Identities Arising from a Certain Family of Special Combinatorial Numbers and Polynomials

1. Introduction

Special numbers and polynomials have attracted sustained attention in recent years because of their central role in combinatorics, analytic number theory, complex analysis, and the theory of special functions. In particular, generating functions and the functional equations they satisfy constitute powerful tools for deriving explicit formulas, recurrence relations, derivative identities, and combinatorial relations. Among the most extensively studied families are the Apostol–Bernoulli, Apostol–Euler, Frobenius–Euler, Fubini, and Stirling numbers and polynomials.

Interpolation functions associated with special numbers and polynomials also occupy an important place in analytic number theory and complex analysis. They provide analytic continuations of discrete sequences and reveal deep connections between combinatorial quantities and zeta-type functions. In many cases, the values of such functions at negative integers reproduce the corresponding special numbers or polynomials, thereby exposing structural properties that are not immediately visible from generating functions alone.

Although numerous generalizations of classical special numbers and polynomials have been introduced, the higher-order structure of the family $y_{9,n}(\lambda ;a)$ and its associated polynomials has not yet been developed in a unified framework. In particular, a systematic treatment of the simultaneous connections of this family with Apostol–Euler, Apostol–Bernoulli, Euler–Frobenius, Fubini, and Stirling numbers, together with the corresponding interpolation functions, is still missing.

Motivated by this observation, we introduce the higher-order numbers $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$ and the higher-order polynomials $y_{9,n}^{\left(\alpha \right)}(x,\lambda ;a)$ via their generating functions. We establish explicit formulas, identities, and recurrence relations, and derive several connections with classical and modern families of special numbers and polynomials. In addition, we define an interpolation function associated with the higher-order numbers and show that its values at negative integers recover these numbers up to an explicit normalization factor. By means of a residue-class decomposition of this interpolation function, we further introduce a generalized hypergeometric Hurwitz–Lerch-type zeta function and investigate some of its special cases and structural properties.

The present paper may therefore be regarded as a higher-order extension of the family introduced in [32]. Several identities established for the original numbers are recovered as special cases when $\alpha =1$, while the present framework yields additional links with Apostol–Euler, Apostol–Bernoulli, Euler–Frobenius, and generalized Hurwitz–Lerch-type zeta functions.

To present our main results in a self-contained manner, we first recall the notation, definitions, and preliminary relations that will be used throughout the paper. Let $\mathbb{N}$ and $\mathbb{C}$ denote the sets of natural numbers and complex numbers, respectively, and set ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$.

The Stirling numbers of the second kind, $S_2(n,k)$, are defined by the generating function

$${\displaystyle \frac{{\left(e^t-1\right)}^k}{k!}}={\displaystyle \sum _{n=0}^{\infty }}{S}_2\left(n,k\right){\displaystyle \frac{t^n}{n!}}$$

(1)

(See, for example, [1,3,32,33] and the references therein.)

By (1), one has

$$S_2\left(n,k\right)={\displaystyle \frac{1}{k!}}{\displaystyle \sum _{j=0}^k}{(-1)}^{k-j}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)j^n,$$

(2)

which immediately implies that $S_2(n,k)=0$ whenever $k>n$.

1.1. Apostol–Bernoulli and Apostol–Euler-Type Polynomials

The Apostol–Bernoulli polynomials ${\mathcal{B}}_n(x;\lambda )$ are defined by the generating function

$$b(t,x;\lambda )={\displaystyle \frac{t}{\lambda e^t-1}}{e}^{xt}={\displaystyle \sum _{n=0}^{\infty }}{\mathcal{B}}_n\left(x;\lambda \right){\displaystyle \frac{t^n}{n!}},$$

(3)

where $\lambda$ is an arbitrary real or complex parameter. Here, $\left|t\right|<2\pi$ when $\lambda =1$, and $\left|t\right|<|log\lambda |$ when $\lambda \ne 1$; see, for example, [3,4,6,8,10,11,13,15,18,19,20,21,22,23,29,31,32,33].

By setting $x=0$ in (3), we obtain the Apostol–Bernoulli numbers:

$${\mathcal{B}}_n\left(0;\lambda \right)={\mathcal{B}}_n\left(\lambda \right),$$

see [3,12,29,31,32,33].

Moreover, taking $\lambda =1$ in (3) yields

$${\mathcal{B}}_n\left(1\right)=B_n,$$

where $B_n$ denotes the classical Bernoulli numbers; see [3,29,31,32,33].

Similarly, the Apostol–Euler polynomials ${\mathcal{E}}_n(x;\lambda )$ are defined by

$$h(t;x,\lambda )={\displaystyle \frac{2}{\lambda e^t+1}}{e}^{xt}={\displaystyle \sum _{n=0}^{\infty }}{\mathcal{E}}_n\left(x;\lambda \right){\displaystyle \frac{t^n}{n!}},$$

(4)

where $\lambda$ is again an arbitrary real or complex parameter. In this case, $\left|t\right|<\pi$ when $\lambda =1$, and $\left|t\right|<|ln(-\lambda \left)\right|$ when $\lambda \ne 1$; see [29,31,32,33].

Setting $x=0$ in (4), we obtain the Apostol–Euler numbers:

$${\mathcal{E}}_n\left(0;\lambda \right)={\mathcal{E}}_n\left(\lambda \right)$$

see [29,31,32,33].

Finally, taking $\lambda =1$ in (4) gives

$${\mathcal{E}}_n\left(1\right)=E_n,$$

where $E_n$ denotes the classical Euler numbers; see [3,29,31,32,33].

1.2. The Polynomials $Y_n\left(x,u;a\right)$ and Euler–Frobenius-Type Polynomials

The polynomials $Y_n\left(x,u;a\right)$ are defined by the generating function:

$${\displaystyle \frac{1}{a^t-u}}{a}^{xt}={\displaystyle \sum _{n=0}^{\infty }}{Y}_n\left(x,u;a\right){\displaystyle \frac{t^n}{n!}},$$

(5)

where $\left|tln\left(a\right)+ln\left({\displaystyle \frac{1}{u}}\right)\right|<2\pi$ (cf. [24,25,26,30,34]; see also [2,28,32]).

Setting $x=0$ in (5), we obtain the numbers $Y_n\left(u;a\right)$ defined by

$$Y_n\left(0,u;a\right)=Y_n\left(u;a\right)$$

which generating function

$${\displaystyle \frac{1}{a^t-u}}={\displaystyle \sum _{n=0}^{\infty }}{Y}_n\left(u;a\right){\displaystyle \frac{t^n}{n!}}$$

(6)

(cf. [30]; and see also [2,28,32]).

Let $a,b,c\in {\mathbb{R}}^{+}$$\left(a\ne b\right),x\in \mathbb{R},\lambda \in \mathbb{C}$ and $u\in \mathbb{C}\setminus \left\{\lambda \right\}$. The generalized Euler-Frobenius type polynomials ${\mathcal{H}}_n\left(x;u;a,b,c;\lambda \right)$ are defined by

$$F_{\lambda }\left(t,x;u,a,b,c;\lambda \right)={\displaystyle \frac{\left(a^t-u\right)c^{xt}}{\lambda b^t-u}}={\displaystyle \sum _{n=0}^{\infty }}{\mathcal{H}}_n\left(x;u;a,b,c;\lambda \right){\displaystyle \frac{t^n}{n!}},$$

(7)

where $a,b,c\in {\mathbb{R}}^{+}$ with $a\ne b$, $x\in \mathbb{R}$, and $\lambda ,u\in \mathbb{C}$ with $\lambda \ne u$. The generating function is considered in a neighborhood of $t=0$ where $\lambda b^t-u\ne 0$.

When $x=0$ in (7), we have the generalized Euler-Frobenius type numbers:

$${\mathcal{H}}_n\left(0;u;a,b,c;\lambda \right)={\mathcal{H}}_n\left(u;a,b,c;\lambda \right),$$

which are generated by

$${\displaystyle \frac{a^t-u}{\lambda b^t-u}}={\displaystyle \sum _{n=0}^{\infty }}{\mathcal{H}}_n\left(u;a,b,c;\lambda \right){\displaystyle \frac{t^n}{n!}},$$

(8)

(cf. [30]; and also see [27,28]).

In the special case $\lambda =a=1$ and $b=c=e$, the generalized Euler–Frobenius-type polynomials reduce to the Euler–Frobenius polynomials, which are defined by

$${\displaystyle \frac{1-u}{e^t-u}}{e}^{xt}={\displaystyle \sum _{n=0}^{\infty }}{H}_n\left(x;u\right){\displaystyle \frac{t^n}{n!}}$$

(9)

(cf. [3,29,31,32,33].

Moreover, setting $x=0$ in (9), we obtain the Euler–Frobenius numbers:

$$H_n\left(0;u\right)=H_n\left(u\right)$$

(cf. [3,29,31,32,33].

Further, substituting $u=-1$ and $x=0$ into (9) yields the Euler numbers:

$$H_n\left(0;-1\right)=H_n\left(-1\right)=E_n.$$

The Fubini numbers, $w_g\left(n\right)$, are defined by the generating function:

$${\displaystyle \frac{1}{2-e^t}}={\displaystyle \sum _{n=0}^{\infty }}{w}_g\left(n\right){\displaystyle \frac{t^n}{n!}},$$

(10)

where $\left|t\right|<\left|ln\left({\displaystyle \frac{1}{2}}\right)\right|$ (cf. [7,16,17,32]).

Combining (1) with (10), we obtain

$$w_g\left(n\right)={\displaystyle \sum _{j=0}^n}j!S_2(n,j)$$

(11)

(cf. [7,16,17,32]).

Finally, by substituting

$$f(t,x;\lambda ,a)={\left(\lambda +a^t\right)}^x,\lambda ,x,t\in {\mathbb{Z}}_p.$$

Building on the p-adic integral representations on ${\mathbb{Z}}_p$, the ring of p-adic integers, Simsek [32] introduced the numbers $y_{9,n}(\lambda ;a)$ and the polynomials $y_{9,n}(x,\lambda ;a)$ through the generating functions

$$Y_1(t;a,\lambda )={\displaystyle \frac{2}{a^t+\lambda }}={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}(\lambda ;a){\displaystyle \frac{t^n}{n!}},$$

(12)

and

$$\begin{array}{cc}\hfill Y_2(t,x;a,\lambda )& ={(1+t)}^x{Y}_1(t;a,\lambda )\hfill \\ \hfill & ={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}(x,\lambda ;a){\displaystyle \frac{t^n}{n!}}.\hfill \end{array}$$

(13)

Here, $\lambda$ is a real or complex parameter, and the parameters are assumed to lie in a domain where the corresponding power-series expansions are convergent.

In particular, the family ${\left\{y_{9,n}(\lambda ;a)\right\}}_{n\ge 0}$ includes the Fubini numbers as a special case.

Theorem 1 (cf.

[32]). Let $n\in {\mathbb{N}}_0$. Then

$$y_{9,n}(x,\lambda ;a)={\displaystyle \sum _{j=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right){\left(x\right)}_{n-j}{y}_{9,j}(\lambda ;a),$$

(14)

where ${\left(x\right)}_n=x(x-1)\dots (x-n+1)$ denotes the falling factorial.

We now record several specializations of the generating functions (12) and (13). Setting $a=1$ in (12), provided that $\lambda \ne -1$, yields

$$Y_1(t;1,\lambda )={\displaystyle \frac{2}{1+\lambda }}.$$

Consequently,

$$y_{9,0}(\lambda ;1)={\displaystyle \frac{2}{1+\lambda }},y_{9,n}(\lambda ;1)=0(n\ge 1).$$

Likewise, setting $a=1$ into (13), we obtain

$$Y_2(t,x;1,\lambda )={\displaystyle \frac{2}{1+\lambda }}{(1+t)}^x.$$

Since

$${(1+t)}^x={\displaystyle \sum _{n=0}^{\infty }}{\left(x\right)}_n{\displaystyle \frac{t^n}{n!}},$$

it follows that

$$y_{9,n}(x,\lambda ;1)={\displaystyle \frac{2}{1+\lambda }}{\left(x\right)}_n,n\in {\mathbb{N}}_0.$$

Another noteworthy specialization is obtained by taking $a=e$ and $\lambda =-2$ in into (12). In this case,

$$y_{9,n}(-2;e)=-2w_g\left(n\right),$$

(15)

where $w_g\left(n\right)$ denotes the nth Fubini number (cf. [32]).

Moreover, since

$${(-1)}^{n}n!=(n+1){\displaystyle \sum _{j=0}^n}{B}_j{S}_1(n,j)$$

(cf. [14,31]), we obtain the following identity involving the numbers $y_{9,n}(\lambda ;1)$, the Bernoulli numbers, and the Stirling numbers of the first kind:

$$y_{9,n}\left(\lambda ;1\right)=2(n+1){\displaystyle \sum _{j=0}^n}{B}_j{S}_1(n,j).$$

Using (11) and (15), we obtain

$$y_{9,n}\left(-2;e\right)=-2{\displaystyle \sum _{j=0}^n}j!S_2(n,j)$$

(cf. [32]).

Furthermore, combining $\lambda$ with $-\lambda$ in (12), we find

$$y_{9,n}\left(-\lambda ;a\right)=2Y_n\left(\lambda ;a\right)$$

in agreement with cf. [32].

In the sequel, we study higher-order analogues of the numbers $y_{9,n}(\lambda ;a)$ and the polynomials $y_{9,n}(x,\lambda ;a)$, together with their generating functions. We also introduce interpolation functions associated with these quantities and establish their relations with the Lerch zeta function.

2. Generating Functions for the Higher-Order Numbers $y_{9,n}\left(\lambda ;a\right)$ and Polynomials $y_{9,n}\left(x,\lambda ;a\right)$

In this section, we introduce the generating functions associated with the higher-order numbers $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$ and the corresponding polynomials $y_{9,n}^{\left(\alpha \right)}(x,\lambda ;a)$. We then derive several of their basic properties using these generating functions and the functional relations they satisfy.

Let $a\in {\mathbb{R}}^{+}$, $x\in \mathbb{R}$, and $\lambda ,\alpha \in \mathbb{C}$. We define

$$f(t;\lambda ,\alpha ,a)={\left({\displaystyle \frac{2}{a^t+\lambda }}\right)}^{\alpha }={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right){\displaystyle \frac{t^n}{n!}}$$

(16)

and

$$g(t,x;\lambda ,\alpha ,a)=f(t;\lambda ,\alpha ,a){(1+t)}^x={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}\left(x,\lambda ;a\right){\displaystyle \frac{t^n}{n!}},$$

(17)

where

$$\left|tlna+ln\left({\displaystyle \frac{1}{\lambda }}\right)\right|<\pi .$$

We now briefly explain the motivation behind the generating functions (16) and (17). The function $f(t;\lambda ,\alpha ,a)$ provides a unified generating mechanism for the family of numbers $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$, while the multiplication by ${(1+t)}^x$ naturally extends this construction to a polynomial sequence in the variable x. This formulation allows us to derive recurrence relations, special values, and other structural identities in a systematic way. In particular, the choice of the factor ${(2/(a^t+\lambda ))}^{\alpha }$ reflects the analytic structure underlying the associated higher-order numbers, and the additional factor ${(1+t)}^x$ produces the polynomial extension in a manner analogous to classical generating-function constructions.

2.1. Motivation for the Generating Functions

Several special cases of the generating functions defined in eq16) and (17) are noteworthy:

$$y_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right)=y_{9,n}^{\left(\alpha \right)}\left(0,\lambda ;a\right),$$

$$y_{9,n}\left(x,\lambda ;a\right)=y_{9,n}^{\left(1\right)}\left(x,\lambda ;a\right),$$

$$y_{9,n}\left(\lambda ;a\right)=y_{9,n}^{\left(1\right)}\left(\lambda ;a\right),$$

$$y_{9,0}^{\left(\alpha \right)}\left(\lambda ;1\right)={\displaystyle \frac{2^{\alpha }}{{\left(1+\lambda \right)}^{\alpha }}}$$

and, for $n\ge 1$,

$$y_{9,n}^{\left(\alpha \right)}\left(\lambda ;1\right)=0.$$

By replacing t with $e^t-1$, setting $\alpha =k$ with $k\in \mathbb{N}$, and substituting $\lambda$ by $-u$ in (16) , we obtain

$${\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(k\right)}\left(-u;a\right){\displaystyle \frac{t^n}{n!}}=2^k{\displaystyle \sum _{n=0}^{\infty }}{\mathsf{Y}}_n^{\left(k\right)}\left(0,u;a\right){\displaystyle \frac{t^n}{n!}}.$$

Therefore, by equating the coefficients of ${\displaystyle \frac{t^n}{n!}}$ on both sides, we deduce

$$y_{9,n}^{\left(k\right)}\left(-u;a\right)=2^k{\mathsf{Y}}_n^{\left(k\right)}\left(0,u;a\right)$$

in agreement with [28].

Moreover, from (16), we derive the functional equation

$${\lambda }^{\alpha }f(t;\lambda ,\alpha ,a)={\left({\displaystyle \frac{2}{\frac{1}{\lambda }{e}^{tlna}+1}}\right)}^{\alpha }.$$

Combining this identity with the generating function of the higher-order Apostol–Euler numbers yields

$${\lambda }^{\alpha }{\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right){\displaystyle \frac{t^n}{n!}}={\displaystyle \sum _{n=0}^{\infty }}{\mathcal{E}}_n^{\left(\alpha \right)}\left({\displaystyle \frac{1}{\lambda }}\right){\left(lna\right)}^n{\displaystyle \frac{t^n}{n!}}.$$

Consequently, comparing the coefficients of ${\displaystyle \frac{t^n}{n!}}$ gives the following corollary:

Corollary 1.

Let $a\in {\mathbb{R}}^{+}$ and $\lambda ,\alpha \in \mathbb{C}$ with $\lambda \ne 0$. Then

$$y_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right)={\displaystyle \frac{1}{{\lambda }^{\alpha }}}{\mathcal{E}}_n^{\left(\alpha \right)}\left({\displaystyle \frac{1}{\lambda }}\right){\left(lna\right)}^n.$$

The generalizes corresponding relation for $y_{9,n}(\lambda ;a)$ obtained in [32] to the higher-order setting.

Using (3) with $x=0$ and (16) with $\alpha =1$, we obtain the functional equation:

$$b(t,0;\lambda )={\displaystyle \frac{t}{2\lambda }}f\left(t;-{\displaystyle \frac{1}{\lambda }},1,e\right).$$

Consequently,

$${\displaystyle \frac{1}{2\lambda }}{\displaystyle \sum _{n=1}^{\infty }}n{y}_{9,n-1}\left(-{\displaystyle \frac{1}{\lambda }};e\right){\displaystyle \frac{t^n}{n!}}={\displaystyle \sum _{n=0}^{\infty }}{\mathcal{B}}_n\left(\lambda \right){\displaystyle \frac{t^n}{n!}}.$$

Comparing the coefficients of $t^n/n!$ on both sides yields the following relation between the Apostol–Bernoulli numbers and the numbers $y_{9,n-1}\left(-{\displaystyle \frac{1}{\lambda }};e\right)$:

Theorem 2.

Let $\lambda \in \mathbb{C}$ with $\lambda \ne 0$ and $n\in \mathbb{N}$. Then

$${\mathcal{B}}_n\left(\lambda \right)={\displaystyle \frac{n}{2\lambda }}{y}_{9,n-1}\left(-{\displaystyle \frac{1}{\lambda }};e\right).$$

Using (16), we also obtain

$${\left({\displaystyle \frac{2}{1+\lambda }}\right)}^{\alpha }{\left({\displaystyle \frac{1+\lambda }{e^{tlna}-(-\lambda )}}\right)}^{\alpha }={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right){\displaystyle \frac{t^n}{n!}}.$$

Hence,

$${\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right){\displaystyle \frac{t^n}{n!}}={\left({\displaystyle \frac{2}{1+\lambda }}\right)}^{\alpha }{\displaystyle \sum _{n=0}^{\infty }}{\left(lna\right)}^n{H}_n^{\left(\alpha \right)}\left(-\lambda \right){\displaystyle \frac{t^n}{n!}}.$$

By comparing the coefficients of $t^n/n!$, we arrive at the following corollary:

Corollary 2.

Let $\lambda ,\alpha \in \mathbb{C}$ with $\lambda \ne -1$ and $a>0$. We have

$$y_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right)={\left({\displaystyle \frac{2}{1+\lambda }}\right)}^{\alpha }{\left(lna\right)}^n{H}_n^{\left(\alpha \right)}\left(-\lambda \right).$$

Finally, using (16), we have

$$2^{\alpha }={\left(a^t+\lambda \right)}^{\alpha }{\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right){\displaystyle \frac{t^n}{n!}}.$$

Applying the binomial theorem under the condition $\left|{\displaystyle \frac{a^t}{\lambda }}\right|<1$, we obtain

$${\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }={\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right){\displaystyle \frac{a^{kt}}{{\lambda }^k}}{\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right){\displaystyle \frac{t^n}{n!}}.$$

Since

$$a^{kt}=e^{ktlna}={\displaystyle \sum _{j=0}^{\infty }}{\left(klna\right)}^j{\displaystyle \frac{t^j}{j!}},$$

the Cauchy product formula yields

$${\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }={\displaystyle \sum _{m=0}^{\infty }}\left[{\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right){\displaystyle \frac{1}{{\lambda }^k}}{\displaystyle \sum _{n=0}^m}\left({\displaystyle \genfrac{}{}{0pt}{}{m}{n}}\right){\left(klna\right)}^{m-n}{y}_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right)\right]{\displaystyle \frac{t^m}{m!}}.$$

Comparing coefficients of ${\displaystyle \frac{t^m}{m!}}$ on both sides of the above equation, we obtain the following computational formula for the numbers $y_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right)$:

Theorem 3.

Let $a\in {\mathbb{R}}^{+}$, $\lambda ,\alpha \in \mathbb{C}$, and $\left|{\displaystyle \frac{1}{\lambda }}\right|<1$. For $m\in \mathbb{N}$, we have

$${\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right){\displaystyle \frac{1}{{\lambda }^k}}{\displaystyle \sum _{n=0}^m}\left({\displaystyle \genfrac{}{}{0pt}{}{m}{n}}\right){\left(klna\right)}^{m-n}{y}_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right)=0.$$

For $m=0$, the constant terms satisfies

$${\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right){\displaystyle \frac{1}{{\lambda }^k}}{y}_{9,0}^{\left(\alpha \right)}\left(\lambda ;a\right)={\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }.$$

Assume now that $\left|{\displaystyle \frac{a^t}{\lambda }}\right|<1$. By the binomial theorem applied to (16), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right){\displaystyle \frac{t^n}{n!}}& =& {\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{-\alpha }{k}}\right){\left({\displaystyle \frac{a^t}{\lambda }}\right)}^k\hfill \\ & =& {\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{\displaystyle \sum _{k=0}^{\infty }}{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\left({\displaystyle \frac{a^t}{\lambda }}\right)}^k.\hfill \end{array}$$

Since

$$a^{kt}=e^{ktlna}={\displaystyle \sum _{n=0}^{\infty }}{\left(klna\right)}^n{\displaystyle \frac{t^n}{n!}},$$

it follows that

$${\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right){\displaystyle \frac{t^n}{n!}}={\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{\displaystyle \sum _{k=0}^{\infty }}{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\displaystyle \frac{1}{{\lambda }^k}}{\displaystyle \sum _{n=0}^{\infty }}{\left(klna\right)}^n{\displaystyle \frac{t^n}{n!}}.$$

Comparing the coefficients of ${\displaystyle \frac{t^n}{n!}}$ on both sides of the above equation, we obtain another explicit computational formula for the numbers $y_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right)$:

Theorem 4.

Let $a\in {\mathbb{R}}^{+}$ and $\lambda ,\alpha \in \mathbb{C}$ with $\left|{\displaystyle \frac{1}{\lambda }}\right|<1$. For $n\in \mathbb{N}$, we have

$$y_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right)={\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{\displaystyle \sum _{k=1}^{\infty }}{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\displaystyle \frac{{\left(klna\right)}^n}{{\lambda }^k}}.$$

(18)

Combining (16) and (17), we obtain

$${(1+t)}^x{\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right){\displaystyle \frac{t^n}{n!}}={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}\left(x,\lambda ;a\right){\displaystyle \frac{t^n}{n!}}.$$

Assuming that $\left|t\right|<1$ and applying the binomial theorem to to the left-hand side, we get

$${\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{j=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right){\left(x\right)}_{n-j}{y}_{9,j}^{\left(\alpha \right)}\left(\lambda ;a\right){\displaystyle \frac{t^n}{n!}}={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}\left(x,\lambda ;a\right){\displaystyle \frac{t^n}{n!}}.$$

Comparing the coefficients of ${\displaystyle \frac{t^n}{n!}}$ on both sides of the above equation, we obtain the following theorem:

Theorem 5.

Let $a\in {\mathbb{R}}^{+}$, $x\in \mathbb{R}$, $\lambda ,\alpha \in \mathbb{C}$, and $n\in {\mathbb{N}}_0$. We have

$$y_{9,n}^{\left(\alpha \right)}\left(x,\lambda ;a\right)={\displaystyle \sum _{j=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right){\left(x\right)}_{n-j}{y}_{9,j}^{\left(\alpha \right)}\left(\lambda ;a\right).$$

(19)

Remark 1.

Equation (19) extends Theorem 1, which corresponds to the case $\alpha =1$. Therefore, Theorem 5 provides a higher-order generalization of the polynomial identity established in [32].

3. Interpolation Function for the Numbers $y_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right)$

In this section, using (18), we introduce an interpolation function for the numbers $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$ by

$$Z_y\left(s,\lambda ;\alpha ,a\right)={\displaystyle \sum _{k=1}^{\infty }}{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\displaystyle \frac{1}{{\lambda }^k{\left(klna\right)}^s}}.$$

(20)

Here, $a\in {\mathbb{R}}^{+}$ with $a\ne 1$, and $\lambda ,\alpha \in \mathbb{C}$ satisfy $\left|{\lambda }^{-1}\right|<1$.

3.1. Residue-Class Decomposition of the Interpolation Function

We now derive a residue-class decomposition of

$$Z_y\left(s,\lambda ;\alpha ,a\right)={\displaystyle \sum _{k=1}^{\infty }}{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\displaystyle \frac{1}{{\lambda }^k{\left(klna\right)}^s}},$$

where $a\in {\mathbb{R}}^{+}$ with $a\ne 1$ and $\lambda ,\alpha ,s\in \mathbb{C}$ with $\left|{\displaystyle \frac{1}{\lambda }}\right|<1$.

Let $d\in \mathbb{N}$. Every positive integer k can be written uniquely as

$$k=dn+r,n\in {\mathbb{N}}_0,1\le r\le d.$$

Hence, we may rewrite the series as

$$\begin{array}{cc}\hfill Z_y\left(s,\lambda ;\alpha ,a\right)& ={\displaystyle \sum _{r=1}^d}{\displaystyle \sum _{n=0}^{\infty }}{(-1)}^{dn+r}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +dn+r-1}{dn+r}}\right)\hfill \\ \hfill & \times {\displaystyle \frac{1}{{\lambda }^{dn+r}{\left((dn+r)lna\right)}^s}}.\hfill \end{array}$$

Since

$$dn+r=d\left(n+{\displaystyle \frac{r}{d}}\right),$$

it follows that

$${\left((dn+r)lna\right)}^s={\left(dlna\right)}^s{\left(n+{\displaystyle \frac{r}{d}}\right)}^s,$$

Therefore,

$$\begin{array}{cc}\hfill Z_y\left(s,\lambda ;\alpha ,a\right)& ={\displaystyle \frac{1}{{\left(dlna\right)}^s}}{\displaystyle \sum _{r=1}^d}{\displaystyle \sum _{n=0}^{\infty }}{(-1)}^{dn+r}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +dn+r-1}{dn+r}}\right)\hfill \\ \hfill & \times {\displaystyle \frac{1}{{\lambda }^{dn+r}{\left(n+\frac{r}{d}\right)}^s}}.\hfill \end{array}$$

Using the Pochhammer representation

$$\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +m-1}{m}}\right)={\displaystyle \frac{{\left(\alpha \right)}_m}{m!}},$$

with $m=dn+r$, we obtain

$$\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +dn+r-1}{dn+r}}\right)={\displaystyle \frac{{\left(\alpha \right)}_{dn+r}}{(dn+r)!}}.$$

Moreover,

$${\left(\alpha \right)}_{dn+r}={\left(\alpha \right)}_r{(\alpha +r)}_{dn},(dn+r)!=r!{(r+1)}_{dn},$$

Thus,

$$\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +dn+r-1}{dn+r}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right){\displaystyle \frac{{(\alpha +r)}_{dn}}{{(r+1)}_{dn}}}.$$

Substituting this into the preceding identity gives

$$\begin{array}{cc}\hfill Z_y\left(s,\lambda ;\alpha ,a\right)& ={\displaystyle \frac{1}{{\left(dlna\right)}^s}}{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right)\hfill \\ \hfill & \times {\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{(\alpha +r)}_{dn}}{{(r+1)}_{dn}}}{\displaystyle \frac{{(-1)}^{dn}}{{\lambda }^{dn}}}{\displaystyle \frac{1}{{\left(n+\frac{r}{d}\right)}^s}}.\hfill \end{array}$$

Finally, by the multiplication formula for the Pochhammer symbol,

$${\left(b\right)}_{dn}=d^{dn}{\displaystyle \prod _{j=0}^{d-1}}{\left({\displaystyle \frac{b+j}{d}}\right)}_n,$$

we have

$${(\alpha +r)}_{dn}=d^{dn}{\displaystyle \prod _{j=0}^{d-1}}{\left({\displaystyle \frac{\alpha +r+j}{d}}\right)}_n$$

and

$${(r+1)}_{dn}=d^{dn}{\displaystyle \prod _{j=0}^{d-1}}{\left({\displaystyle \frac{r+1+j}{d}}\right)}_n.$$

Hence,

$${\displaystyle \frac{{(\alpha +r)}_{dn}}{{(r+1)}_{dn}}}={\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +r+j}{d}\right)}_n}{{\left(\frac{r+1+j}{d}\right)}_n}}.$$

Consequently, we obtain the following theorem.

Theorem 6.

Let $d\in \mathbb{N}$, $a\in {\mathbb{R}}^{+}$ with $a\ne 1$, and $\lambda ,\alpha ,s\in \mathbb{C}$ with $\left|{\displaystyle \frac{1}{\lambda }}\right|<1$. Then the interpolation function $Z_y\left(s,\lambda ;\alpha ,a\right)$ admits the decomposition

$$\begin{array}{cc}\hfill Z_y\left(s,\lambda ;\alpha ,a\right)& ={\displaystyle \frac{1}{{\left(dlna\right)}^s}}{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right)\hfill \\ \hfill & \times {\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +r+j}{d}\right)}_n}{{\left(\frac{r+1+j}{d}\right)}_n}}\right]\hfill \\ \hfill & \times {\displaystyle \frac{{\left(\frac{{(-1)}^d}{{\lambda }^d}\right)}^n}{{\left(n+\frac{r}{d}\right)}^s}}.\hfill \end{array}$$

This representation shows that $Z_y(s,\lambda ;\alpha ,a)$ can be written as a weighted Hurwitz–Lerch-type series with hypergeometric coefficients.

3.2. A generalized Hypergeometric Hurwitz–Lerch-Type Zeta function

Motivated by the residue-class decomposition obtained in the preceding subsection, we introduce the following family of zeta-type functions.

Definition 1.

Let $d\in \mathbb{N}$, $\alpha ,\beta ,s,z\in \mathbb{C}$, and $x\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$. Assume that

$${\displaystyle \frac{\beta +j}{d}}\notin {\mathbb{Z}}_0^{-},j=0,1,\dots ,d-1,$$

and let $\left|z\right|<1$. We define the generalized hypergeometric Hurwitz–Lerch-type zeta function by

$${\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}_n}{{\left(\frac{\beta +j}{d}\right)}_n}}\right]{\displaystyle \frac{z^n}{{(n+x)}^s}}.$$

(21)

Proposition 1.

Under the assumptions of Definition 1, the series defining ${\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)$ converges absolutely for $\left|z\right|<1$.

Proof. 

Let

$$u_n=\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}_n}{{\left(\frac{\beta +j}{d}\right)}_n}}\right]{\displaystyle \frac{z^n}{{(n+x)}^s}}.$$

Hence, by the ratio test, the series converges absolutely for $\left|z\right|<1$. □

Remark 2.

The preceding proposition justifies the condition $\left|z\right|<1$ imposed in Definition 1. The convergence behavior on the boundary $\left|z\right|=1$ depends on the parameters α, β, and s, and requires a separate analysis.

Corollary 3.

Let $d\in \mathbb{N}$, $a\in {\mathbb{R}}^{+}$ with $a\ne 1$, and $\lambda ,\alpha ,s\in \mathbb{C}$ with

$$\left|{\displaystyle \frac{1}{\lambda }}\right|<1.$$

Then

$$\begin{array}{cc}\hfill Z_y(s,\lambda ;\alpha ,a)& ={\displaystyle \frac{1}{{(dlna)}^s}}{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right)\hfill \\ \hfill & \times {\mathcal{Z}}_d^{(\alpha +r,r+1)}\left(s,{\displaystyle \frac{r}{d}};{\displaystyle \frac{{(-1)}^d}{{\lambda }^d}}\right).\hfill \end{array}$$

(22)

Proof. 

From the residue-class decomposition theorem, we have

$$\begin{array}{cc}\hfill Z_y(s,\lambda ;\alpha ,a)& ={\displaystyle \frac{1}{{(dlna)}^s}}{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right)\hfill \\ \hfill & \times {\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +r+j}{d}\right)}_n}{{\left(\frac{r+1+j}{d}\right)}_n}}\right]\hfill \\ \hfill & \times {\displaystyle \frac{{\left(\frac{{(-1)}^d}{{\lambda }^d}\right)}^n}{{\left(n+\frac{r}{d}\right)}^s}}.\hfill \end{array}$$

By the definition of ${\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)$, the inner series is

$${\mathcal{Z}}_d^{(\alpha +r,r+1)}\left(s,{\displaystyle \frac{r}{d}};{\displaystyle \frac{{(-1)}^d}{{\lambda }^d}}\right).$$

Substituting this expression into the preceding identity yields (22). □

Remark 3.

For $d=1$, we obtain

$${\mathcal{Z}}_1^{(\alpha ,\beta )}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{\left(\alpha \right)}_n}{{\left(\beta \right)}_n}}{\displaystyle \frac{z^n}{{(n+x)}^s}}.$$

In particular, when $\beta =1$, we have

$${\mathcal{Z}}_1^{(\alpha ,1)}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{\left(\alpha \right)}_n}{n!}}{\displaystyle \frac{z^n}{{(n+x)}^s}},$$

which is a hypergeometric Hurwitz–Lerch-type series.

Remark 4.

Setting $\alpha =\beta$, we obtain

$${\mathcal{Z}}_d^{(\alpha ,\alpha )}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}_n}{{\left(\frac{\alpha +j}{d}\right)}_n}}\right]{\displaystyle \frac{z^n}{{(n+x)}^s}}.$$

Therefore,

$${\mathcal{Z}}_d^{(\alpha ,\alpha )}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{z^n}{{(n+x)}^s}}=\Phi (z,s,x),$$

where $\Phi (z,s,x)$ denotes the classical Lerch transcendent.

Remark 5.

Let $m\in {\mathbb{N}}_0$. Setting $s=-m$ in the definition of ${\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)$, we obtain

$${\mathcal{Z}}_d^{(\alpha ,\beta )}(-m,x;z)={\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}_n}{{\left(\frac{\beta +j}{d}\right)}_n}}\right]z^n{(n+x)}^m.$$

Setting $\alpha =1$ and $a=e$ in (20), we obtain

$$Z_y\left(s,\lambda ;1,e\right)={\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{{(-1)}^k}{{\lambda }^k{k}^s}}=-{\displaystyle \frac{1}{\lambda }}\Phi \left(-{\displaystyle \frac{1}{\lambda }},s,1\right),$$

where $\Phi \left(z,s,x\right)$ denotes the Lerch transcendent defined by

$$\Phi \left(z,s,x\right)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{z^n}{{(n+x)}^s}}$$

(cf. [5,33]). Substituting $s=-n$, where $n\in \mathbb{N}$, into (20), we obtain the following theorem:

Theorem 7.

Let $n\in \mathbb{N}$. Then

$$Z_y\left(-n,\lambda ;\alpha ,a\right)={\left({\displaystyle \frac{\lambda }{2}}\right)}^{\alpha }{y}_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right).$$

Remark 6.

Theorem 7 extends the interpolation property of $y_{9,n}(\lambda ;a)$ given in [32] from the first-order case to arbitrary complex order α.

Theorem 8.

Let $m\in \mathbb{N}$. Then

$$\begin{array}{cc}\hfill Z_y(-m,\lambda ;\alpha ,a)& ={(dlna)}^m{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right)\hfill \\ \hfill & \times {\mathcal{Z}}_d^{(\alpha +r,r+1)}\left(-m,{\displaystyle \frac{r}{d}};{\displaystyle \frac{{(-1)}^d}{{\lambda }^d}}\right).\hfill \end{array}$$

(23)

Consequently,

$$\begin{array}{cc}\hfill y_{9,m}^{\left(\alpha \right)}(\lambda ;a)& ={\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{(dlna)}^m{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right)\hfill \\ \hfill & \times {\mathcal{Z}}_d^{(\alpha +r,r+1)}\left(-m,{\displaystyle \frac{r}{d}};{\displaystyle \frac{{(-1)}^d}{{\lambda }^d}}\right).\hfill \end{array}$$

(24)

Proof. 

From the residue-class decomposition, we have

$$\begin{array}{cc}\hfill Z_y(s,\lambda ;\alpha ,a)& ={\displaystyle \frac{1}{{(dlna)}^s}}{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right)\hfill \\ \hfill & \times {\mathcal{Z}}_d^{(\alpha +r,r+1)}\left(s,{\displaystyle \frac{r}{d}};{\displaystyle \frac{{(-1)}^d}{{\lambda }^d}}\right).\hfill \end{array}$$

Replacing s by $-m$, where $m\in \mathbb{N}$, yields

$$\begin{array}{cc}\hfill Z_y(-m,\lambda ;\alpha ,a)& ={(dlna)}^m{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right)\hfill \\ \hfill & \times {\mathcal{Z}}_d^{(\alpha +r,r+1)}\left(-m,{\displaystyle \frac{r}{d}};{\displaystyle \frac{{(-1)}^d}{{\lambda }^d}}\right).\hfill \end{array}$$

On the other hand, from the interpolation property of $Z_y(s,\lambda ;\alpha ,a)$, we have

$$Z_y(-m,\lambda ;\alpha ,a)={\left({\displaystyle \frac{\lambda }{2}}\right)}^{\alpha }{y}_{9,m}^{\left(\alpha \right)}(\lambda ;a).$$

Equating the two representations of $Z_y(-m,\lambda ;\alpha ,a)$, we obtain

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{\lambda }{2}}\right)}^{\alpha }{y}_{9,m}^{\left(\alpha \right)}(\lambda ;a)& ={(dlna)}^m{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right)\hfill \\ \hfill & \times {\mathcal{Z}}_d^{(\alpha +r,r+1)}\left(-m,{\displaystyle \frac{r}{d}};{\displaystyle \frac{{(-1)}^d}{{\lambda }^d}}\right).\hfill \end{array}$$

Multiplying both sides by ${\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }$ gives (24). □

Remark 7.

For $d=1$, Theorem (8) reduces to

$$Z_y(-m,\lambda ;\alpha ,a)={(lna)}^m{\displaystyle \frac{(-1)}{\lambda }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{1}}\right)Z_1^{(\alpha +1,2)}\left(-m,1;-{\displaystyle \frac{1}{\lambda }}\right).$$

Combining this relation with the interpolation identity

$$Z_y(-m,\lambda ;\alpha ,a)={\left({\displaystyle \frac{\lambda }{2}}\right)}^{\alpha }{y}_{9,m}^{\left(\alpha \right)}(\lambda ;a),$$

we obtain the corresponding $d=1$ representation of the higher-order numbers. Hence Theorem 8 generalizes the interpolation formula from the single-class case $d=1$ to an arbitrary residue-class decomposition.

Theorem 9.

Let $d\in \mathbb{N}$, $\alpha ,\beta ,s,z\in \mathbb{C}$, and $x\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$. For $\left|z\right|<1$, the generalized hypergeometric Hurwitz–Lerch-type zeta function ${\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)$ satisfies the differential relation

$${\displaystyle \frac{\partial }{\partial x}}{\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)=-s{\mathcal{Z}}_d^{(\alpha ,\beta )}(s+1,x;z).$$

(25)

Proof. 

From the definition, we have

$${\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}_n}{{\left(\frac{\beta +j}{d}\right)}_n}}\right]{\displaystyle \frac{z^n}{{(n+x)}^s}}.$$

Differentiating term-by-term with respect to x, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial x}}{\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)& ={\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}_n}{{\left(\frac{\beta +j}{d}\right)}_n}}\right]z^n{\displaystyle \frac{\partial }{\partial x}}{(n+x)}^{-s}.\hfill \end{array}$$

Using

$${\displaystyle \frac{\partial }{\partial x}}{(n+x)}^{-s}=-s{(n+x)}^{-s-1},$$

we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial x}}{\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)& =-s{\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}_n}{{\left(\frac{\beta +j}{d}\right)}_n}}\right]{\displaystyle \frac{z^n}{{(n+x)}^{s+1}}}.\hfill \end{array}$$

Recognizing the resulting series, we arrive at

$${\displaystyle \frac{\partial }{\partial x}}{\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)=-s{\mathcal{Z}}_d^{(\alpha ,\beta )}(s+1,x;z).$$

This completes the proof. □

3.3. A Functional-Differential Relation

Let

$$\vartheta =z{\displaystyle \frac{\partial }{\partial z}}$$

denote the Euler differential operator.

Theorem 10.

Let $d\in \mathbb{N}$, $\alpha ,\beta ,s,z\in \mathbb{C}$, and $x\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$. Assume that $\left|z\right|<1$ and that the denominator parameters do not produce poles. Then

$${\displaystyle \prod _{j=0}^{d-1}}\left(\vartheta +{\displaystyle \frac{\beta +j}{d}}-1\right)Z_d^{(\alpha ,\beta )}(s,x;z)$$

$$=\left[{\displaystyle \prod _{j=0}^{d-1}}\left({\displaystyle \frac{\beta +j}{d}}-1\right)\right]{\displaystyle \frac{1}{x^s}}+z{\displaystyle \prod _{j=0}^{d-1}}\left(\vartheta +{\displaystyle \frac{\alpha +j}{d}}\right)Z_d^{(\alpha ,\beta )}(s,x+1;z).$$

Proof. 

Set

$$A_n={\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}_n}{{\left(\frac{\beta +j}{d}\right)}_n}}.$$

Then

$$Z_d^{(\alpha ,\beta )}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}{A}_n{\displaystyle \frac{z^n}{{(n+x)}^s}}.$$

Since

$$\vartheta z^n=n{z}^n,$$

we obtain

$${\displaystyle \prod _{j=0}^{d-1}}\left(\vartheta +{\displaystyle \frac{\beta +j}{d}}-1\right)Z_d^{(\alpha ,\beta )}(s,x;z)$$

$$={\displaystyle \sum _{n=0}^{\infty }}{A}_n{\displaystyle \prod _{j=0}^{d-1}}\left(n+{\displaystyle \frac{\beta +j}{d}}-1\right){\displaystyle \frac{z^n}{{(n+x)}^s}}.$$

For $n\ge 1$, the quotient of consecutive coefficients satisfies

$$A_n=A_{n-1}{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{n-1+\frac{\alpha +j}{d}}{n-1+\frac{\beta +j}{d}}}.$$

Consequently,

$$A_n{\displaystyle \prod _{j=0}^{d-1}}\left(n+{\displaystyle \frac{\beta +j}{d}}-1\right)=A_{n-1}{\displaystyle \prod _{j=0}^{d-1}}\left(n-1+{\displaystyle \frac{\alpha +j}{d}}\right).$$

Separating the term $n=0$ and shifting the index in the remaining series, we obtain

$${\displaystyle \prod _{j=0}^{d-1}}\left(\vartheta +{\displaystyle \frac{\beta +j}{d}}-1\right)Z_d^{(\alpha ,\beta )}(s,x;z)$$

$$=\left[{\displaystyle \prod _{j=0}^{d-1}}\left({\displaystyle \frac{\beta +j}{d}}-1\right)\right]{\displaystyle \frac{1}{x^s}}+z{\displaystyle \sum _{n=0}^{\infty }}{A}_n{\displaystyle \prod _{j=0}^{d-1}}\left(n+{\displaystyle \frac{\alpha +j}{d}}\right){\displaystyle \frac{z^n}{{(n+x+1)}^s}}.$$

Recognizing the final series in terms of the Euler operator $\vartheta$, we arrive at

$${\displaystyle \prod _{j=0}^{d-1}}\left(\vartheta +{\displaystyle \frac{\beta +j}{d}}-1\right)Z_d^{(\alpha ,\beta )}(s,x;z)$$

$$=\left[{\displaystyle \prod _{j=0}^{d-1}}\left({\displaystyle \frac{\beta +j}{d}}-1\right)\right]{\displaystyle \frac{1}{x^s}}+z{\displaystyle \prod _{j=0}^{d-1}}\left(\vartheta +{\displaystyle \frac{\alpha +j}{d}}\right)Z_d^{(\alpha ,\beta )}(s,x+1;z).$$

This completes the proof. □

Corollary 4.

When $\alpha =\beta$, one has

$$Z_d^{(\alpha ,\alpha )}(s,x;z)=\Phi (z,s,x).$$

In this case, the corresponding classical shift identity is

$$\Phi (z,s,x)={\displaystyle \frac{1}{x^s}}+z\Phi (z,s,x+1).$$

Proof. 

When $\alpha =\beta$, the Pochhammer factors cancel and hence

$$Z_d^{(\alpha ,\alpha )}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{z^n}{{(n+x)}^s}}=\Phi (z,s,x).$$

Separating the $n=0$ term and shifting the index gives

$$\Phi (z,s,x)={\displaystyle \frac{1}{x^s}}+z\Phi (z,s,x+1).$$

□

3.4. An Integral Representation

Theorem 11.

Let $d\in \mathbb{N}$, $\alpha ,\beta ,z\in \mathbb{C}$, $\Re \left(s\right)>0$, $\Re \left(x\right)>0$, and $\left|z\right|<1$. Assume that

$${\displaystyle \frac{\beta +j}{d}}\notin {\mathbb{Z}}_0^{-},j=0,1,\dots ,d-1.$$

Then

$$Z_d^{(\alpha ,\beta )}(s,x;z)={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }{t}^{s-1}{e}^{-xt}{}_{d+1}\mathbf{F}_{\mathrm{d}}\left(\begin{array}{c}1,{\displaystyle \frac{\alpha }{d}},{\displaystyle \frac{\alpha +1}{d}},\dots ,{\displaystyle \frac{\alpha +d-1}{d}}\\ {\displaystyle \frac{\beta }{d}},{\displaystyle \frac{\beta +1}{d}},\dots ,{\displaystyle \frac{\beta +d-1}{d}}\end{array};z{e}^{-t}\right)dt.$$

Proof. 

Using the Gamma-integral identity

$${\displaystyle \frac{1}{{(n+x)}^s}}={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }{t}^{s-1}{e}^{-(n+x)t}dt,\Re \left(s\right)>0,\Re \left(x\right)>0,$$

in the defining series of $Z_d^{(\alpha ,\beta )}(s,x;z)$, we obtain

$$Z_d^{(\alpha ,\beta )}(s,x;z)={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}_n}{{\left(\frac{\beta +j}{d}\right)}_n}}\right]z^n{\int }_0^{\infty }{t}^{s-1}{e}^{-(n+x)t}dt.$$

Since $\left|z\right|<1$, the sum and the integral may be interchanged under the stated conditions. Hence

$$Z_d^{(\alpha ,\beta )}(s,x;z)={\displaystyle \frac{1}{\Gamma \left(s\right)}}{\int }_0^{\infty }{t}^{s-1}{e}^{-xt}{\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}_n}{{\left(\frac{\beta +j}{d}\right)}_n}}\right]{\left(z{e}^{-t}\right)}^{n}dt.$$

Using

$${\displaystyle \frac{{\left(1\right)}_n}{n!}}=1,$$

the inner series is recognized as

$${}_{d+1}\mathbf{F}_{\mathrm{d}}\left(\begin{array}{c}1,{\displaystyle \frac{\alpha }{d}},{\displaystyle \frac{\alpha +1}{d}},\dots ,{\displaystyle \frac{\alpha +d-1}{d}}\\ {\displaystyle \frac{\beta }{d}},{\displaystyle \frac{\beta +1}{d}},\dots ,{\displaystyle \frac{\beta +d-1}{d}}\end{array};z{e}^{-t}\right).$$

This proves the result. □

3.5. Numerical Illustrations

In this subsection, we present two simple examples illustrating the computational formulas and interpolation identities established in the previous sections.

Example 1.

Consider the parameter values

$$a=e,\lambda =2,\alpha =2.$$

Then the generating function in (16) becomes

$${\left({\displaystyle \frac{2}{e^t+2}}\right)}^2={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(2\right)}(2;e){\displaystyle \frac{t^n}{n!}}.$$

The first few values are given in Table 1.

These values illustrate how the generating function and the explicit formula in Theorem 4 can be used for the effective computation of the higher-order numbers.

Example 2.

We now consider a special case related to the Fubini numbers. Setting

$$\alpha =1,\lambda =-2,a=e$$

in Theorem 7, we obtain

$$Z_y(-n,-2;1,e)=-y_{9,n}(-2;e).$$

Using the relation

$$y_{9,n}(-2;e)=-2w_g\left(n\right),$$

Table 1. The first values of $y_{9,n}^{\left(2\right)}(2;e)$.

n	$y_{9,n}^{\left(2\right)}(2;e)$
0	${\displaystyle \frac{4}{9}}$
1	$-{\displaystyle \frac{8}{27}}$
2	0
3	${\displaystyle \frac{16}{81}}$
4	${\displaystyle \frac{16}{243}}$
5	$-{\displaystyle \frac{112}{243}}$

Table 1. The first values of $y_{9,n}^{\left(2\right)}(2;e)$.

n	$y_{9,n}^{\left(2\right)}(2;e)$
0	${\displaystyle \frac{4}{9}}$
1	$-{\displaystyle \frac{8}{27}}$
2	0
3	${\displaystyle \frac{16}{81}}$
4	${\displaystyle \frac{16}{243}}$
5	$-{\displaystyle \frac{112}{243}}$

we arrive at

$$Z_y(-n,-2;1,e)=2w_g\left(n\right),n\in \mathbb{N}.$$

Thus, up to the normalization factor 2, the interpolation function recovers the Fubini numbers at negative integers. Some numerical values are presented in Table 2.

Table 2 confirms that the values of the interpolation function at negative integers agree with twice the corresponding Fubini numbers, as predicted by Theorem 7.

4. Conclusions

In this paper, we introduced higher-order combinatorial numbers and polynomials associated with the family $y_{9,n}(\lambda ;a)$ via their generating functions. Within this framework, we derived several explicit formulas, identities, and recurrence-type relations, and we established new connections with Apostol–Euler, Apostol–Bernoulli, Euler–Frobenius, Fubini, and Stirling numbers.

A central contribution of this work is the construction of an interpolation function for the higher-order numbers $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$. We showed that its values at negative integers recover these numbers, up to an explicit normalization factor, and we obtained a residue-class decomposition for this interpolation function. This decomposition yields a new representation in terms of a generalized hypergeometric Hurwitz–Lerch-type zeta function.

Another main contribution is the introduction of this generalized hypergeometric Hurwitz–Lerch-type zeta function, which extends the classical Lerch transcendent and provides a unified analytic framework linking generating functions, interpolation theory, and higher-order combinatorial numbers. We established several of its fundamental structural and analytic properties, including its relation with the classical Lerch transcendent, a differential identity, a functional-differential relation, and a Mellin-type integral representation.

The results obtained here reveal new and meaningful connections among generating functions, higher-order combinatorial numbers, interpolation functions, and zeta-type functions. They also open several directions for future research. In particular, it would be of interest to investigate the analytic continuation, singularities, functional equations, special values, and asymptotic behavior of the newly introduced generalized hypergeometric Hurwitz–Lerch-type zeta function.

From the standpoint of analytic number theory, one may ask whether this function leads to new series representations, summation formulas, or evaluations involving classical zeta functions, polylogarithms, and Apostol-type numbers. In complex analysis, its analytic continuation, singularity structure, zero distribution, and possible associated integral and differential operators deserve further study. From a combinatorial perspective, the explicit formulas derived in this paper may serve to develop finite sums, convolution identities, recurrence relations, and potential enumerative interpretations for the associated higher-order numbers and polynomials. Finally, the construction of efficient numerical algorithms for evaluating the new zeta-type function and analyzing its dependence on the parameters also appears to be a natural and promising direction for future work.