_{\left(n-1\right),p_3\left(x\right)}\left(x\right)+R_{\left(n-2\right),p_3\left(x\right)}\left(x\right)+p_3\left(x\right)$$

(30)

with the initial condition $R_{0,p_3\left(x\right)}\left(x\right)=0$ and $R_{1,p_3\left(x\right)}\left(x\right)=2x+2$ where $p_3\left(x\right)$ is a polynomial in $x.$

(d) 

Generalized $p_4\left(x\right)-$John polynomial is defined as

$$H_{n,p_4\left(x\right)}\left(x\right)=2x{H}_{\left(n-1\right),p_4\left(x\right)}\left(x\right)+H_{\left(n-2\right),p_4\left(x\right)}\left(x\right)+p_4\left(x\right)$$

(31)

with the initial condition $H_{0,p_4\left(x\right)}\left(x\right)=3$ and $H_{1,p_4\left(x\right)}\left(x\right)=2x+1$ where $p_4\left(x\right)$ is a polynomial in $x.$

It is clear that, for taking each polynomial $p\left(x\right)$ in the recurrence relations (28)-(31) generate distinct $p\left(x\right)-$Leonardo-Pell polynomials.

Theorem 1.

For $n\ge 2$, we give closed form solution of generalized $p\left(x\right)-$John polynomials as follows:

$$L_{n,p\left(x\right)}\left(x\right)=L_{n,p\left(x\right)}^{\left(h\right)}\left(x\right)+L_{n,p\left(x\right)}^{\left(p\right)}\left(x\right)$$

(32)

where the particular solution is

$$L_{n,p\left(x\right)}^{\left(p\right)}\left(x\right)=A\left(x\right)=-{\displaystyle \frac{p\left(x\right)}{2x}}$$

and homogeneous solution is

$$L_{n,p\left(x\right)}^{\left(h\right)}\left(x\right)={\displaystyle \frac{L_{1,p\left(x\right)}^{\left(h\right)}\left(x\right)-\beta L_{0,p\left(x\right)}^{\left(h\right)}\left(x\right)}{\alpha -\beta }}{\alpha }^n-{\displaystyle \frac{L_{1,p\left(x\right)}^{\left(h\right)}\left(x\right)-\alpha L_{0,p\left(x\right)}^{\left(h\right)}\left(x\right)}{\alpha -\beta }}{\beta }^n$$

with the arbitrary initial condition $L_{0,p\left(x\right)}^{\left(h\right)}\left(x\right)=L_{0,p\left(x\right)}\left(x\right)-A\left(x\right),$ $L_{1,p\left(x\right)}^{\left(h\right)}\left(x\right)=L_{1,p\left(x\right)}\left(x\right)-A\left(x\right).$

Proof. The generalized $p\left(x\right)-$John polynomials defined by (27) can be given with homogeneous solution denoted as $L_{n,p\left(x\right)}^{\left(h\right)}\left(x\right)$ and a particular solution denoted as $L_{n,p\left(x\right)}^{\left(p\right)}\left(x\right).$ So $L_{n,p\left(x\right)}\left(x\right)$ can be given as

$$L_{n,p\left(x\right)}\left(x\right)=L_{n,p\left(x\right)}^{\left(h\right)}\left(x\right)+L_{n,p\left(x\right)}^{\left(p\right)}\left(x\right).$$

The homogeneous part has the second order recurrence relation given as follow:

$$L_{n,p\left(x\right)}^{\left(h\right)}\left(x\right)=2x{L}_{n-1,p\left(x\right)}^{\left(h\right)}\left(x\right)+L_{n-2,p\left(x\right)}^{\left(h\right)}\left(x\right)$$

(33)

with the initial condition $L_{0,p\left(x\right)}^{\left(h\right)}\left(x\right)$ and $L_{1,p\left(x\right)}^{\left(h\right)}\left(x\right).$

Since the non-homogeneous term of (27) is independent of the index $n,$ the particular solution $L_{n,p\left(x\right)}^{\left(p\right)}\left(x\right)$ is the form of a polynomial $A\left(x\right).$ Substituting this constant particular solution into the recurrence relation (32) , we obtain $A\left(x\right)=$ -${\displaystyle \frac{p\left(x\right)}{2x}}.$ (33) leads to the characteristic equation

$$y^2-2xy-1=0$$

where the roots are $\alpha =x+\sqrt{x^2+1}$ and $\beta =x-\sqrt{x^2+1}$ where $\left|x\right|\ne i.$ Using [[15], Theorem 1] we obtain

$$L_{n,p\left(x\right)}^{\left(h\right)}\left(x\right)={\displaystyle \frac{L_{1,p\left(x\right)}^{\left(h\right)}\left(x\right)-\beta L_{0,p\left(x\right)}^{\left(h\right)}\left(x\right)}{\alpha -\beta }}{\alpha }^n-{\displaystyle \frac{L_{1,p\left(x\right)}^{\left(h\right)}\left(x\right)-\alpha L_{0,p\left(x\right)}^{\left(h\right)}\left(x\right)}{\alpha -\beta }}{\beta }^n$$

with the initial condition $L_{0,p\left(x\right)}^{\left(h\right)}\left(x\right)=L_{0,p\left(x\right)}\left(x\right)-A\left(x\right)$ and $L_{1,p\left(x\right)}^{\left(h\right)}\left(x\right)=L_{1,p\left(x\right)}\left(x\right)-A\left(x\right).$ This completes the proof. □

Having established the closed-form solution of $L_{n,p\left(x\right)}\left(x\right)$ in Theorem 1, we now refine this result into a more explicit representation, as formulated in the following lemma.

Lemma 1.

The identity (32) can be written as follows.

$$L_{n,p\left(x\right)}\left(x\right)=L_{1,p\left(x\right)}^{\left(h\right)}\left(x\right)P_n\left(x\right)-L_{0,p\left(x\right)}^{\left(h\right)}\left(x\right)P_{n-1}\left(x\right)-{\displaystyle \frac{p\left(x\right)}{2x}}.$$

(34)

Proof. Using the Binet’s formula of Pell polynomial, [16], identity (1.7)],

$$P_n\left(x\right)={\displaystyle \frac{{\alpha }^n-{\beta }^n}{\alpha -\beta }}$$

the proof follows easily. □

Using Theorem 1, we present four distinct cases of generalized $p\left(x\right)-$John polynomials recursively as follows.

Corollary 1.

For $n\ge 2,$ we have

(a) 

$$M_{n,p_1\left(x\right)}\left(x\right)={\displaystyle \frac{M_{1,p_1\left(x\right)}^{\left(h\right)}\left(x\right)-\beta M_{0,p_1\left(x\right)}^{\left(h\right)}\left(x\right)}{\alpha -\beta }}{\alpha }^n-{\displaystyle \frac{M_{1,p_1\left(x\right)}^{\left(h\right)}\left(x\right)-\alpha M_{0,p_1\left(x\right)}^{\left(h\right)}\left(x\right)}{\alpha -\beta }}{\beta }^n-{\displaystyle \frac{p_1\left(x\right)}{2x}}$$

with the initial condition $M_{0,p_1\left(x\right)}^{\left(h\right)}\left(x\right)={\displaystyle \frac{p_1\left(x\right)}{2x}}$ and $M_{1,p_1\left(x\right)}^{\left(h\right)}\left(x\right)=1+{\displaystyle \frac{p_1\left(x\right)}{2x}}.$

(b) 

$$N_{n,p\left(x\right)}\left(x\right)={\displaystyle \frac{N_{1,p_2\left(x\right)}^{\left(h\right)}\left(x\right)-\beta N_{0,p_2\left(x\right)}^{\left(h\right)}\left(x\right)}{\alpha -\beta }}{\alpha }^n-{\displaystyle \frac{N_{1,p_2\left(x\right)}^{\left(h\right)}\left(x\right)-\alpha N_{0,p_2\left(x\right)}^{\left(h\right)}\left(x\right)}{\alpha -\beta }}{\beta }^n-{\displaystyle \frac{p_2\left(x\right)}{2x}}$$

with the initial condition $N_{0,p_2\left(x\right)}^{\left(h\right)}\left(x\right)=1+{\displaystyle \frac{p_2\left(x\right)}{2x}}$ and $N_{1,p_2\left(x\right)}^{\left(h\right)}\left(x\right)=2x+1+{\displaystyle \frac{p_2\left(x\right)}{2x}}.$

(c) 

$$R_{n,p_3\left(x\right)}\left(x\right)={\displaystyle \frac{R_{1,p_3\left(x\right)}^{\left(h\right)}\left(x\right)-\beta R_{0,p_3\left(x\right)}^{\left(h\right)}\left(x\right)}{\alpha -\beta }}{\alpha }^n-{\displaystyle \frac{R_{1,p_3\left(x\right)}^{\left(h\right)}\left(x\right)-\alpha R_{0,p_3\left(x\right)}^{\left(h\right)}\left(x\right)}{\alpha -\beta }}{\beta }^n-{\displaystyle \frac{p_3\left(x\right)}{2x}}$$

with the initial condition $R_{0,p_3\left(x\right)}^{\left(h\right)}\left(x\right)={\displaystyle \frac{p_3\left(x\right)}{2x}}$ and $R_{1,p_3\left(x\right)}^{\left(h\right)}\left(x\right)=2x+2+{\displaystyle \frac{p_3\left(x\right)}{2x}}.$

(d) 

$$H_{n,p_4\left(x\right)}\left(x\right)={\displaystyle \frac{H_{1,p_4\left(x\right)}^{\left(h\right)}\left(x\right)-\beta H_{0,p_4\left(x\right)}^{\left(h\right)}\left(x\right)}{\alpha -\beta }}{\alpha }^n-{\displaystyle \frac{H_{1,p_4\left(x\right)}^{\left(h\right)}\left(x\right)-\alpha H_{0,p_4\left(x\right)}^{\left(h\right)}\left(x\right)}{\alpha -\beta }}{\beta }^n-{\displaystyle \frac{p_4\left(x\right)}{2x}}$$

with the initial condition $H_{0,p_4\left(x\right)}^{\left(h\right)}\left(x\right)=3+{\displaystyle \frac{p_4\left(x\right)}{2x}}$ and $H_{1,p_4\left(x\right)}^{\left(h\right)}\left(x\right)=2x+1+{\displaystyle \frac{p_4\left(x\right)}{2x}}.$

Now, we define five special types of $p\left(x\right)-$John polynomial with assuming special cases of $p\left(x\right)$. Here $\left\{M_n\left(x\right)\right\}$, $\left\{N_n\left(x\right)\right\}$, $\left\{{\mathfrak{L}}_n\left(x\right)\right\}$, and $\left\{{\mathfrak{R}}_n\left(x\right)\right\}$ are defined by A.F Horadam in [14] but in this study we define John-Lucas polynomial denoted as $\left\{H_n\left(x\right)\right\}.$

Notation: We denote $M_{n,1}\left(x\right),$ $N_{n,2}\left(x\right),$ $M_{n,2}\left(x\right),$ ${\mathfrak{R}}_{n,4}\left(x\right),$ $H_{n,\left(-2x\right)}\left(x\right)$ as $M_n\left(x\right)$, $N_n\left(x\right)$, ${\mathfrak{L}}_n\left(x\right)$, ${\mathfrak{R}}_n\left(x\right),$ $H_n\left(x\right),$ respectively.

Definition 3.

For $n\ge 2,$ we have

(a) 

By taking $p_1\left(x\right)=1$in Definition 2, (a) MinMax polynomial $M_n\left(x\right)$ can be defined as ,

$$M_n\left(x\right)=2x{M}_{n-1}\left(x\right)+M_{n-2}\left(x\right)+1$$

with the initial condition $M_0\left(x\right)=0$ and $M_1\left(x\right)=1.$ Note that: MinMax polynomial $M_n\left(x\right)$ can be called as John polynomial.

(b) 

By taking $p_2\left(x\right)=2$in Definition 2, (b) Subsidiary MinMax polynomial $N_n\left(x\right)$ can be defined as, ,

$$N_n\left(x\right)=2x{N}_{n-1}\left(x\right)+N_{n-2}\left(x\right)+2$$

with the initial condition $N_0\left(x\right)=1$ and $N_1\left(x\right)=2x+1.$

(c) 

By taking $p_1\left(x\right)=2$ in Definition 2, (a) MinMax polynomial ${\mathfrak{L}}_n\left(x\right)$ can be defined as,

$${\mathfrak{L}}_n\left(x\right)=2x{\mathfrak{L}}_{n-1}\left(x\right)+{\mathfrak{L}}_{n-2}\left(x\right)+2$$

with the initial condition ${\mathfrak{L}}_0\left(x\right)=0$ and ${\mathfrak{L}}_1\left(x\right)=1.$

(d) 

By taking $p_3\left(x\right)=4$in Definition 2, (c) Subsidiary MinMax polynomial ${\mathfrak{R}}_n\left(x\right)$ can be defined as,

$${\mathfrak{R}}_n\left(x\right)=2x{\mathfrak{R}}_{n-1}\left(x\right)+{\mathfrak{R}}_{n-2}\left(x\right)+4$$

with the initial condition ${\mathfrak{R}}_0\left(x\right)=0$ and ${\mathfrak{R}}_1\left(x\right)=2x+2.$

(e) 

By taking $p_4\left(x\right)=-2x$in Corollary 2, (d) John-Lucas polynomial $H_n\left(x\right)$ can be defined as, ,

$$H_n\left(x\right)=2x{H}_{n-1}\left(x\right)+H_{n-2}\left(x\right)-2x$$

with the initial condition $H_0\left(x\right)=3$ and $H_1\left(x\right)=2x+1.$

The sequence of $p\left(x\right)-$John polynomials satisfies the third order homogeneous recurrence relation given in the following lemma.

Lemma 2.

For $n\ge 3,$ $p\left(x\right)-$John polynomial can be written as third order recurrence relation:

$$L_{n,p\left(x\right)}\left(x\right)=(2x+1)L_{\left(n-1\right),p\left(x\right)}\left(x\right)-(2x-1)L_{\left(n-2\right),p\left(x\right)}\left(x\right)-L_{\left(n-3\right),p\left(x\right)}\left(x\right)$$

with the initial condition $L_{0,p\left(x\right)}\left(x\right)$, $L_{1,p\left(x\right)}\left(x\right)$ and $L_{2,p\left(x\right)}\left(x\right)=2x{L}_{1,p\left(x\right)}\left(x\right)+L_{0,p\left(x\right)}\left(x\right)+p\left(x\right).$

Proof. Using (27), the identity is proved easily. □

Note that in Lemma 2, the third initial condition $L_{2,p\left(x\right)}\left(x\right)$ is strictly determined by $L_{0,p\left(x\right)}\left(x\right)$, $L_{1,p\left(x\right)}\left(x\right)$, and the polynomial $p\left(x\right)$. By relaxing this structural dependency and treating the third initial term as an arbitrary polynomial, we can capture a much broader family of sequences. This motivates the introduction of the generalized sequence $W_n\left(x\right)$, defined as follows.

Definition 4.

For $n\ge 3$ the generalized John polynomial is defined as, recursively,

$$W_n\left(x\right)=(2x+1)W_{\left(n-1\right)}\left(x\right)-(2x-1)W_{\left(n-2\right)}\left(x\right)-W_{\left(n-3\right)}\left(x\right)$$

(35)

with the arbitrary initial condition $W_0\left(x\right)$, $W_1\left(x\right)$ and $W_2\left(x\right).$

It is worth noting that the recurrence (35) can be extended to negative subscripts, i.e,

$$W_{-n}\left(x\right)=(2x+1)W_{-\left(n-1\right)}\left(x\right)-(2x-1)W_{-\left(n-2\right)}\left(x\right)-W_{-\left(n-3\right)}\left(x\right)$$

However, in such a case, $W_{-n}\left(x\right)$ may not represent a polynomial in the usual sense. Hence, the recurrence (35) holds for all integer $n.$

This recurrence relation (35) gives following characteristic equation.

$$z^3-(2x+1)z^2+(2x-1)z+1=0$$

where roots of this equation are $\alpha =x+\sqrt{x^2+1},$ $\beta =x-\sqrt{x^2+1}$ and $\gamma =1$ such that $\left|x\right|\ne i.$

Hence, polynomial sequences $\left\{M_n\left(x\right)\right\}$, $\left\{N_n\left(x\right)\right\}$, $\left\{{\mathfrak{L}}_n\left(x\right)\right\}$, $\left\{{\mathfrak{R}}_n\left(x\right)\right\},$ $\left\{J_n\left(x\right)\right\}$ and $\left\{H_n\left(x\right)\right\}$ satisfy third order homogeneous recurrence relation given below.

Corollary 2.

For $n\ge 3,$ we have

(a) 

$$M_n\left(x\right)=(2x+1)M_{\left(n-1\right)}\left(x\right)-(2x-1)M_{\left(n-2\right)}\left(x\right)-M_{\left(n-3\right)}\left(x\right)$$

with the initial condition $M_0\left(x\right)=0,$ $M_1\left(x\right)=1$ and $M_2\left(x\right)=2x+1.$

(b) 

$$N_n\left(x\right)=(2x+1)N_{\left(n-1\right)}\left(x\right)-(2x-1)N_{\left(n-2\right)}\left(x\right)-N_{\left(n-3\right)}\left(x\right)$$

with the initial condition $N_0\left(x\right)=1,$ $N_1\left(x\right)=2x+1$ and $N_2\left(x\right)=4x^2+2x+3.$

(c) 

$${\mathfrak{L}}_n\left(x\right)=(2x+1){\mathfrak{L}}_{\left(n-1\right)}\left(x\right)-(2x-1){\mathfrak{L}}_{\left(n-2\right)}\left(x\right)-{\mathfrak{L}}_{\left(n-3\right)}\left(x\right)$$

with the initial condition ${\mathfrak{L}}_0\left(x\right)=0,$ ${\mathfrak{L}}_1\left(x\right)=1$ and ${\mathfrak{L}}_2\left(x\right)=2x+2.$

(d) 

$${\mathfrak{R}}_n\left(x\right)=(2x+1){\mathfrak{R}}_{\left(n-1\right)}\left(x\right)-(2x-1){\mathfrak{R}}_{\left(n-2\right)}\left(x\right)-{\mathfrak{R}}_{\left(n-3\right)}\left(x\right)$$

with the initial condition ${\mathfrak{R}}_0\left(x\right)=0,$ ${\mathfrak{R}}_1\left(x\right)=2x+2$ and ${\mathfrak{R}}_2\left(x\right)=4x^2+4x+4.$

(e) 

$$H_n\left(x\right)=(2x+1)H_{n-1}\left(x\right)-(2x-1)H_{n-2}\left(x\right)-h_{n-3}\left(x\right)$$

with the initial condition $H_0\left(x\right)=3,$ $H_1\left(x\right)=2x+1$ and $H_2\left(x\right)=4x^2+3.$

Here, we present the first few values of polynomial sequences $\left\{M_n\left(x\right)\right\}$, $\left\{N_n\left(x\right)\right\}$, $\left\{{\mathfrak{L}}_n\left(x\right)\right\}$, $\left\{{\mathfrak{R}}_n\left(x\right)\right\},$ $\left\{J_n\left(x\right)\right\}$ and $\left\{H_n\left(x\right)\right\}$ as follow, respectively.

• $M_0=0,$ $M_1=1,$ $M_2=2x+1,$ $M_3=4x^2+2x+2$, $M_4=8x^3+4x^2+6x+2,$ $M_5=16x^4+8x^3+16x^2+6x+3,$
• $N_0=1,$ $N_1=2x+1$, $N_2=4x^2+2x+3$, $N_3=8x^3+4x^2+8x+3$, $N_4=16x^4+8x^3+20x^2+8x+5$, $N_5=32x^5+16x^4+48x^3+20x^2+18x+5,$
• ${\mathfrak{L}}_0=0$, ${\mathfrak{L}}_1=1,$ ${\mathfrak{L}}_2=2x+2,$ ${\mathfrak{L}}_3=4x^2+4x+3$, ${\mathfrak{L}}_4=8x^3+8x^2+8x+4$, ${\mathfrak{L}}_5=16x^4+16x^3+20x^2+12x+5$
• ${\mathfrak{R}}_0=0$, ${\mathfrak{R}}_1=$ $2x+2$, ${\mathfrak{R}}_2=4x^2+4x+4$, ${\mathfrak{R}}_3=8x^3+8x^2+10x+6$, ${\mathfrak{R}}_4=16x^4+16x^3+24x^2+16x+8$, ${\mathfrak{R}}_5=32x^5+32x^4+56x^3+40x^2+26x+10,$
• $H_0=3$, $H_1=2x+1,$ $H_2=4x^2+3$, $H_3=8x^3+6x+1$, $H_4=16x^4+16x^2+3$, $H_5=32x^5+40x^3+10x+1.$

In the subsequent theorem, we establish the Binet-type representation for the generalized John polynomials, thereby providing a closed-form expression that encapsulates their structural properties.

Theorem 2.

For all integer $n,$ the Binet’s formula of generalized John polynomial can be given as follow.

$$W_n\left(x\right)=A_1{\alpha }^n+A_2{\beta }^n+A_3{\gamma }^n$$

where

$$\begin{array}{cc}\hfill A_1& ={\displaystyle \frac{p_1}{(\alpha -\beta )(\alpha -\gamma )}}={\displaystyle \frac{W_2\left(x\right)-(\beta +\gamma )W_1\left(x\right)+\beta \gamma W_0\left(x\right)}{(\alpha -\beta )(\alpha -\gamma )}},\hfill \\ \hfill A_2& ={\displaystyle \frac{p_2}{(\beta -\alpha )(\beta -\gamma )}}={\displaystyle \frac{W_2\left(x\right)-(\alpha +\gamma )W_1\left(x\right)+\alpha \gamma W_0\left(x\right)}{(\beta -\alpha )(\beta -\gamma )}},\hfill \\ \hfill A_3& ={\displaystyle \frac{p_3}{(\gamma -\alpha )(\gamma -\beta )}}={\displaystyle \frac{W_2\left(x\right)-(\alpha +\beta )W_1\left(x\right)+\alpha \beta W_0\left(x\right)}{(\gamma -\alpha )(\gamma -\beta )}}.\hfill \end{array}$$

Proof. Set $r=2x+1,$ $s=-2x+1,$ $t=-1$ in [[17], Theorem 1].

By applying the preceding theorem, or alternatively Theorem 1, we are able to derive the following set of results, which further clarify the structural properties of the generalized John polynomials.

Corollary 3.

For all integer $n,$ we have following formulas.

(a) 

$M_n\left(x\right)={\displaystyle \frac{{\alpha }^{n+1}}{(\alpha -\beta )(\alpha -1)}}+{\displaystyle \frac{{\beta }^{n+1}}{(\beta -\alpha )(\beta -1)}}+{\displaystyle \frac{1}{(1-\alpha )(1-\beta )}}.$

(b) 

$N_n\left(x\right)={\displaystyle \frac{({\alpha }^2+1){\alpha }^n}{(\alpha -\beta )(\alpha -1)}}+{\displaystyle \frac{({\beta }^2+1){\beta }^n}{(\beta -\alpha )(\beta -1)}}+{\displaystyle \frac{2}{(1-\alpha )(1-\beta )}}.$

(c) 

${\mathfrak{L}}_n\left(x\right)={\displaystyle \frac{(\alpha +1){\alpha }^n}{(\alpha -\beta )(\alpha -1)}}+{\displaystyle \frac{(\beta +1){\beta }^n}{(\beta -\alpha )(\beta -1)}}+{\displaystyle \frac{2}{(1-\alpha )(1-\beta )}}.$

(d) 

${\mathfrak{R}}_n\left(x\right)={\displaystyle \frac{(\left(2x+2\right)\alpha -2x+2){\alpha }^n}{(\alpha -\beta )(\alpha -1)}}+{\displaystyle \frac{(\left(2x+2\right)\beta -2x+2){\beta }^n}{(\beta -\alpha )(\beta -1)}}+{\displaystyle \frac{4}{(1-\alpha )(1-\beta )}}.$

(e) 

$H_n\left(x\right)={\alpha }^n+{\beta }^n+1.$

In the next theorem we present generating function of generalized John polynomials.

Theorem 3.

Suppose that $f_{W_n\left(x\right)}\left(z\right)={\displaystyle \sum _{n=0}^{\infty }}{W}_n\left(x\right)z^n$ is the ordinary generating function of the generalized John (sequence of) polynomials. Then ${\displaystyle \sum _{n=0}^{\infty }}{W}_n\left(x\right)z^n$ is given by

$${\displaystyle \sum _{n=0}^{\infty }}{W}_n\left(x\right)z^n={\displaystyle \frac{W_0\left(x\right)+(W_1\left(x\right)-(2x+1)W_0\left(x\right))z+(W_2\left(x\right)-(2x+1)W_1\left(x\right)+(2x-1)W_0\left(x\right))z^2}{1-(2x+1)z+(2x-1)z^2+z^3}}.$$

(36)

Proof. Set $r=2x+1,$ $s=-2x+1,$ $t=-1$ in [17], Lemma 9]. □

Theorem 3 gives in the following corollary.

Corollary 4.

For generalized John polynomials, we have,

(a) 

${\displaystyle \sum _{n=0}^{\infty }}{M}_n\left(x\right)z^n={\displaystyle \frac{z}{1-(2x+1)z+(2x-1)z^2+z^3}}.$

(b) 

${\displaystyle \sum _{n=0}^{\infty }}{N}_n\left(x\right)z^n={\displaystyle \frac{1+z^2}{1-(2x+1)z+(2x-1)z^2+z^3}}.$

(c) 

${\displaystyle \sum _{n=0}^{\infty }}{\mathfrak{L}}_n\left(x\right)z^n={\displaystyle \frac{z+z^2}{1-(2x+1)z+(2x-1)z^2+z^3}}.$

(d) 

${\displaystyle \sum _{n=0}^{\infty }}{\mathfrak{R}}_n\left(x\right)z^n={\displaystyle \frac{(2x+2)z+(2-2x)z^2}{1-(2x+1)z+(2x-1)z^2+z^3}}.$

(e) 

${\displaystyle \sum _{n=0}^{\infty }}{H}_n\left(x\right)z^n={\displaystyle \frac{3-2(2x+1)z+(2x-1)z^2}{1-(2x+1)z+(2x-1)z^2+z^3}}.$

In the forthcoming theorem, we derive the exponential generating function associated with the generalized John polynomials, thereby providing a powerful analytic tool for studying their structural and combinatorial properties.

Theorem 4.

Suppose that $g_{W_n\left(x\right)}\left(z\right)={\displaystyle \sum _{n=0}^{\infty }}{W}_n\left(x\right){\displaystyle \frac{z^n}{n!}}$ is the exponential generating function of the generalized Leonardo-Pell (sequence of) polynomials. Then ${\displaystyle \sum _{n=0}^{\infty }}{W}_n\left(x\right){\displaystyle \frac{z^n}{n!}}$ is given by

$${\displaystyle \sum _{n=0}^{\infty }}{W}_n\left(x\right){\displaystyle \frac{z^n}{n!}}=A_1{e}^{\alpha t}+A_2{e}^{\alpha t}+A_3{e}^{\gamma t}$$

where $A_1,$ $A_2$ and $A_3$ as stated in the Theorem 2.

Proof. For the proof, we derive following equalities.

$$\begin{array}{cc}\hfill g_{W_n\left(x\right)}^{\prime }\left(z\right)& ={\displaystyle \sum _{n=0}^{\infty }}{W}_{n+1}\left(x\right){\displaystyle \frac{z^n}{n!}},\hfill \\ \hfill g_{W_n\left(x\right)}^{\prime \prime }\left(z\right)& ={\displaystyle \sum _{n=0}^{\infty }}{W}_{n+2}\left(x\right){\displaystyle \frac{z^n}{n!}},\hfill \\ \hfill g_{W_n\left(x\right)}^{\prime \prime \prime }\left(z\right)& ={\displaystyle \sum _{n=0}^{\infty }}{W}_{n+3}\left(x\right){\displaystyle \frac{z^n}{n!}}.\hfill \end{array}$$

Then we obtain

$$g_{W_n\left(x\right)}^{‴}\left(z\right)=(2x+1)g_{W_n\left(x\right)}^{″}\left(z\right)-(2x-1)g_{W_n\left(x\right)}^{\prime }\left(z\right)-g_{W_n\left(x\right)}\left(z\right).$$

The solution of this diferantial equation is

$$g_{W_n\left(x\right)}\left(z\right)=C_1{e}^{\alpha t}+C_2{e}^{\alpha t}+C_3{e}^{\gamma t}$$

For $t=0,$ we obtain

$$\begin{array}{cc}\hfill g_{W_n\left(x\right)}\left(z\right)& =C_1+C_2+C_3=W_0\left(x\right),\hfill \\ \hfill g_{W_n\left(x\right)}^{\prime }\left(z\right)& =C_1\alpha +C_2\beta +C_3\gamma =W_1\left(x\right),\hfill \\ \hfill g_{W_n\left(x\right)}^{″}\left(z\right)& =C_1{\alpha }^2+C_2{\beta }^2+C_3{\gamma }^2=W_2\left(x\right).\hfill \end{array}$$

This implies $C_1=A_1$, $C_2=A_2$, $C_3=A_3.$ This completes the proof. □

In the following lemma, we present the third-order homogeneous representation of second-order linear non-homogeneous Horadam-Leonardo polynomial sequences, that is necessary to construct the general determinantal representations presented later in this paper.

Definition 5.

For all integers n, we define generalized $p\left(x\right)-$Horadam Leonardo polynomials with non-homogeneous second order recurrence relation as follows:

$${\tilde{V}}_n\left(x\right)=r{\tilde{V}}_{n-1}\left(x\right)+s{\tilde{V}}_{n-2}\left(x\right)+p\left(x\right)$$

(37)

with arbitrary initial condition ${\tilde{V}}_0\left(x\right)$, ${\tilde{V}}_1\left(x\right)$ and $p\left(x\right),r$, s are polynomials in x.

Based on the considerations above, while the ${\tilde{V}}_2\left(x\right)$ term in the standard $p\left(x\right)-$Horadam Leonardo polynomials depends on the first two terms, we now introduce a more generalized framework by assuming ${\tilde{V}}_2\left(x\right)$ as an additional arbitrary initial condition. We believe that this selection enriches the scope of our study by providing a more comprehensive and versatile mathematical structure. Hence, next definition we present generalized Horodam-Leonardo polynomials.

Definition 6.

The generalized Horadam-Leonardo polynomials ${\tilde{V}}_n\left(x\right)$ for $n\ge 3$ are given by the following homogeneous recurrence relation:

$${\tilde{V}}_n\left(x\right)=\left(r+1\right){\tilde{V}}_{\left(n-1\right)}\left(x\right)+(s-r){\tilde{V}}_{\left(n-2\right)}\left(x\right)-s{\tilde{V}}_{\left(n-3\right)}\left(x\right),[[18],(2.1)]$$

(38)

with arbitrary initial condition ${\tilde{V}}_0\left(x\right),$ ${\tilde{V}}_1\left(x\right)$ and ${\tilde{V}}_2\left(x\right).$

Note: ${\tilde{V}}_n\left(x\right)$ satisfying (37) can be written in the form (38).

A combinatorial formula is often the most direct tool for exploring the exact structure of a sequence. With this in mind, the next theorem establishes an explicit representation for the generalized Horadam-Leonardo polynomials.

Theorem 5.

For $n>2,$ we have the following formula.

$${\tilde{V}}_n\left(x\right)=\tilde{A}\left(x\right)+\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \frac{\tilde{B}\left(x\right)(n-2i)+r\tilde{C}\left(x\right)(n-i)}{(n-i)}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right)r^{n-2i-1}{s}^i$$

where

$$\begin{array}{cc}\hfill \tilde{A}\left(x\right)& ={\displaystyle \frac{1}{\left(1-r-s\right)}}({\tilde{V}}_2\left(x\right)-r{\tilde{V}}_1\left(x\right)-s{\tilde{V}}_0\left(x\right))\hfill \\ \hfill \tilde{B}\left(x\right)& ={\displaystyle \frac{1}{\left(1-r-s\right)}}(-{\tilde{V}}_2\left(x\right)+r{\tilde{V}}_1\left(x\right)+(1-r){\tilde{V}}_0\left(x\right))\hfill \\ \hfill \tilde{C}\left(x\right)& ={\displaystyle \frac{1}{\left(1-r-s\right)}}((r-1){\tilde{V}}_2\left(x\right)-(r^2+rs+s-r-1){\tilde{V}}_1\left(x\right)+(r^2+s-1){\tilde{V}}_0\left(x\right)).\hfill \end{array}$$

Proof. For the proof, we use some basic properties of power series and combinatorial formulation. First we can write

$${\displaystyle \frac{1}{1-rz-s{z}^2}}={\displaystyle \frac{1}{1-(rz+s{z}^2)}}=\underset{m=0}{{\displaystyle \sum }^{\infty }}{(rz+s{z}^2)}^m$$

and

$${(2xz-z^2)}^m=\underset{i=0}{{\displaystyle \sum }^m}\left({\displaystyle \genfrac{}{}{0pt}{}{m}{i}}\right){\left(rz\right)}^{m-i}{\left(s{z}^2\right)}^i=\underset{i=0}{{\displaystyle \sum }^m}\left({\displaystyle \genfrac{}{}{0pt}{}{m}{i}}\right)r^{m-i}{s}^i{z}^{m+i}.$$

If we take $m+i=n$ we obtain $0\le i\le ⌊{\displaystyle \frac{n}{2}}⌋$ since $0\le i\le m.$ So we have

$${\displaystyle \frac{1}{1-rz-s{z}^2}}=\underset{n=0}{{\displaystyle \sum }^{\infty }}\left(\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right)r^{n-2i}{s}^i\right)z^n.$$

For the next step of the proof we denote

$$C_n=\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right)r^{n-2i}{s}^i$$

Then assuming $C_{-1}=0$ we can write

$${\displaystyle \frac{z}{1-rz-s{z}^2}}=\stackrel{\infty }{{\displaystyle \sum }_{n=0}}{C}_n{z}^{n+1}=\stackrel{\infty }{{\displaystyle \sum }_{n=1}}{C}_{n-1}{z}^n=\stackrel{\infty }{{\displaystyle \sum }_{n=0}}{C}_{n-1}{z}^n.$$

This gives us

$${\displaystyle \frac{z}{1-rz-s{z}^2}}=\underset{n=0}{{\displaystyle \sum }^{\infty }}\left(\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i-1}{i}}\right)r^{n-2i}{s}^i\right)z^n.$$

Then, we decompose the generating function of the generalized Horadam-Leonardo polynomials given in [[17], Lemma 9] into partial fractions, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \sum _{n=0}^{\infty }}{\tilde{V}}_n\left(x\right)z^n& ={\displaystyle \frac{{\tilde{V}}_0+({\tilde{V}}_1-(r+1){\tilde{V}}_0)+({\tilde{V}}_2-(r+1){\tilde{V}}_1-(s-r){\tilde{V}}_0)}{1-\left(r+1\right)z-\left(s-r\right)z^2+s{z}^3}},\hfill \\ \hfill & ={\displaystyle \frac{A\left(x\right)}{1-z}}+{\displaystyle \frac{B\left(x\right)z}{1-rz-s{z}^2}}+{\displaystyle \frac{C\left(x\right)}{1-rz-s{z}^2}}\hfill \end{array}$$

where $A\left(x\right),$ $B\left(x\right),$ $C\left(x\right)$ as stated in the theorem. So we obtain,

$$\begin{array}{cc}\hfill {\displaystyle \sum _{n=0}^{\infty }}{\tilde{V}}_n\left(x\right)z^n& =\tilde{A}\left(x\right)\underset{n=0}{{\displaystyle \sum }^{\infty }}{z}^n+\tilde{B}\left(x\right)\underset{n=0}{{\displaystyle \sum }^{\infty }}\left(\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i-1}{i}}\right)r^{n-2i-1}{s}^i\right)z^n\hfill \\ \hfill & +\tilde{C}\left(x\right)\underset{n=0}{{\displaystyle \sum }^{\infty }}\left(\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right)r^{n-2i}{s}^i\right)z^n\hfill \end{array}$$

Therefore, we get

$$\begin{array}{cc}\hfill {\tilde{V}}_n\left(x\right)& =\tilde{A}\left(x\right)+\tilde{B}\left(x\right)\left(\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i-1}{i}}\right)r^{n-2i-1}{s}^i\right)+\tilde{C}\left(x\right)\left(\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right)r^{n-2i}{s}^i\right)\hfill \\ \hfill & =\tilde{A}\left(x\right)+\tilde{B}\left(x\right)\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i-1}{i}}\right)r^{n-2i-1}{s}^i+\tilde{C}\left(x\right)\left(\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}{\displaystyle \frac{(n-i)}{(n-2i)}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i-1}{i}}\right)r^{n-2i}{s}^i\right)\hfill \end{array}$$

This gives us,

$$\begin{array}{cc}\hfill {\tilde{V}}_n\left(x\right)& =\tilde{A}\left(x\right)+\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \frac{\tilde{B}\left(x\right)}{r}}+{\displaystyle \frac{\tilde{C}\left(x\right)(n-i)}{(n-2i)}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-i-1}{i}}\right)r^{n-2i}{s}^i\hfill \\ \hfill & =\tilde{A}\left(x\right)+\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \frac{\tilde{B}\left(x\right)(n-2i)+r\tilde{C}\left(x\right)(n-i)}{r(n-2i)}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-i-1}{i}}\right)r^{n-2i}{s}^i.\hfill \\ \hfill & =\tilde{A}\left(x\right)+\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \frac{\tilde{B}\left(x\right)(n-2i)+r\tilde{C}\left(x\right)(n-i)}{(n-i)}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right)r^{n-2i-1}{s}^i.\hfill \end{array}$$

This completes the proof. □

In the subsequent theorem, we present combinatorial presentation of generalized John polynomials.

Theorem 6.

For $n>2,$ we have the following formulas.

$$W_n\left(x\right)=\widehat{A}\left(x\right)+\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \frac{\widehat{B}\left(x\right)(n-2i)+\widehat{C}\left(x\right)2x(n-i)}{(n-i)}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right){\left(2x\right)}^{n-2i-1}$$

where

$$\begin{array}{cc}\hfill \widehat{A}\left(x\right)& ={\displaystyle \frac{W_0\left(x\right)+2x{W}_1\left(x\right)-W_2\left(x\right)}{2x}}\hfill \\ \hfill \widehat{B}\left(x\right)& ={\displaystyle \frac{-\left(4x^2-2x+1\right)W_0\left(x\right)+4x^2{W}_1\left(x\right)-(2x-1)W_2\left(x\right)}{2x}}\hfill \\ \hfill \widehat{C}\left(x\right)& ={\displaystyle \frac{(2x-1)W_0\left(x\right)-2x{W}_1\left(x\right)+W_2\left(x\right)}{2x}}.\hfill \end{array}$$

Proof. For the proof, set ${\tilde{V}}_n\left(x\right)=W_n\left(x\right),$ $r=2x,$ $s=1$ in Theorem 5. □

As a consequence of the previous theorem, we obtain the following result.

Corollary 5.

For $n>2,$ we have the following formula.

(a) 

$$M_n\left(x\right)=-{\displaystyle \frac{1}{2x}}+\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \frac{\left(2x+1\right)n-\left(2x+2\right)i}{n-i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right){\left(2x\right)}^{n-2i-2}.$$

(b) 

$$N_n\left(x\right)=-{\displaystyle \frac{1}{x}}+\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}{\displaystyle \frac{\left(2x^2+x+1\right)n-\left(2x^2+2\right)i}{\left(n-i\right)x}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right){\left(2x\right)}^{n-2i-1}.$$

(c) 

$${\mathfrak{L}}_n\left(x\right)=-{\displaystyle \frac{1}{x}}+\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \frac{(x+1)n-2i}{x(n-i)}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right){\left(2x\right)}^{n-2i-1}.$$

(d) 

$${\mathfrak{R}}_n\left(x\right)=-{\displaystyle \frac{2}{x}}+\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \frac{\left(2x^2+2x+2\right)n-\left(4x^2+4\right)i}{x(n-i)}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right){\left(2x\right)}^{n-2i-1}{(-1)}^i.$$

(e) 

$$H_n\left(x\right)=1+\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \frac{n}{n-i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right){\left(2x\right)}^{n-2i}.$$

Hence, the following theorem provides a determinantal presentation for the generalized Horodam-Leonardo polynomials.

Theorem 7.

For$n\ge 3$ and ${\tilde{V}}_1\left(x\right)\ne 0,$ we have following formula.

$${\tilde{V}}_n\left(x\right)=det{D}_n$$

where

$$D_n=\left(d_{ij}\right)={\left(\begin{array}{cccccccc}{\tilde{V}}_1\left(x\right)& r{\tilde{V}}_1\left(x\right)-{\tilde{V}}_2\left(x\right)& s{\tilde{V}}_0\left(x\right)& & \dots & & & \\ 1& r& (s-r)& -s& & & & \\ & -1& r+1& (s-r)& \ddots & & ⋮& \\ & & \ddots & r+1& \ddots & (s-r)& -s& \\ & & & -1& \ddots & r+1& (s-r)& -s\\ & & & & \dots & -1& r+1& (s-r)\\ & & & & \dots & & -1& r+1\end{array}\right)}_{n\times n}$$

Note that, we read all the nonmentioned entries as zero.

Proof. For the proof we use LU decomposition using Doolittle algorithm. Before proceeding the proof, we recall $LU$ decomposition using Doolittle algorithm [19]. We assume that the lower and upper triangular matrices L and U of size $n,$ respectively, is given as follow

$$\begin{array}{cc}\hfill L& =\left(l_{ij}\right),\hfill \\ \hfill U& =\left(u_{ij}\right),\hfill \end{array}$$

$l_{ij}=\left\{\begin{array}{c}1\\ 0\end{array}\right.\begin{array}{c}\mathrm{if}i=j\\ \mathrm{if}i>j\end{array},$ and $u_{ij}=0$, if $i<j.$ The matrix $\mathcal{A}=\left(a_{ij}\right)$ of size n can be factor as follow

$$\mathcal{A}=LU$$

where using Dooolittle algorithm, we obtain the following equalities:

$$u_{1j}=a_{1j},j=1,2,\dots ,n$$

$$l_{i1}={\displaystyle \frac{a_{i1}}{u_{11}}},i=2,\dots ,n$$

$$u_{ij}=a_{ij}-{\displaystyle \sum _{k=1}^{i-1}}{l}_{ik}{u}_{kj},j=i,\dots ,n.$$

$$l_{ij}={\displaystyle \frac{a_{ij}-{\sum }_{k=1}^{j-1}{l}_{ik}{u}_{kj}}{u_{jj}}},i=j+1,\dots ,n.$$

Therefore, we deduce the following equalities:

$$u_{1j}=\left\{\begin{array}{c}{\tilde{V}}_1\left(x\right)\\ r{\tilde{V}}_1\left(x\right)-{\tilde{V}}_2\left(x\right)\\ s{\tilde{V}}_0\left(x\right)\\ 0\end{array}\right.\begin{array}{c}\mathrm{if}j=1\\ \mathrm{if}j=2\\ \mathrm{if}j=3\\ \mathrm{else}\end{array},$$

$$l_{i1}=\left\{\begin{array}{c}1\\ {\displaystyle \frac{1}{{\tilde{V}}_1\left(x\right)}}\\ 0\end{array}\right.\begin{array}{c}\mathrm{if}i=1\\ \mathrm{if}i=2\\ \mathrm{else}\end{array},$$

$$u_{ij}=\left\{\begin{array}{c}{\displaystyle \frac{{\tilde{V}}_i\left(x\right)}{{\tilde{V}}_{i-1}\left(x\right)}}\\ (s-r)-{\displaystyle \frac{s{\tilde{V}}_{j-3}\left(x\right)}{{\tilde{V}}_{j-2}\left(x\right)}}\\ -s\\ 0\end{array}\right.\begin{array}{c}\mathrm{if}i=j\ge 2\\ j-i=1,i\ge 2\\ j-i=2,i\ge 2\\ \mathrm{otherwise}\end{array},$$

$$l_{ij}=\left\{\begin{array}{c}1\\ {\displaystyle \frac{1}{{\tilde{V}}_1\left(x\right)}}\\ -{\displaystyle \frac{{\tilde{V}}_{i-2}\left(x\right)}{{\tilde{V}}_{i-1}\left(x\right)}}\\ 0\end{array}\right.\begin{array}{c}\mathrm{if}i=j\\ i=2,j=1\\ i-j=1,i\ge 3\\ \mathrm{otherwise}\end{array}.$$

This implies

$L={\left(\begin{array}{ccccccc}1& & & & \dots & & \\ {\displaystyle \frac{1}{{\tilde{V}}_1\left(x\right)}}& 1& & & & & \\ & -{\displaystyle \frac{{\tilde{V}}_1\left(x\right)}{{\tilde{V}}_2\left(x\right)}}& 1& & \ddots & & ⋮\\ & & -{\displaystyle \frac{{\tilde{V}}_2\left(x\right)}{{\tilde{V}}_3\left(x\right)}}& 1& \ddots & & \\ & & & \ddots & \ddots & \ddots & \\ & & & & \ddots & 1& \\ & & & & & -{\displaystyle \frac{{\tilde{V}}_{n-2}\left(x\right)}{{\tilde{V}}_{n-1}\left(x\right)}}& 1\end{array}\right)}_{n\times n},$

$U={\left(\begin{array}{cccccccc}{\tilde{V}}_1\left(x\right)& 2x{\tilde{V}}_1\left(x\right)-{\tilde{V}}_2\left(x\right)& s{\tilde{V}}_0\left(x\right)& & \dots & & & \\ & {\displaystyle \frac{{\tilde{V}}_2\left(x\right)}{{\tilde{V}}_1\left(x\right)}}& (r-s)-{\displaystyle \frac{s{\tilde{V}}_0\left(x\right)}{{\tilde{V}}_1\left(x\right)}}& -s& & & & \\ & & {\displaystyle \frac{{\tilde{V}}_3\left(x\right)}{{\tilde{V}}_2\left(x\right)}}& (r-s)-{\displaystyle \frac{s{\tilde{V}}_1\left(x\right)}{{\tilde{V}}_2\left(x\right)}}& -s& & & ⋮\\ & & & \ddots & \ddots & \ddots & & \\ & & & & & & -s& \\ & & & & \ddots & & (r-s)-{\displaystyle \frac{s{\tilde{V}}_{n-4}\left(x\right)}{{\tilde{V}}_{n-3}\left(x\right)}}& -s\\ & & & & & & {\displaystyle \frac{{\tilde{V}}_{n-1}\left(x\right)}{{\tilde{V}}_{n-2}\left(x\right)}}& (r-s)-{\displaystyle \frac{s{\tilde{V}}_{n-3}\left(x\right)}{{\tilde{V}}_{n-2}\left(x\right)}}\\ & & & & & & & {\displaystyle \frac{{\tilde{V}}_n\left(x\right)}{{\tilde{V}}_{n-1}\left(x\right)}}\end{array}\right)}_{n\times n}.$

Since $detL=1$ and

$$detU={\tilde{V}}_1\left(x\right){\displaystyle \frac{{\tilde{V}}_2\left(x\right)}{{\tilde{V}}_1\left(x\right)}}{\displaystyle \frac{{\tilde{V}}_3\left(x\right)}{{\tilde{V}}_2\left(x\right)}}...{\displaystyle \frac{{\tilde{V}}_n\left(x\right)}{{\tilde{V}}_{n-1}\left(x\right)}}={\tilde{V}}_n\left(x\right),$$

we obtain

$$det{D}_n=detLdetU={\tilde{V}}_n\left(x\right).\square$$

By applying this general determinantal result to our specific sequence, we immediately obtain the following theorem.

Theorem 8.

For$n\ge 3$ and $W_1\left(x\right)\ne 0,$ we have the following formula.

$$W_n\left(x\right)=det{D}_n$$

where

$$D_n=\left(d_{ij}\right)={\left(\begin{array}{cccccccc}{W}_1\left(x\right)& 2x{W}_1\left(x\right)-W_2\left(x\right)& W_0\left(x\right)& & \dots & & & \\ 1& 2x& -\left(2x-1\right)& -1& & & & \\ & -1& \left(2x+1\right)& -\left(2x-1\right)& \ddots & & ⋮& \\ & & \ddots & \left(2x+1\right)& \ddots & -\left(2x-1\right)& -1& \\ & & & -1& \ddots & \left(2x+1\right)& -\left(2x-1\right)& -1\\ & & & & \dots & -1& \left(2x+1\right)& -\left(2x-1\right)\\ & & & & \dots & & -1& \left(2x+1\right)\end{array}\right)}_{n\times n}$$

Note that, we read all the nonmentioned entries as zero.

Proof. For the proof, replace $r=2x,$ $s=1$ in Theorem 7. □

The following corollary gives special cases of Theorem 8.

Corollary 6.

For $n\ge 3,$ we have the following formulas:

(a) 

$M_n\left(x\right)={\left|\begin{array}{ccccccc}1& -1& 0& & & \dots & \\ 1& 2x& -\left(2x-1\right)& -1& & & \\ & -1& \left(2x+1\right)& -\left(2x-1\right)& \ddots & & ⋮\\ & & \ddots & \left(2x+1\right)& \ddots & -\left(2x-1\right)& -1\\ & & & -1& \ddots & \left(2x+1\right)& -\left(2x-1\right)\\ & & & & \dots & -1& \left(2x+1\right)\end{array}\right|}_{n\times n}.$

(b) 

$N_n\left(x\right)={\left|\begin{array}{ccccccc}2x+1& -3& 1& & & \dots & \\ 1& 2x& -\left(2x-1\right)& -1& & & \\ & -1& \left(2x+1\right)& -\left(2x-1\right)& \ddots & & ⋮\\ & & \ddots & \left(2x+1\right)& \ddots & -\left(2x-1\right)& -1\\ & & & -1& \ddots & \left(2x+1\right)& -\left(2x-1\right)\\ & & & & \dots & -1& \left(2x+1\right)\end{array}\right|}_{n\times n}.$

(c) 

${\mathfrak{L}}_n\left(x\right)={\left|\begin{array}{ccccccc}1& -2& 0& & & \dots & \\ 1& 2x& -\left(2x-1\right)& -1& & & \\ & -1& \left(2x+1\right)& -\left(2x-1\right)& \ddots & & ⋮\\ & & \ddots & \left(2x+1\right)& \ddots & -\left(2x-1\right)& -1\\ & & & -1& \ddots & \left(2x+1\right)& -\left(2x-1\right)\\ & & & & \dots & -1& \left(2x+1\right)\end{array}\right|}_{n\times n}.$

(d) 

${\mathfrak{R}}_n\left(x\right)={\left|\begin{array}{ccccccc}2x+2& -4& 0& & & \dots & \\ 1& 2x& -\left(2x+1\right)& 1& & & \\ & -1& \left(2x+1\right)& -\left(2x+1\right)& \ddots & & ⋮\\ & & \ddots & \left(2x+1\right)& \ddots & -\left(2x+1\right)& 1\\ & & & -1& \ddots & \left(2x+1\right)& -\left(2x+1\right)\\ & & & & \dots & -1& \left(2x+1\right)\end{array}\right|}_{n\times n}.$

(e) 

$H_n\left(x\right)={\left|\begin{array}{ccccccc}2x+1& 2x-3& 3& & & \dots & \\ 1& 2x& -\left(2x-1\right)& -1& & & \\ & -1& \left(2x+1\right)& -\left(2x-1\right)& \ddots & & ⋮\\ & & \ddots & \left(2x+1\right)& \ddots & -\left(2x-1\right)& -1\\ & & & -1& \ddots & \left(2x+1\right)& -\left(2x-1\right)\\ & & & & \dots & -1& \left(2x+1\right)\end{array}\right|}_{n\times n}.$

We proceed by formulating Catalan’s identity for the generalized John polynomials.

Theorem 9

(Catalan’s identity). For all integer $n,m,$ we have

$$\begin{array}{cc}\hfill W_{n+m}\left(x\right)W_{n-m}\left(x\right)-W_n^2\left(x\right)& ={\displaystyle \frac{p_1{p}_2}{\left(\alpha -\gamma \right)\left(\gamma -\beta \right)}}{U}_{m-1}^2\left(x\right)+{\displaystyle \frac{p_1{p}_3{\alpha }^{n-m}}{\left(\alpha -\beta \right)\left(\beta -\gamma \right)}}{\displaystyle \frac{{\left(\gamma -{\alpha }^m\right)}^2}{{\left(\gamma -\alpha \right)}^2}}\hfill \\ \hfill & +{\displaystyle \frac{p_2{p}_3{\beta }^{n-m}}{\left(\alpha -\beta \right)\left(\gamma -\alpha \right)}}{\displaystyle \frac{{\left(\gamma -{\beta }^m\right)}^2}{{\left(\gamma -\beta \right)}^2}}.\hfill \end{array}$$

Proof. Using Binet’s formula of generalized John polynomials, we have,

$$\begin{array}{cc}\hfill W_{n+m}\left(x\right)W_{n-m}\left(x\right)-W_n^2\left(x\right)& =A_1{A}_2{\left({\alpha }^m-{\beta }^m\right)}^2+{\alpha }^{n-m}{A}_1{A}_3{\left(1-{\alpha }^m\right)}^2+{\beta }^{n-m}{A}_2{A}_3{\left(1-{\beta }^m\right)}^2\hfill \\ \hfill & ={\displaystyle \frac{p_1{p}_2}{\left(\alpha -\gamma \right)\left(\gamma -\beta \right)}}{\displaystyle \frac{{\left({\alpha }^m-{\beta }^m\right)}^2}{{\left(\alpha -\beta \right)}^2}}+{\displaystyle \frac{p_1{p}_3}{\left(\alpha -\beta \right)\left(\beta -\gamma \right)}}{\alpha }^{n-m}{\displaystyle \frac{{\left(\gamma -{\alpha }^m\right)}^2}{{\left(\gamma -\alpha \right)}^2}}\hfill \\ \hfill & +{\displaystyle \frac{p_2{p}_3}{\left(\alpha -\beta \right)\left(\gamma -\alpha \right){\left(\gamma -\beta \right)}^2}}{\beta }^{n-m}{\left(\gamma -{\beta }^m\right)}^2.\square \hfill \end{array}$$

If we take $m=1$ in the above theorem we have the following corollary.

Corollary 7.

For $n\ge 1$ we have

$$W_{n+1}\left(x\right)W_{n-1}\left(x\right)-W_n^2\left(x\right)={\displaystyle \frac{p_1{p}_2}{\left(\alpha -\gamma \right)\left(\gamma -\beta \right)}}+{\displaystyle \frac{p_1{p}_3}{\left(\alpha -\beta \right)\left(\beta -\gamma \right)}}{\alpha }^{n-m}+{\displaystyle \frac{p_2{p}_3}{\left(\alpha -\beta \right)\left(\gamma -\alpha \right)}}{\beta }^{n-m}.$$

Our next objective is to state d’Ocagne’s identity for the generalized John polynomials.

Theorem 10

(d’Ocagne’s identity). For all integer $n,m,$ then we have,

$$\begin{array}{cc}\hfill W_{m+1}\left(x\right)W_n\left(x\right)-W_m\left(x\right)W_{n+1}\left(x\right)& ={\displaystyle \frac{p_1{p}_2}{\left(\alpha -\beta \right)\left(\alpha -\gamma \right)\left(\gamma -\beta \right)}}{P}_{m-n}\left(x\right)+{\displaystyle \frac{p_1{p}_3}{\left(\alpha -\beta \right)\left(\alpha -\gamma \right)\left(\beta -\gamma \right)}}\left({\alpha }^n-{\alpha }^m\right)\hfill \\ \hfill & +{\displaystyle \frac{p_2{p}_3}{\left(\alpha -\beta \right)\left(\alpha -\gamma \right)\left(\beta -\gamma \right)}}\left({\beta }^n-{\beta }^m\right).\hfill \end{array}$$

Proof. Using Binet’s formula of generalized John polynomials, we have,

$$\begin{array}{cc}\hfill W_{m+1}\left(x\right)W_n\left(x\right)-W_m\left(x\right)W_{n+1}\left(x\right)& =A_1{A}_2\left(\alpha -\beta \right)\left({\alpha }^{m-n}-{\beta }^{m-n}\right)+A_1{A}_3\left(\gamma -\alpha \right)\left({\alpha }^n-{\alpha }^m\right)+A_2{A}_3\left(1-\beta \right)\left({\beta }^n-{\beta }^m\right)\hfill \\ \hfill & ={\displaystyle \frac{p_1{p}_2}{\left(\alpha -\beta \right)\left(\alpha -\gamma \right)\left(\gamma -\beta \right)}}{\displaystyle \frac{\left({\alpha }^{m-n}-{\beta }^{m-n}\right)}{\left(\alpha -\beta \right)}}+{\displaystyle \frac{p_1{p}_3}{\left(\alpha -\beta \right)\left(\alpha -\gamma \right)\left(\beta -\gamma \right)}}\left({\alpha }^n-{\alpha }^m\right)\hfill \\ \hfill & +{\displaystyle \frac{p_2{p}_3}{\left(\alpha -\beta \right)\left(\alpha -\gamma \right)\left(\beta -\gamma \right)}}\left({\beta }^n-{\beta }^m\right).\square \hfill \end{array}$$

We now aim to determine Vajda’s identity for the generalized John polynomials.

Theorem 11

(Vajda’s identity). For all integer $n,m,$ we have,

$$\begin{array}{cc}\hfill W_{n+m}\left(x\right)W_{n+k}\left(x\right)-W_n\left(x\right)W_{n+m+k}\left(x\right)& ={\beta }^n{A}_2{A}_3\left({\gamma }^k-{\beta }^k\right)\left({\beta }^m-{\gamma }^m\right)+{\displaystyle \frac{p_1{p}_2}{(\beta -\gamma )(\alpha -\gamma )}}{P}_k\left(x\right)P_m\left(x\right)\hfill \\ \hfill & +{\alpha }^n{A}_1{A}_3\left({\gamma }^k-{\alpha }^k\right)\left({\alpha }^m-{\gamma }^m\right).\hfill \end{array}$$

Proof. Using Binet’s formula of generalized John polynomials, we have, the required identity.

Let us now explore Honsberger’s identity in the context of the generalized John polynomials.

Theorem 12

(Honsberger’s identity). For all integer $n,m,$ we have the following formulas.

$$W_{n+m}\left(x\right)=W_n\left(x\right)M_{m+1}\left(x\right)+W_{n-1}\left(x\right)\left(-(2x-1)M_m\left(x\right)-M_{m-1}\left(x\right)\right)-W_{n-2}\left(x\right)M_m\left(x\right)$$

Proof. For the proof set $r=2x+1,$ $s=-2x+1,$ $t=-1$ in [[17], Theorem 53]. □

Also, the last theorem gives following corollary.

Corollary 8.

For all integer n, we have

$$\begin{array}{cc}\hfill W_{n+2}\left(x\right)& =W_2\left(x\right)M_{n+1}\left(x\right)+W_1\left(x\right)\left(-(2x-1)M_n\left(x\right)-M_{n-1}\left(x\right)\right)-W_0\left(x\right)M_n\left(x\right).\hfill \\ \hfill W_{2n}\left(x\right)& =W_n\left(x\right)M_{n+1}\left(x\right)+W_{n-1}\left(x\right)\left(-(2x-1)M_n\left(x\right)-M_{n-1}\left(x\right)\right)-W_{n-2}\left(x\right)M_n\left(x\right).\hfill \\ \hfill W_{2n+1}\left(x\right)& =W_n\left(x\right)M_{n+2}\left(x\right)+W_{n-1}\left(x\right)\left(-(2x-1)M_{n+1}\left(x\right)-M_n\left(x\right)\right)-W_{n-2}\left(x\right)M_{n+1}\left(x\right).\hfill \end{array}$$

In the following theorem, we determine an identity that describes the relationship between generalized John polynomials and polynomial $M_n\left(x\right)$.

Remark 1.

[[4], page 320 identity (7.21)]Let $G\left(z\right)={\displaystyle {\displaystyle \sum }_{n=0}^{\infty }}$ $a_n{z}^n$ be the generating function of sequence of $\left\{a_n\right\}$ then we have

$${\displaystyle \frac{1}{1-z}}G\left(z\right)=\stackrel{\infty }{{\displaystyle \sum }_{n=0}}\left(\stackrel{n}{{\displaystyle \sum }_{k=0}}{a}_n{z}^n\right).$$

Theorem 13.

For $n\ge 0,$ we have the following formula.

$$\begin{array}{cc}\hfill W_n\left(x\right)& =(W_2\left(x\right)-2x{W}_1\left(x\right)-W_0\left(x\right))\stackrel{n}{{\displaystyle \sum }_{k=0}}{P}_k\left(x\right)+W_0\left(x\right)P_{n+1}\left(x\right)\hfill \\ \hfill & -(W_2\left(x\right)-(2x+1)W_1\left(x\right)+(2x-1)W_0\left(x\right))P_n\left(x\right)\hfill \end{array}$$

(39)

Proof. Using Corollary 5, and Remark 1, we get

$${\displaystyle \frac{1}{\left(1-z\right)}}{\displaystyle \frac{1}{\left(1-2xz-z^2\right)}}=\stackrel{\infty }{{\displaystyle \sum }_{n=0}}\left(\stackrel{n}{{\displaystyle \sum }_{k=0}}{P}_{k+1}\left(x\right)\right)z^n.$$

(40)

Using (40) and (36) we obtain,

$$\begin{array}{cc}\hfill {\displaystyle \sum _{n=0}^{\infty }}{W}_n\left(x\right)z^n& ={\displaystyle \frac{W_0\left(x\right)+(W_1\left(x\right)-(2x+1)W_0\left(x\right))z+(W_2\left(x\right)-(2x+1)W_1\left(x\right)+(2x-1)W_0\left(x\right))z^2}{1-(2x+1)z+(2x-1)z^2+z^3}}\hfill \\ \hfill & =W_0\left(x\right){\displaystyle \frac{1}{\left(1-z\right)}}{\displaystyle \frac{1}{\left(1-2xz-z^2\right)}}+(W_1\left(x\right)-(2x+1)W_0\left(x\right))z{\displaystyle \frac{1}{\left(1-z\right)}}{\displaystyle \frac{1}{\left(1-2xz-z^2\right)}}\hfill \\ \hfill & +(W_2\left(x\right)-(2x+1)W_1\left(x\right)+(2x-1)W_0\left(x\right))z^2{\displaystyle \frac{1}{\left(1-z\right)}}{\displaystyle \frac{1}{\left(1-2xz-z^2\right)}}\hfill \\ \hfill & =W_0\left(x\right)\stackrel{\infty }{{\displaystyle \sum }_{n=0}}\left(\stackrel{n}{{\displaystyle \sum }_{k=0}}{P}_{k+1}\left(x\right)\right)z^n+(W_1\left(x\right)-(2x+1)W_0\left(x\right))\stackrel{\infty }{{\displaystyle \sum }_{n=0}}\left(\stackrel{n}{{\displaystyle \sum }_{k=0}}{P}_{k+1}\left(x\right)\right)z^{n+1}\hfill \\ \hfill & +(W_2\left(x\right)-(2x+1)W_1\left(x\right)+(2x-1)W_0\left(x\right))\stackrel{\infty }{{\displaystyle \sum }_{n=0}}\left(\stackrel{n}{{\displaystyle \sum }_{k=0}}{P}_{k+1}\left(x\right)\right)z^{n+2}\hfill \\ \hfill & =W_0\left(x\right)\stackrel{\infty }{{\displaystyle \sum }_{n=0}}\left(\stackrel{n}{{\displaystyle \sum }_{k=0}}{P}_{k+1}\left(x\right)\right)z^n+(W_1\left(x\right)-(2x+1)W_0\left(x\right))\stackrel{\infty }{{\displaystyle \sum }_{n=0}}\left(\stackrel{n}{{\displaystyle \sum }_{k=0}}{P}_{k+1}\left(x\right)-P_{n+1}\left(x\right)\right)z^n\hfill \\ \hfill & +(W_2\left(x\right)-(2x+1)W_1\left(x\right)+(2x-1)W_0\left(x\right))\stackrel{\infty }{{\displaystyle \sum }_{n=0}}\left(\stackrel{n}{{\displaystyle \sum }_{k=0}}{P}_{k+1}\left(x\right)-P_{n+1}\left(x\right)-P_n\left(x\right)\right)z^n\hfill \\ \hfill & ={\displaystyle \sum _{n=0}^{\infty }}(W_0\left(x\right)\stackrel{n}{{\displaystyle \sum }_{k=0}}{P}_{k+1}\left(x\right)+(W_1\left(x\right)-(2x+1)W_0\left(x\right))\stackrel{n}{{\displaystyle \sum }_{k=0}}{P}_{k+1}\left(x\right)+(W_2\left(x\right)-(2x+1)W_1\left(x\right)\hfill \\ \hfill & +(2x-1)W_0\left(x\right))\stackrel{n}{{\displaystyle \sum }_{k=0}}{P}_{k+1}\left(x\right)-(W_1\left(x\right)-(2x+1)W_0\left(x\right))P_{n+1}\left(x\right)-(W_2\left(x\right)-(2x+1)W_1\left(x\right)\hfill \\ \hfill & +(2x-1)W_0\left(x\right))P_{n+1}\left(x\right)-(W_2\left(x\right)-(2x+1)W_1\left(x\right)+(2x-1)W_0\left(x\right))P_n\left(x\right))z^n.\hfill \end{array}$$

Hence we have

$$\begin{array}{cc}\hfill {\displaystyle \sum _{n=0}^{\infty }}{W}_n\left(x\right)z^n& ={\displaystyle \sum _{n=0}^{\infty }}(W_2\left(x\right)-2x{W}_1\left(x\right)-W_0\left(x\right))\stackrel{n}{{\displaystyle \sum }_{k=0}}{P}_k\left(x\right)+W_0\left(x\right)P_{n+1}\left(x\right)\hfill \\ \hfill & -(W_2\left(x\right)-(2x+1)W_1\left(x\right)+(2x-1)W_0\left(x\right))P_n\left(x\right))z^n.\hfill \end{array}$$

Thus, the proof is completed. □

The last theorem gives us following Theorem.

Theorem 14.

For $n\ge 0,$ we have

$$\begin{array}{cc}\hfill W_n\left(x\right)& ={\displaystyle \frac{(W_2\left(x\right)-2x{W}_1\left(x\right)+(2x-1)W_0\left(x\right))}{2x}}{P}_{n+1}\left(x\right)+{\displaystyle \frac{(1-2x)W_2\left(x\right)+4x^2{W}_1\left(x\right)-(4x^2-2x+1)W_0\left(x\right)}{2x}}{P}_n\left(x\right)\hfill \\ \hfill & -{\displaystyle \frac{(W_2\left(x\right)-2x{W}_1\left(x\right)-W_0\left(x\right))}{2x}}.\hfill \end{array}$$

Proof. Using [[15], Theorem15] and and using identity (15), we have

$$\begin{array}{cc}\hfill W_n\left(x\right)& =(W_2\left(x\right)-2x{W}_1\left(x\right)-W_0\left(x\right))\left({\displaystyle \frac{P_{n+1}\left(x\right)+P_n\left(x\right)-1}{2x}}\right)+W_0\left(x\right)P_{n+1}\left(x\right)-(W_2\left(x\right)-(2x+1)W_1\left(x\right)\hfill \\ \hfill & +(2x-1)W_0\left(x\right))P_n\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{(W_2\left(x\right)-2x{W}_1\left(x\right)+(2x-1)W_0\left(x\right))}{2x}}{P}_{n+1}\left(x\right)+{\displaystyle \frac{(1-2x)W_2\left(x\right)+4x^2{W}_1\left(x\right)-(4x^2-2x+1)W_0\left(x\right)}{2x}}{P}_n\left(x\right)\hfill \\ \hfill & -{\displaystyle \frac{(W_2\left(x\right)-2x{W}_1\left(x\right)-W_0\left(x\right))}{2x}}.\square \hfill \end{array}$$

Then, Theorem 14 gives the following theorem which provides an alternative proof of Theorem 5.

Theorem 15.

For $n\ge 0,$ we have

$$W_n\left(x\right)=a_1\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right){\left(2x\right)}^{n-2i}+a_2\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n-1}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i-1}{i}}\right){\left(2x\right)}^{n-2i-1}+a_3$$

where

$$\begin{array}{cc}\hfill a_1& ={\displaystyle \frac{(W_2\left(x\right)-2x{W}_1\left(x\right)+(2x-1)W_0\left(x\right))}{2x}},\hfill \\ \hfill a_2& ={\displaystyle \frac{(1-2x)W_2\left(x\right)+4x^2{W}_1\left(x\right)-(4x^2-2x+1)W_0\left(x\right)}{2x}},\hfill \\ \hfill a_3& =-{\displaystyle \frac{(W_2\left(x\right)-2x{W}_1\left(x\right)-W_0\left(x\right))}{2x}}.\hfill \end{array}$$

Proof. Using Theorem 14 and identity (21), the proof can be done. □

Now, using Theorem 15 and Theorem 14, we derive the following corollary.

Corollary 9.

For $n\ge 0,$ we have the following equalities.

(a) 

$$\begin{array}{cc}\hfill M_n\left(x\right)& ={\displaystyle \frac{P_{n+1}\left(x\right)+P_n\left(x\right)-1}{2x}},\hfill \\ \hfill & ={\displaystyle \frac{1}{2x}}\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right){\left(2x\right)}^{n-2i}+{\displaystyle \frac{1}{2x}}\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n-1}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i-1}{i}}\right){\left(2x\right)}^{n-2i-1}-{\displaystyle \frac{1}{2x}}.\hfill \end{array}$$

(b) 

$$\begin{array}{cc}\hfill N_n\left(x\right)& ={\displaystyle \frac{\left(x+1\right)P_{n+1}\left(x\right)-\left(x-1\right)P_n\left(x\right)-1}{x}},\hfill \\ \hfill & ={\displaystyle \frac{\left(x+1\right)}{x}}\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right){\left(2x\right)}^{n-2i}-{\displaystyle \frac{\left(x-1\right)}{x}}\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n-1}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i-1}{i}}\right){\left(2x\right)}^{n-2i-1}-{\displaystyle \frac{1}{x}}.\hfill \end{array}$$

(c) 

$$\begin{array}{cc}\hfill {\mathfrak{L}}_n\left(x\right)& ={\displaystyle \frac{P_{n+1}\left(x\right)-\left(x-1\right)P_n\left(x\right)-1}{x}}\hfill \\ \hfill & ={\displaystyle \frac{1}{x}}\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right){\left(2x\right)}^{n-2i}-{\displaystyle \frac{\left(x-1\right)}{x}}\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n-1}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i-1}{i}}\right){\left(2x\right)}^{n-2i-1}-{\displaystyle \frac{1}{x}}.\hfill \end{array}$$

(d) 

$$\begin{array}{cc}\hfill {\mathfrak{R}}_n\left(x\right)& ={\displaystyle \frac{2P_{n+1}\left(x\right)+\left(2x^2-2x+2\right)P_n\left(x\right)-2}{x}},\hfill \\ \hfill & ={\displaystyle \frac{2}{x}}\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right){\left(2x\right)}^{n-2i}+{\displaystyle \frac{\left(2x^2-2x+2\right)}{x}}\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n-1}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i-1}{i}}\right){\left(2x\right)}^{n-2i-1}-{\displaystyle \frac{2}{x}}.\hfill \end{array}$$

(e) 

$$\begin{array}{cc}\hfill H_n\left(x\right)& =2P_{n+1}\left(x\right)-2x{P}_n\left(x\right)+1,\hfill \\ \hfill & =2\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{i}}\right){\left(2x\right)}^{n-2i}-2x\underset{i=0}{{\displaystyle \sum }^{⌊{\displaystyle \frac{n-1}{2}}⌋}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i-1}{i}}\right){\left(2x\right)}^{n-2i-1}+1.\hfill \end{array}$$

Next, we give several identities that are relevant to matrix theory. To become more specific, we use the generalized John polynomials to construct a family of square matrices of order three. These matrices are built such that their entries can be expressed in relation to the polynomial sequence, permitting the exploration of structural properties, recurrence relations, and determinant formulas that result from the interplay between polynomial theory and linear algebra.

$$\begin{array}{cc}\hfill Q\left(x\right)& =\left(\begin{array}{ccc}2x+1& -(2x-1)& -1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right).\hfill \\ \hfill K_n\left(x\right)& =\left(\begin{array}{ccc}{W}_{n+2}\left(x\right)& W_{n+1}\left(x\right)& W_n\left(x\right)\\ W_{n+1}\left(x\right)& W_n\left(x\right)& W_{n-1}\left(x\right)\\ W_n\left(x\right)& W_{n-1}\left(x\right)& W_{n-2}\left(x\right)\end{array}\right).\hfill \\ \hfill B_n\left(x\right)& =\left(\begin{array}{ccc}{M}_{n+1}\left(x\right)& -(2x-1)M_n\left(x\right)-M_{n-1}\left(x\right)& -M_n\left(x\right)\\ M_n\left(x\right)& -(2x-1)M_{n-1}\left(x\right)-M_{n-2}\left(x\right)& -M_{n-1}\left(x\right)\\ M_{n-1}\left(x\right)& -(2x-1)M_{n-2}\left(x\right)-M_{n-3}\left(x\right)& -M_{n-2}\left(x\right)\end{array}\right).\hfill \\ \hfill D_n\left(x\right)& =\left(\begin{array}{ccc}{W}_{n+1}\left(x\right)& -(2x-1)W_n\left(x\right)-W_{n-1}\left(x\right)& -W_n\left(x\right)\\ W_n\left(x\right)& -(2x-1)W_{n-1}\left(x\right)-W_{n-2}\left(x\right)& -W_{n-1}\left(x\right)\\ W_{n-1}\left(x\right)& -(2x-1)W_{n-2}\left(x\right)-W_{n-3}\left(x\right)& -W_{n-2}\left(x\right)\end{array}\right).\hfill \end{array}$$

As a result, we may apply the matrix representations given above to find an important structural relationship for the generalized John polynomials. We can now use this framework to establish the following theorems.

Theorem 16.

For all integer $n,$ we have

$$Q^n\left(x\right)K_0\left(x\right)=K_n\left(x\right)$$

Proof. The proof can be done by using strong induction on $n\ge 0.$ First, if we take $n=0,$ the statement of the theorem is hold. Next assuming the statement of the theorem is true for $0\le n\le k,$ then for $n=k+1$, we obtain

$$Q^{k+1}\left(x\right)K_0\left(x\right)=Q\left(x\right)Q^k\left(x\right)K_0\left(x\right)=Q\left(x\right)K_k\left(x\right)=K_{k+1}\left(x\right).$$

The other case $n<0$ can be proof similarly. □

If we take determinant of the statement given in the Theorem 16, we obtain the following corollary.

Corollary 10

(Simpson’s formula). For all integer $n,$ we have

$$\left|\begin{array}{ccc}{W}_{n+2}\left(x\right)& W_{n+1}\left(x\right)& W_n\left(x\right)\\ W_{n+1}\left(x\right)& W_n\left(x\right)& W_{n-1}\left(x\right)\\ W_n\left(x\right)& W_{n-1}\left(x\right)& W_{n-2}\left(x\right)\end{array}\right|={(-1)}^n\left|\begin{array}{ccc}{W}_2\left(x\right)& W_1\left(x\right)& W_0\left(x\right)\\ W_1\left(x\right)& W_0\left(x\right)& W_{-1}\left(x\right)\\ W_0\left(x\right)& W_{-1}\left(x\right)& W_{-2}\left(x\right)\end{array}\right|.$$

Using $B_n\left(x\right),$ $Q\left(x\right)$ and $D_n\left(x\right),$ we can state the following theorem.

Theorem 17.

The following properties hold: for all integers $m,n$,

(a) 

$B_n\left(x\right)={\left(Q\left(x\right)\right)}^n$.

(b) 

$D_1\left(x\right){\left(Q\left(x\right)\right)}^n={\left(Q\left(x\right)\right)}^n{D}_1\left(x\right).$

(c) 

$D_{n+m}\left(x\right)=D_n\left(x\right)B_m\left(x\right)=B_m\left(x\right)D_n\left(x\right).$

Proof. Set $r=2x+1,$ $s=-2x+1,$ $t=-1$ in [17], Theorem 51]. □

In the next lemma we give formula related to derivative of the power of the square matrix.

Lemma 3.

Let $A\left(x\right)$ be a square matrix function in variable $x,$ and $n\ge 1,$ then we have

$${\displaystyle \frac{d{A}^n\left(x\right)}{dx}}=\underset{k=0}{{\displaystyle \sum }^{n-1}}{A}^{n-1-k}\left(x\right){\displaystyle \frac{dA\left(x\right)}{dx}}{A}^k\left(x\right).$$

Proof. For the proof, see [[21], p. 207]. □

In the next theorem, we state the derivative of the generalized John polynomials.

Theorem 18.

For $n\ge 1,$ we have the following equality.

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{W}_n\left(x\right)}{dx}}& =\underset{k=0}{{\displaystyle \sum }^{n-1}}\left(\left(2W_{k+2}\left(x\right)-2W_{k+1}\left(x\right)\right)g_{n-k-2}\left(x\right)\right)+{\displaystyle \frac{d{W}_2\left(x\right)}{dx}}{g}_{n-1}\left(x\right)\hfill \\ \hfill & +\left(-(2x+1)g_{n-2}\left(x\right)+g_{n-3}\left(x\right)\right){\displaystyle \frac{d{W}_1\left(x\right)}{dx}}+g_{n-2}\left(x\right){\displaystyle \frac{d{W}_0\left(x\right)}{dx}}.\hfill \end{array}$$

Proof. First, we define the vector $v_n\left(x\right)={(W_n\left(x\right),W_{n-1}\left(x\right),W_{n-2}\left(x\right))}^T,$ then we obtain

$$v_{n+2}\left(x\right)=Q^n\left(x\right)v_2\left(x\right).$$

(41)

This identity can be proof using induction on $n.$

Then, if we derive both side of the equation (41) and using Lemma 3 and Theorem 17, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{v}_{n+2}\left(x\right)}{dx}}& ={\displaystyle \frac{d{Q}^n\left(x\right)}{dx}}{v}_2\left(x\right)+Q^n\left(x\right){\displaystyle \frac{d{v}_2\left(x\right)}{dx}}\hfill \\ \hfill & =\left(\underset{k=0}{{\displaystyle \sum }^{n-1}}{Q}^{n-1-k}\left(x\right){\displaystyle \frac{dQ\left(x\right)}{dx}}{Q}^k\left(x\right)\right)v_2\left(x\right)+Q^n\left(x\right){\displaystyle \frac{d{v}_2\left(x\right)}{dx}}\hfill \\ \hfill & =\underset{k=0}{{\displaystyle \sum }^{n-1}}{B}_{n-1-k}\left(x\right)\left(\begin{array}{ccc}2& -2& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)v_{k+2}\left(x\right)+B_n\left(x\right){\displaystyle \frac{d{v}_2\left(x\right)}{dx}}\hfill \\ \hfill & =\underset{k=0}{{\displaystyle \sum }^{n-1}}{B}_{n-1-k}\left(x\right)\left(\begin{array}{c}2W_{k+2}\left(x\right)-2W_{k+1}\left(x\right)\\ 0\\ 0\end{array}\right)+B_n\left(x\right)\left(\begin{array}{c}{\displaystyle \frac{d{W}_2\left(x\right)}{dx}}\\ {\displaystyle \frac{d{W}_1\left(x\right)}{dx}}\\ {\displaystyle \frac{d{W}_0\left(x\right)}{dx}}\end{array}\right)\hfill \\ \hfill & =\underset{k=0}{{\displaystyle \sum }^{n-1}}\left(\begin{array}{c}\left(2W_{k+2}\left(x\right)-2W_{k+1}\left(x\right)\right)M_{n-k}\left(x\right)\\ \left(2W_{k+2}\left(x\right)-2W_{k+1}\left(x\right)\right)M_{n-k-1}\left(x\right)\\ \left(2W_{k+2}\left(x\right)-2W_{k+1}\left(x\right)\right)M_{n-k-2}\left(x\right)\end{array}\right)\hfill \\ \hfill & +\left(\begin{array}{c}{\displaystyle \frac{d{W}_2\left(x\right)}{dx}}{M}_{n+1}\left(x\right)+\left(-(2x-1)M_n\left(x\right)-M_{n-1}\left(x\right)\right){\displaystyle \frac{d{W}_1\left(x\right)}{dx}}-M_n\left(x\right){\displaystyle \frac{d{W}_0\left(x\right)}{dx}}\\ {\displaystyle \frac{d{W}_2\left(x\right)}{dx}}{M}_n\left(x\right)+\left(-(2x-1)M_{n-1}\left(x\right)-M_{n-2}\left(x\right)\right){\displaystyle \frac{d{W}_1\left(x\right)}{dx}}-M_{n-1}\left(x\right){\displaystyle \frac{d{W}_0\left(x\right)}{dx}}\\ {\displaystyle \frac{d{W}_2\left(x\right)}{dx}}{M}_{n-1}\left(x\right)+\left(-(2x-1)M_{n-2}\left(x\right)-M_{n-3}\left(x\right)\right){\displaystyle \frac{d{W}_1\left(x\right)}{dx}}-M_{n-2}\left(x\right){\displaystyle \frac{d{W}_0\left(x\right)}{dx}}\end{array}\right)\hfill \end{array}$$

This gives,

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{W}_n\left(x\right)}{dx}}& =\underset{k=0}{{\displaystyle \sum }^{n-1}}\left(2W_{k+2}\left(x\right)-2W_{k+1}\left(x\right)\right)M_{n-k-2}\left(x\right)+{\displaystyle \frac{d{W}_2\left(x\right)}{dx}}{M}_{n-1}\left(x\right)\hfill \\ \hfill & +\left(-(2x-1)M_{n-2}\left(x\right)-M_{n-3}\left(x\right)\right){\displaystyle \frac{d{W}_1\left(x\right)}{dx}}-M_{n-2}\left(x\right){\displaystyle \frac{d{W}_0\left(x\right)}{dx}}.\square \hfill \end{array}$$

In the next corollary, some derivative formulas are given for the special cases of John polynomials.

Corollary 11.

For $n\ge 1,$ we have the following equalities.

(a) 

${\displaystyle \frac{d{M}_n\left(x\right)}{dx}}={\displaystyle {\displaystyle \sum }_{k=0}^{n-1}}\left(2M_{k+2}\left(x\right)-2M_{k+1}\left(x\right)\right)M_{n-k-2}\left(x\right)+2M_{n-1}\left(x\right).$

(b) 

${\displaystyle \frac{d{N}_n\left(x\right)}{dx}}={\displaystyle {\displaystyle \sum }_{k=0}^{n-1}}\left(2N_{k+2}\left(x\right)-2N_{k+1}\left(x\right)\right)M_{n-k-2}\left(x\right)+\left(8x+2\right)M_{n-1}\left(x\right)+2\left(-(2x-1)M_{n-2}\left(x\right)\right.$$\left.-M_{n-3}\left(x\right)\right).$

(c) 

${\displaystyle \frac{d{\mathfrak{L}}_n\left(x\right)}{dx}}={\displaystyle {\displaystyle \sum }_{k=0}^{n-1}}\left(2{\mathfrak{L}}_{k+2}\left(x\right)-2{\mathfrak{L}}_{k+1}\left(x\right)\right)M_{n-k-2}\left(x\right)+2M_{n-1}\left(x\right).$

(d) 

${\displaystyle \frac{d{\mathfrak{R}}_n\left(x\right)}{dx}}={\displaystyle {\displaystyle \sum }_{k=0}^{n-1}}\left(2{\mathfrak{R}}_{k+2}\left(x\right)-2{\mathfrak{R}}_{k+1}\left(x\right)\right)M_{n-k-2}\left(x\right)+\left(8x+4\right)M_{n-1}\left(x\right)+2\left(-(2x-1)M_{n-2}\left(x\right)\right.$$\left.-M_{n-3}\left(x\right)\right).$

(e) 

${\displaystyle \frac{d{H}_n\left(x\right)}{dx}}={\displaystyle {\displaystyle \sum }_{k=0}^{n-1}}\left(2H_{k+2}\left(x\right)-2H_{k+1}\left(x\right)\right)M_{n-k-2}\left(x\right)+8x{M}_{n-1}\left(x\right)+2\left(-(2x-1)M_{n-2}\left(x\right)-M_{n-3}\left(x\right)\right).$

3. Conclusion

In this study, we propose and examine the "generalized John polynomials," a novel sequence developed by merging the attributes of Leonardo and Pell numbers within a second-order linear non-homogeneous recurrence framework. Additionally, we isolate five specific sub-cases of this polynomial family to analyze their foundational characteristics. The primary theoretical and algebraic contributions of this research are outlined below:

Fundamental Properties: We established the ordinary generating functions, general solutions, and Binet-type formulas for these polynomials, ultimately revealing that they also conform to a third-order homogeneous recurrence relation.

Combinatorial Expressions: By applying partial fraction decompositions alongside power series, we derived exact combinatorial representations for the new polynomial family.

Classical Identities: Prominent mathematical theorems specifically the formulas of Catalan, d’Ocagne, Vajda, and Honsberger were successfully adapted and proven within the context of this generalized structure.

Matrix and Determinantal Formulations: We achieved determinantal representations by utilizing the Doolittle algorithm for LU decomposition. Moreover, the creation of third-order square matrix representations enabled the extraction of Simpson’s identity and other relevant matrix properties.

Analytical Applications: Finally, we utilized matrix differential calculus techniques to formulate derivative identities for the generalized John polynomials.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.