Derivatives, Integrals and Polynomials Arising from the Inhomogeneous Airy’s Equation

1. Introduction

Airy’s differential equation and its associated Airy’s functions date back to the nineteenth century, [1]. They are as relevant today as they were then due to their many applications in mathematical physics, circuit theory, systems theory, signal processing, electromagnetism and in fluid dynamic modeling (cf. [2,3,4] and the references therein). A large number of differential equations in quantum theory can be reduced to Airy’s equation by an appropriate change of variables, thus adding to the importance and relevance of studies of Airy’s functions and other arising and related special functions, [3,4].

Although Airy’s equation has been largely studied in its homogeneous form, we consider it here in its more general, inhomogeneous form, written as, [5]:

$${u\text{'}\text{'}}_i-x{u}_i=f\left(x\right)$$

(1)

wherein the forcing function $f\left(x\right)$ is a continuous function of the non-negative, real variable $x$.

In equation (1), and throughout this work, prime notation denotes ordinary differentiation with respect to the independent variable.

When $f\left(x\right)\equiv 0$ general solution to the homogeneous Airy’s equation (1) can be expressed in the form:

$$u_i=a_1{A}_i\left(x\right)+b_1{B}_i\left(x\right)$$

(2)

where $a_1 \mathrm{a}\mathrm{n}\mathrm{d} b_1$ are arbitrary constants, and $A_i\left(x\right)$ and $B_i\left(x\right)$ are Airy’s functions of the first- and second-kind, respectively, defined by:

$$A_i\left(x\right)={\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }\cos \left(xt+{\displaystyle \frac{t^3}{3}}\right)dt$$

(3)

$$B_i\left(x\right)={\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }\sin \left(xt+{\displaystyle \frac{t^3}{3}}\right)dt+{\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }\exp \left(xt-{\displaystyle \frac{t^3}{3}}\right)dt$$

(4)

The Wronskian of $A_i\left(x\right)$ and $B_i\left(x\right)$ is given by, [2,4]:

$$W\left(A_i\left(x\right),B_i\left(x\right)\right)={\displaystyle \frac{1}{\pi }}$$

(5)

When $f\left(x\right)=\mp {\displaystyle \frac{1}{\pi }}$ , Scorer [6] obtained the following general solutions using variation of parameters. For $f\left(x\right)=-{\displaystyle \frac{1}{\pi }}$, general solution to equation (1) is given by

$$u_i=a_1{A}_i\left(x\right)+b_1{B}_i\left(x\right)+G_i\left(x\right)$$

(6)

and when $f\left(x\right)={\displaystyle \frac{1}{\pi }}$, general solution to equation (1) is given by

$$u_i=a_1{A}_i\left(x\right)+b_1{B}_i\left(x\right)+H_i\left(x\right)$$

(7)

The functions $G_i\left(x\right)$ and $H_i\left(x\right)$ are known as Scorer’s functions, [4], or the inhomogeneous Airy’s functions, and arise in Raman scattering in chemical physics [7,8]. The Scorer functions are defined by:

$$G_i\left(x\right)={\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }\sin \left(xt+{\displaystyle \frac{1}{3}}{t}^3\right)dt$$

(8)

$$H_i\left(x\right)={\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }\exp \left(xt-{\displaystyle \frac{1}{3}}{t}^3\right)dt$$

(9)

and are related to Airy’s functions by

$$G_i\left(x\right)=A_i\left(x\right){\int }_0^x{B}_i\left(t\right)dt+B_i\left(x\right){\int }_x^{\infty }{A}_i\left(t\right)dt$$

(10)

$$H_i\left(x\right)=B_i\left(x\right){\int }_{-\infty }^x{A}_i\left(t\right)dt-A_i\left(x\right){\int }_{-\infty }^x{B}_i\left(t\right)dt$$

(11)

$$G_i\left(x\right)+H_i\left(x\right)=B_i\left(x\right)$$

(12)

In a generalization to the above, when $f\left(x\right)=\kappa$, where $\kappa$ is any constant, general solution to equation (1) has been obtained in the following form, [9]:

$$u_i=a_1{A}_i\left(x\right)+b_1{B}_i\left(x\right)-{\kappa \pi N}_i\left(x\right)$$

(13)

where

$$N_i\left(x\right)=A_i\left(x\right){\int }_0^x{B}_i\left(t\right)dt-B_i\left(x\right){\int }_0^x{A}_i\left(t\right)dt$$

(14)

The function $N_i\left(x\right)$ was introduced by Nield and Kuznetsov, [10], in their analysis of flow in the transition layer where the governing Brinkman’s equation was reduced to the inhomogeneous Airy’s equation by a special choice of the permeability function. The function $N_i\left(x\right)$ is referred to as the standard Nield-Kuznetsov function of the first-kind, and its main properties were studied by Hamdan and Kamel [9].

In order to offer modelling flexibility in the study of flow through the transition layer, Abu Zaytoon et.al. [11] introduced a permeability model that reduced Brinkman’s equation to the generalized inhomogeneous Airy’s equation of index $n$, which takes the form

$${u\text{'}\text{'}}_n-x^n{u}_n=f\left(x\right)$$

(15)

The homogeneous part of (15), that is when $f\left(x\right)\equiv 0$, was studied by Swanson and Headley [12], who expressed its general solution as

$$u_n=a_n{A}_n\left(x\right)+b_n{B}_n\left(x\right)$$

(16)

wherein $a_n$ and $b_n$ are arbitrary constants, and the functions $A_n\left(x\right)$ and $B_n\left(x\right)$ are the generalized Airy’s functions of the first- and second-kind, respectively, defined by:

$$A_n\left(x\right)=p {\left(x\right)}^{1/2} [I_{- p}\left(\zeta \right)-I_p\left(\zeta \right)]$$

(17)

$$B_n\left(x\right)={\left(px\right)}^{1/2} [I_p\left(\zeta \right)+I_{-p}\left(\zeta \right)]$$

(18)

The Wronskian of $A_n\left(x\right)$ and $B_n\left(x\right)$ is given by:

$$W\left(A_n\left(x\right),B_n\left(x\right)\right)={\displaystyle \frac{2}{\pi }}{p}^{{\displaystyle \frac{1}{2}}}\sin \left(p\pi \right)$$

(19)

and the function $I_p$ is the modified Bessel function defined as:

$$I_p\left(\zeta \right)=\sum _{m=1}^{\infty }{\displaystyle \frac{1}{m!\Gamma (m+p+1)}}({\displaystyle \frac{\zeta }{2}}{)}^{2m+p}$$

(20)

with $p={\displaystyle \frac{1}{n+2}}$, $\zeta =2p{\left(x\right)}^{{\displaystyle \frac{1}{2p}}}$, and $\Gamma (.)$ being the gamma function. It should be noted that when the index $n=1$ in equations (15)-(19), we recover Airy’s equation and its solutions (although we use subscript $"i"$ instead of 1 for consistency with notation in the literature).

When $f\left(x\right)=\kappa$, Abu Zaytoon et.al. [11] obtained and expressed the general solution to the inhomogeneous generalized Airy’s equation (15) as

$$u_n=a_n{A}_n\left(x\right)+b_n{B}_n\left(x\right)-{\displaystyle \frac{\kappa \pi }{2\sqrt{p}\mathrm{s}\mathrm{i}\mathrm{n}\left(p\pi \right)}}{N}_n\left(x\right)$$

(21)

where the function $N_n\left(x\right)$ is the generalized Nield-Kuznetsov function of the first-kind, defined by:

$$N_n\left(x\right)=A_n\left(x\right){\int }_0^x{B}_n\left(t\right)dt-B_n\left(x\right){\int }_0^x{A}_n\left(t\right)dt$$

(22)

Analysis of the Nield-Kuznetsov functions were discussed and documented, [9,10,13], and include solution methodologies and methods of computations to the inhomogeneous Airy’s equation, inhomogeneous generalized Airy’s equation, and inhomogeneous Weber’s equations, with initial and boundary conditions. Recent work in this field also includes the elegant work of Dunster, [14], on the Nield-Kuznetsov functions and the use of Laplace transform and uniform asymptotic expansions, and the analysis of Airy’s polynomials that arise when higher derivatives are involved, and has been carried out in the elaborate work of Abramochkin and Razueva, [15]. The same Airy’s polynomials, and other polynomials, have been shown to arise in the higher derivatives of the standard Nield-Kuznetsov function of the first-kind, and are important from both a theoretical and a practical point of view, as discussed by Hamdan et.al. [16].

The above discussion of the importance of the inhomogeneous Airy’s and generalized Airy’s equations motivates the current work in which we derive further properties of the Nield-Kuznetsov functions. In particular, we initiate discussion, derivation and analysis of the generalized Scorer functions and study their properties, and their relationships to the Nield-Kuznetsov functions and to the modified Bessel functions. We also define and analyze properties of the complementary Nield-Kuznetsov functions in the sense defined by Dunster [14]. Higher derivatives of the generalized Airy’s, Nield-Kuznetsov, and Scorer functions, are then discussed and introduced together with their associated generalized polynomials. These generalized functions and polynomials might find their way in analysis of the Stark equation, [17], Schrodinger equation and Tricomi’s inhomogeneous equation.

2. Airy’s Inhomogeneous Equation

Solutions to Airy’s equation (1) are streamlined into the following cases.

Case 1: When $f\left(x\right)\equiv 0$, general solution to Airy’s homogeneous equation is given by equation (2) as a linear combination of Airy’s functions of the first- and second-kind, $A_i\left(x\right)$ and $B_i\left(x\right)$, respectively, that are defined by (3) and (4), and whose nonzero Wronskian is given by (5).

Airy’s functions $A_i\left(x\right)$ and $B_i\left(x\right)$ take the following forms in terms of the modified Bessel functions, [2,4], obtained from (17) and (18) with $n=1$, $p={\displaystyle \frac{1}{n+2}}={\displaystyle \frac{1}{3}}$, and $\zeta =2p{\left(x\right)}^{{\displaystyle \frac{1}{2p}}}={\displaystyle \frac{2}{3}}{x}^{{\displaystyle \frac{3}{2}}}:$

$$A_i\left(x\right)={\displaystyle \frac{\sqrt{x}}{3}}[I_{-{\displaystyle \frac{1}{3}}}\left(\zeta \right)-I_{{\displaystyle \frac{1}{3}}}\left(\zeta \right)]$$

(23)

$$B_i\left(x\right)=\sqrt{{\displaystyle \frac{x}{3}}} [I_{{\displaystyle \frac{1}{3}}}\left(\zeta \right)+I_{-{\displaystyle \frac{1}{3}}}\left(\zeta \right)]$$

(24)

where the function $I_{\mp 1/3}$ is obtained from (20) as:

$$I_{\mp 1/3}\left(\zeta \right)=\sum _{m=1}^{\infty }{\displaystyle \frac{1}{m!\Gamma (m\mp 1/3+1)}}({\displaystyle \frac{\zeta }{2}}{)}^{2m\mp 1/3}$$

(25)

Using (23) and (24), derivatives and integrals of $A_i\left(x\right)$ and $B_i\left(x\right)$, take the following forms in terms of the modified Bessel function:

$${A\text{'}}_i\left(x\right)=-{\displaystyle \frac{x}{3}}\left[I_{- {\displaystyle \frac{2}{3}}}\left(\zeta \right)-I_{{\displaystyle \frac{2}{3}}}\left(\zeta \right)\right]$$

(26)

$${B\text{'}}_i\left(x\right)={\displaystyle \frac{x}{\sqrt{3}}} \left[I_{-{\displaystyle \frac{2}{3}}}\left(\zeta \right)+I_{{\displaystyle \frac{2}{3}}}\left(\zeta \right)\right]$$

(27)

$${\int }_0^x{A}_i\left(t\right)dt={\displaystyle \frac{1}{3}}{\int }_0^{\zeta }\left[I_{-{\displaystyle \frac{1}{3}}}\left(t\right)-I_{{\displaystyle \frac{1}{3}}}\left(t\right)\right]dt$$

(28)

$${\int }_0^x{B}_i\left(t\right)dt={\displaystyle \frac{1}{\sqrt{3}}}{\int }_0^{\zeta }\left[I_{-{\displaystyle \frac{1}{3}}}\left(t\right)+I_{{\displaystyle \frac{1}{3}}}\left(t\right)\right]dt$$

(29)

Furthermore, the following integrals are important properties of Airy’s functions, [4]:

$${\int }_{-\infty }^0{B}_i\left(t\right)dt=0$$

(30)

$${\int }_{-\infty }^{\infty }{A}_i\left(t\right)dt=1$$

(31)

$${\int }_0^{\infty }{A}_i\left(t\right)dt={\displaystyle \frac{1}{3}}$$

(32)

Case 2: When $f\left(x\right)=\mp {\displaystyle \frac{1}{\pi }}$, general solutions to Airy’s inhomogeneous equation (1) are given by equations (6) and (7). The particular solutions are expressed in terms of the Scorer functions: $G_i\left(x\right)$ when $f\left(x\right)=-{\displaystyle \frac{1}{\pi }}$, and $H_i\left(x\right)$ when $f\left(x\right)={\displaystyle \frac{1}{\pi }}$. Scorer functions are defined by equations (8) and (9), and are related to Airy’s functions by equations (10)-(12).

Further relationships between Scorer functions and integrals of Airy’s functions can be obtained by defining the following Wronskians:

$$W_1=W\left(A_i\left(x\right),G_i\left(x\right)\right)=A_i\left(x\right){G^{\text{'}}}_i\left(x\right)-G_i\left(x\right){A^{\text{'}}}_i\left(x\right)$$

(33)

$$W_2=W\left(A_i\left(x\right),H_i\left(x\right)\right)=A_i\left(x\right){H^{\text{'}}}_i\left(x\right)-H_i\left(x\right){A^{\text{'}}}_i\left(x\right)$$

(34)

$$W_3=W\left(B_i\left(x\right),G_i\left(x\right)\right)=B_i\left(x\right){G^{\text{'}}}_i\left(x\right)-G_i\left(x\right){B^{\text{'}}}_i\left(x\right)$$

(35)

$$W_4=W\left(B_i\left(x\right),H_i\left(x\right)\right)=B\left(x\right){H^{\text{'}}}_i\left(x\right)-H_i\left(x\right){B^{\text{'}}}_i\left(x\right)$$

(36)

The right-hand-sides of (33)-(36) have been expressed in terms of ${\int }_0^x{A}_i\left(t\right)dt$ and ${\int }_0^x{B}_i\left(t\right)dt$, [4]. We can then write:

$${\int }_0^x{A}_i\left(t\right)dt={\displaystyle \frac{1}{3}}+\pi \left\{{G_i\left(x\right){A^{\text{'}}}_i\left(x\right)-A}_i\left(x\right){G^{\text{'}}}_i\left(x\right)\right\}={\displaystyle \frac{1}{3}}-\pi W_1$$

(37)

$${\int }_0^x{A}_i\left(t\right)dt=-{\displaystyle \frac{2}{3}}-\pi \left\{{H_i\left(x\right){A^{\text{'}}}_i\left(x\right)-A}_i\left(x\right){H^{\text{'}}}_i\left(x\right)\right\}=-{\displaystyle \frac{2}{3}}+\pi W_2$$

(38)

$${\int }_0^x{B}_i\left(t\right)dt=\pi \left\{{G_i\left(x\right){B^{\text{'}}}_i\left(x\right)-B}_i\left(x\right){G^{\text{'}}}_i\left(x\right)\right\}=-\pi W_3$$

(39)

$${\int }_0^x{B}_i\left(t\right)dt=-\pi \left\{{H_i\left(x\right){B^{\text{'}}}_i\left(x\right)-B}_i\left(x\right){H^{\text{'}}}_i\left(x\right)\right\}=\pi W_4$$

(40)

Case 3: If $f\left(x\right)=\kappa$, where $\kappa$ is any real constant, then a particular solution to (1) can be constructed using variation of parameters and takes the form

$$(u_i{)}_p=-{\kappa \pi N}_i(x)$$

(41)

and general solution to (1) is given by equation (13), wherein $N_i\left(x\right)$ is the standard Nield-Kuznetsov function of the first-kind, defined by equation (14). First derivative of $N_i\left(x\right)$ is obtained from (14) as:

$${N\text{'}}_i\left(x\right)={A\text{'}}_i\left(x\right){\int }_0^x{B}_i\left(t\right)dt-{B\text{'}}_i\left(x\right){\int }_0^x{A}_i\left(t\right)dt$$

(42)

Using (23)-(29), we can express $N_i\left(x\right)$ and its derivative, defined in equations (14) and (42), respectively, in terms of modified Bessel function as

$$N_i\left(x\right)={\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{x}{3}}} \left\{I_{-{\displaystyle \frac{1}{3}}}\left(\zeta \right){\int }_0^{\zeta }{I}_{{\displaystyle \frac{1}{3}}}\left(t\right)dt -I_{{\displaystyle \frac{1}{3}}}\left(\zeta \right){\int }_0^{\zeta }{I}_{-{\displaystyle \frac{1}{3}}}\left(t\right)dt \right\}$$

(43)

$${N\text{'}}_i\left(x\right)={\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{x}{3}}} \left\{I_{{\displaystyle \frac{2}{3}}}\left(\zeta \right){\int }_0^{\zeta }{I}_{{\displaystyle \frac{1}{3}}}\left(t\right)dt -I_{-{\displaystyle \frac{2}{3}}}\left(\zeta \right){\int }_0^{\zeta }{I}_{-{\displaystyle \frac{1}{3}}}\left(t\right)dt \right\}$$

(44)

Connections between $N_i\left(x\right)$ and the Scorer functions $G_i\left(x\right)$ and $H_i\left(x\right)$ are established as follows.

Using (37)-(40) and (32) in (43), we obtain the following expressions for $Ni\left(x\right)$:

$$N_i\left(x\right)=\pi \left\{W_1{B}_i\left(x\right)-W_3{A}_i\left(x\right)\right\}-{\displaystyle \frac{1}{3}}{B}_i\left(x\right)$$

(45)

$$N_i\left(x\right)=\pi \left\{W_4{A}_i\left(x\right)-W_2{B}_i\left(x\right)\right\}+{\displaystyle \frac{2}{3}}{B}_i\left(x\right)$$

(46)

Furthermore, equation (10) can be written as

$$G_i\left(x\right)=A_i\left(x\right){\int }_0^x{B}_i\left(t\right)dt+B_i\left(x\right)[{\int }_0^{\infty }{A}_i\left(t\right)dt-{\int }_0^x{A}_{i}dt]$$

(47)

Upon using (32) in (47), and invoking (14), we obtain

$$G_i\left(x\right)=N_i\left(x\right)+{\displaystyle \frac{1}{3}}{B}_i\left(x\right)$$

(48)

and upon using (14) in (48), we obtain

$$H_i\left(x\right)={\displaystyle \frac{2}{3}}{B}_i\left(x\right)-N_i\left(x\right)$$

(49)

Equations (48) and (49) yield

$$N_i\left(x\right)={\displaystyle \frac{2}{3}}{G}_i\left(x\right)-{{\displaystyle \frac{1}{3}}H}_i\left(x\right)$$

(50)

and upon using (8) and (9) in (50), we obtain

$$N_i\left(x\right)={\displaystyle \frac{2}{3\pi }}{\int }_0^{\infty }\sin \left(xt+{\displaystyle \frac{1}{3}}{t}^3\right)dt-{\displaystyle \frac{1}{3\pi }}{\int }_0^{\infty }\exp \left(xt-{\displaystyle \frac{1}{3}}{t}^3\right)dt$$

(51)

Scorer functions $G_i\left(x\right)$ and $H_i\left(x\right)$ take the following forms in terms of modified Bessel function, obtained using (23), (24), (43), (48) and (49):

$$G_i\left(x\right)={\displaystyle \frac{ 2}{3}}\sqrt{{\displaystyle \frac{x}{3}}}\left\{I_{-{\displaystyle \frac{1}{3}}}\left(\zeta \right){\int }_0^{\zeta }\left[I_{{\displaystyle \frac{1}{3}}}\left(t\right)\right]dt-I_{{\displaystyle \frac{1}{3}}}\left(\zeta \right){\int }_0^{\zeta }\left[I_{-{\displaystyle \frac{1}{3}}}\left(t\right)\right]dt\right\}-{\displaystyle \frac{1}{3}} \sqrt{{\displaystyle \frac{x}{3}}}\left[I_{{\displaystyle \frac{1}{3}}}\left(\zeta \right)+I_{-{\displaystyle \frac{1}{3}}}\left(\zeta \right)\right]$$

(52)

$$H_i\left(x\right)={\displaystyle \frac{2}{3}} \sqrt{{\displaystyle \frac{x}{3}}}\left[I_{{\displaystyle \frac{1}{3}}}\left(\zeta \right)+I_{-{\displaystyle \frac{1}{3}}}\left(\zeta \right)\right]-{\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{x}{3}}}\left\{ I_{-{\displaystyle \frac{1}{3}}}\left(\zeta \right){\int }_0^{\zeta }{I}_{{\displaystyle \frac{1}{3}}}\left(t\right)dt+I_{{\displaystyle \frac{1}{3}}}\left(\zeta \right){\int }_0^{\zeta }{I}_{{\displaystyle \frac{1}{3}}}\left(t\right)dt\right\}$$

(53)

and equations (48)-(53) help furnish the following results.

Result 1: General solution to the inhomogeneous Airy’s equation (1) when $f\left(x\right)=-{\displaystyle \frac{1}{\pi }}$ is given by equation (6), or equivalently by

$$u_i=a_1{A}_i\left(x\right)+c_1{B}_i\left(x\right)+N_i\left(x\right)$$

(54)

where ${c_1=b}_1+{\displaystyle \frac{1}{3}}$.

Result 2: General solution to the inhomogeneous Airy’s equation (1) when $f\left(x\right)={\displaystyle \frac{1}{\pi }}$ is given by equation (7), or equivalently by

$$u_i=a_1{A}_i\left(x\right)+{c_{1}B}_i\left(x\right)-N_i\left(x\right)$$

(55)

where ${c_1=b}_1+{\displaystyle \frac{2}{3}}$.

Result 3: The standard Nield-Kuznetsov function of the first-kind, $N_i\left(x\right)$, is defined in terms of Scorer functions by equation (50), and in terms of the improper integral definitions of Airy’s functions by equation (51).

Result 4: Scorer functions $G_i\left(x\right)$ and $H_i\left(x\right)$ are defined in terms of modified Bessel functions by equations (52) and (53).

We conclude this section by tabulating values at zero of $A_i\left(x\right)$, $B_i\left(x\right)$, $G_i\left(x\right), H_i\left(x\right)$, and $N_i\left(x\right)$, and their derivatives:

Table 1. Values of the integral functions and their derivatives at x=0.

Function Value at $\mathit{x}=0$	First Derivative Value at $\mathit{x}=0$
$A_i\left(0\right)=\sqrt{3}{G}_i\left(0\right)={\displaystyle \frac{\sqrt{3}}{3^{\frac{7}{6}}\mathsf{\Gamma }\left(\frac{2}{3}\right)}}$	${A\text{'}}_i\left(0\right)=-\sqrt{3}{G^{\text{'}}}_i\left(0\right)={\displaystyle \frac{-\sqrt{3}}{3^{\frac{5}{6}}\mathsf{\Gamma }\left(\frac{1}{3}\right)}}$
$B_i\left(0\right)=3G_i\left(0\right)={\displaystyle \frac{3}{3^{\frac{7}{6}}\mathsf{\Gamma }\left(\frac{2}{3}\right)}}$	${B\text{'}}_i\left(0\right)={{3G}^{\text{'}}}_i\left(0\right)={\displaystyle \frac{3}{3^{\frac{5}{6}}\mathsf{\Gamma }\left(\frac{1}{3}\right)}}$
$G_i\left(0\right)={\displaystyle \frac{A_i\left(0\right)}{\sqrt{3}}}={\displaystyle \frac{1}{3^{\frac{7}{6}}\mathsf{\Gamma }\left(\frac{2}{3}\right)}}$	${G\text{'}}_i\left(0\right)=-{\displaystyle \frac{1}{\sqrt{3}}}{A^{\text{'}}}_i\left(0\right)={\displaystyle \frac{1}{3^{\frac{5}{6}}\mathsf{\Gamma }\left(\frac{1}{3}\right)}}$
$H_i\left(0\right)={\displaystyle \frac{2}{3}}{B}_i\left(0\right)={\displaystyle \frac{2}{3^{\frac{7}{6}}\mathsf{\Gamma }\left(\frac{2}{3}\right)}}$	${H\text{'}}_i\left(0\right)={\displaystyle \frac{2}{3}}{B\text{'}}_i\left(0\right)={\displaystyle \frac{2}{3^{\frac{5}{6}}\mathsf{\Gamma }\left(\frac{1}{3}\right)}}$
$N_i\left(0\right)=0$	${N^{\text{'}}}_i\left(0\right)=0$

Table 1. Values of the integral functions and their derivatives at x=0.

Function Value at $\mathit{x}=0$	First Derivative Value at $\mathit{x}=0$
$A_i\left(0\right)=\sqrt{3}{G}_i\left(0\right)={\displaystyle \frac{\sqrt{3}}{3^{\frac{7}{6}}\mathsf{\Gamma }\left(\frac{2}{3}\right)}}$	${A\text{'}}_i\left(0\right)=-\sqrt{3}{G^{\text{'}}}_i\left(0\right)={\displaystyle \frac{-\sqrt{3}}{3^{\frac{5}{6}}\mathsf{\Gamma }\left(\frac{1}{3}\right)}}$
$B_i\left(0\right)=3G_i\left(0\right)={\displaystyle \frac{3}{3^{\frac{7}{6}}\mathsf{\Gamma }\left(\frac{2}{3}\right)}}$	${B\text{'}}_i\left(0\right)={{3G}^{\text{'}}}_i\left(0\right)={\displaystyle \frac{3}{3^{\frac{5}{6}}\mathsf{\Gamma }\left(\frac{1}{3}\right)}}$
$G_i\left(0\right)={\displaystyle \frac{A_i\left(0\right)}{\sqrt{3}}}={\displaystyle \frac{1}{3^{\frac{7}{6}}\mathsf{\Gamma }\left(\frac{2}{3}\right)}}$	${G\text{'}}_i\left(0\right)=-{\displaystyle \frac{1}{\sqrt{3}}}{A^{\text{'}}}_i\left(0\right)={\displaystyle \frac{1}{3^{\frac{5}{6}}\mathsf{\Gamma }\left(\frac{1}{3}\right)}}$
$H_i\left(0\right)={\displaystyle \frac{2}{3}}{B}_i\left(0\right)={\displaystyle \frac{2}{3^{\frac{7}{6}}\mathsf{\Gamma }\left(\frac{2}{3}\right)}}$	${H\text{'}}_i\left(0\right)={\displaystyle \frac{2}{3}}{B\text{'}}_i\left(0\right)={\displaystyle \frac{2}{3^{\frac{5}{6}}\mathsf{\Gamma }\left(\frac{1}{3}\right)}}$
$N_i\left(0\right)=0$	${N^{\text{'}}}_i\left(0\right)=0$

3. Complementary Function of ${\mathit{N}}_{\mathit{i}}\left(\mathit{x}\right)$:

Dunster [14] introduced the concept of a complementary function to the standard Nield-Kuznetsov parametric function by extending its definition to $[x,+\infty )$. Aspnes [18,19] introduced a function $A_{i1}\left(x\right)$, defined by, [4]:

$$A_{i1}\left(x\right)={\int }_x^{\infty }{A}_i\left(x\right)=\pi \left\{A_i\left(x\right){G^{\text{'}}}_i\left(x\right)-{A^{\text{'}}}_i\left(x\right)G_i\left(x\right)\right\}$$

(56)

Following the concept introduced by Dunster, [14], we define the complement of $N_i\left(x\right)$ as

$${\tilde{N}}_i\left(x\right)=A_i\left(x\right){\int }_x^{\infty }{B}_i\left(t\right)dt-B_i\left(x\right){\int }_x^{\infty }{A}_i\left(t\right)dt$$

(57)

Now, using (37), we can express (56) as

$${\int }_x^{\infty }{A}_i\left(t\right)dt=\pi W_1$$

(58)

and use (58) to write (57) as

$${\tilde{N}}_i\left(x\right)=A_i\left(x\right){\int }_x^{\infty }{B}_i\left(t\right)dt-{\pi W_{1}B}_i\left(x\right)$$

(59)

The sum of $N_i\left(x\right)$ and its complement ${\tilde{N}}_i\left(x\right)$, given by

$$N_i\left(x\right)+{\tilde{N}}_i\left(x\right)=A_i\left(x\right){\int }_0^{\infty }{B}_i\left(t\right)dt-B_i\left(x\right){\int }_0^{\infty }{A}_i\left(t\right)dt$$

(60)

can be written, with the help of (32), as

$$N_i\left(x\right)+{\tilde{N}}_i\left(x\right)=A_i\left(x\right){\int }_0^{\infty }{B}_i\left(t\right)dt-{{\displaystyle \frac{1}{3}}B}_i\left(x\right)$$

(61)

Furthermore, upon implementing (48), (49) and (50), we obtain the following three expressions for ${\tilde{N}}_i\left(x\right)$, respectively:

$${\tilde{N}}_i\left(x\right)=A_i\left(x\right){\int }_0^{\infty }{B}_i\left(t\right)dt-G_i\left(x\right)$$

(62)

$${\tilde{N}}_i\left(x\right)=A_i\left(x\right){\int }_0^{\infty }{B}_i\left(t\right)dt-B_i\left(x\right)+H_i\left(x\right)$$

(63)

$${\tilde{N}}_i\left(x\right)=A_i\left(x\right){\int }_0^{\infty }{B}_i\left(t\right)dt-{\displaystyle \frac{1}{3}}\left\{B_i\left(x\right)+2G_i\left(x\right)-H_i\left(x\right)\right\}$$

(64)

Upon using (3), (4), (51) and (61), we obtain the following two expressions for ${\tilde{N}}_i\left(x\right)$:

$${\tilde{N}}_i\left(x\right)=A_i\left(x\right){\int }_0^{\infty }{B}_i\left(t\right)dt-{\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }\sin \left(xt+{\displaystyle \frac{1}{3}}{t}^3\right)dt$$

(65)

$${\tilde{N}}_i\left(x\right)=\left\{{\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }\cos \left(xt+{\displaystyle \frac{t^3}{3}}\right)dt\right\}{\int }_0^{\infty }{B}_i\left(t\right)dt-{\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }\sin \left(xt+{\displaystyle \frac{1}{3}}{t}^3\right)dt$$

(66)

The above derivations yield the following result.

Result 5: The complementary function ${\tilde{N}}_i\left(x\right)$ to the standard Nield-Kuznetsov function of the first-kind, $N_i\left(x\right)$, is defined by the equivalent expressions (57) and (59)-(66).

4. Generalized Airy’s Inhomogeneous Equation

4.1. Generalized Airy’s and Nield-Kuznetsov Functions

Consider the generalized, inhomogeneous Airy’s equation (15) in which $n>-2$ is an integer and $f\left(x\right)$ is a smooth function of its real, non-negative variable $x$. Solution to the homogeneous part of this equation is given by (16), with the generalized Airy’s functions given by (17) and (18). The following two cases arise.

Case 1: If $f\left(x\right)=\kappa$, where $\kappa$ is any real constant, then a particular solution to (15) can be constructed using variation of parameters and takes the form:

$${(u}_n{)}_{particular}=-{\displaystyle \frac{\kappa \pi }{2\sqrt{p}\mathrm{s}\mathrm{i}\mathrm{n}\left(p\pi \right)}}{N}_n\left(x\right)$$

(66)

where $N_n\left(x\right)$ is the generalized Nield-Kuznetsov function of the first-kind, defined by equation (22). General solution to (15) thus takes the form given by (21).

From equations (17), (18) and (22) the following derivatives of the functions $A_n\left(x\right)$, $B_n\left(x\right)$ and $N_n\left(x\right)$ are obtained, respectively:

$${A\text{'}}_n\left(x\right)=-p {\left(x\right)}^{{\displaystyle \frac{n+1}{2}}}\left[\right(I_{1-p}\left(\zeta \right)-I_{p-1}\left(\zeta \right)\left)\right]$$

(68)

$${B\text{'}}_n\left(x\right)=p^{1/2} {\left(x\right)}^{{\displaystyle \frac{n+1}{2}}}[I_{p-1}\left(\zeta \right)+I_{1-p}\left(\zeta \right)]$$

(69)

$${N\text{'}}_n\left(x\right)={A\text{'}}_n\left(x\right){\int }_0^x{B}_n\left(t\right)dt-{B\text{'}}_n\left(x\right){\int }_0^x{A}_n\left(t\right)dt$$

(70)

Integrals of the generalized Airy’s functions are obtained from (17) and (18) as

$${\int }_0^x{A}_n\left(t\right)dt=p{\int }_0^{\zeta }[I_{-p}\left(t\right)-I_p\left(t\right)]dt$$

(71)

$${\int }_0^x{B}_n\left(t\right)dt=\sqrt{p}{\int }_0^{\zeta }\left[I_p\left(t\right)+I_{-p}\left(t\right)\right]dt$$

(72)

Upon using (17), (18), (68), (69), (71) and (72) in (22) and (70), we obtain the following expressions for $N_n\left(x\right)$ and ${N^{\text{'}}}_n\left(x\right)$:

$$N_n\left(x\right)=2p\sqrt{p} {\left(x\right)}^{{\displaystyle \frac{1}{2}}}\left\{I_{- p}\left(\zeta \right){\int }_0^{\zeta }\left[I_p\left(t\right)\right]dt-I_p\left(\zeta \right){\int }_0^{\zeta }\left[I_{-p}\left(t\right)\right]dt \right\}$$

(73)

$${N^{\text{'}}}_n\left(x\right)=2p\sqrt{p} {\left(x\right)}^{{\displaystyle \frac{n+1}{2}}}\left\{I_{p-1}\left(\zeta \right){\int }_0^{\zeta }{I}_p\left(t\right)dt-I_{1-p}\left(\zeta \right){\int }_0^{\zeta }{I}_{-p}\left(t\right)dt\right\}$$

(74)

Complementary function to the generalized Nield-Kuznetsov function of the first-kind is defined by:

$${\tilde{N}}_n\left(x\right)=A_n\left(x\right){\int }_x^{\infty }{B}_n\left(t\right)dt-B_n\left(x\right){\int }_x^{\infty }{A}_n\left(t\right)dt$$

(75)

and the sum of functions $N_n\left(x\right)+{\tilde{N}}_n\left(x\right)$ is given by:

$$N_n\left(x\right)+{\tilde{N}}_n\left(x\right)=A_n\left(x\right){\int }_0^{\infty }{B}_n\left(t\right)dt-B_n\left(x\right){\int }_0^{\infty }{A}_n\left(t\right)dt$$

(76)

4.2. Generalized Scorer Functions

For the special cases of $f\left(x\right)=\kappa =\mp {\displaystyle \frac{1}{\pi }}$, general solutions to the inhomogeneous Airy’s equation (1) are expressed in terms of Airy’s and Scorer’s functions, as given by (6) and (7). Scorer [6] obtained these solutions using variation of parameters and defining the functions $G_i\left(x\right)$ and $H_i\left(x\right)$ using (10)-(12). Following Scorer’s approach, we assume that the general solutions to (15) are given as follows. When $\kappa =-{\displaystyle \frac{1}{\pi }}$, general solution to (15) is given by

$$u_n=a_n{A}_n\left(x\right)+b_n{B}_n\left(x\right)+G_n\left(x\right)$$

(77)

and when $\kappa ={\displaystyle \frac{1}{\pi }}$, general solution to (15) is given by

$$u_n=a_n{A}_n\left(x\right)+b_n{B}_n\left(x\right)+H_n\left(x\right)$$

(78)

The functions $G_n\left(x\right)$ and $H_n\left(x\right)$ are termed the generalized Scorer functions, defined and related to the generalized Airy’s functions by

$$G_n\left(x\right)=A_n\left(x\right){\int }_0^x{B}_n\left(t\right)dt+B_n\left(x\right){\int }_x^{\infty }{A}_n\left(t\right)dt$$

(79)

$$H_n\left(x\right)=B_n\left(x\right){\int }_{-\infty }^x{A}_n\left(t\right)dt-A_n\left(x\right){\int }_{-\infty }^x{B}_n\left(t\right)dt$$

(80)

We can relate $G_n\left(x\right)$ and $H_n\left(x\right)$ to $N_n\left(x\right)$, defined by (22) as follows. When $f\left(x\right)=\kappa =-{\displaystyle \frac{1}{\pi }}$, solution to (15) as given by (21) takes the form

$$u_n=a_n{A}_n\left(x\right)+c_n{B}_n\left(x\right)+{\displaystyle \frac{1}{2\sqrt{p}\mathrm{s}\mathrm{i}\mathrm{n}\left(p\pi \right)}}{N}_n\left(x\right)$$

(81)

When $f\left(x\right)=\kappa ={\displaystyle \frac{1}{\pi }}$, solution to (15) as given by (21) takes the form

$$u_n=a_n{A}_n\left(x\right)+c_n{B}_n\left(x\right)-{\displaystyle \frac{1}{2\sqrt{p}\mathrm{s}\mathrm{i}\mathrm{n}\left(p\pi \right)}}{N}_n\left(x\right)$$

(82)

In order for solutions (81) and (82) to reduce to the corresponding solutions when $n=1$, namely solutions (54) and (55), we select $c_n=b_n+p$ in (81) and $c_n=b_n+2p$ in (82). General solutions (81) and (82) can thus be written as:

$$u_n=a_n{A}_n\left(x\right)+b_n{B}_n\left(x\right)+{\displaystyle \frac{1}{2\sqrt{p}\sin \left(p\pi \right)}}{N}_n\left(x\right)+p{B}_n\left(x\right)$$

(83)

$$u_n=a_n{A}_n\left(x\right)+c_n{B}_n\left(x\right)-{\displaystyle \frac{1}{2\sqrt{p}\sin \left(p\pi \right)}}{N}_n\left(x\right)+2p{B}_n\left(x\right)$$

(84)

The particular solutions in (83) and (84) can thus be compared to the particular solutions in (77) and (78) to yield:

$$G_n\left(x\right)={\displaystyle \frac{1}{2\sqrt{p}\sin \left(p\pi \right)}}{N}_n\left(x\right)+p{B}_n\left(x\right)$$

(85)

$$H_n\left(x\right)=2p{B}_n\left(x\right)-{\displaystyle \frac{1}{2\sqrt{p}\mathrm{s}\mathrm{i}\mathrm{n}\left(p\pi \right)}}{N}_n\left(x\right)$$

(86)

with first derivatives given by

$${G\text{'}}_n\left(x\right)={\displaystyle \frac{1}{2\sqrt{p}\sin \left(p\pi \right)}}{N\text{'}}_n\left(x\right)+p{B\text{'}}_n\left(x\right)$$

(87)

$${H\text{'}}_n\left(x\right)=2p{B\text{'}}_n\left(x\right)-{\displaystyle \frac{1}{2\sqrt{p}\mathrm{s}\mathrm{i}\mathrm{n}\left(p\pi \right)}}{N\text{'}}_n\left(x\right)$$

(88)

Adding (85) and (86), we get

$$G_n\left(x\right)+H_n\left(x\right)=3p{B}_n\left(x\right)$$

(89)

Solving (85) and (86) for $N_n\left(x\right)$, gives:

$$N_n\left(x\right)=\left\{{\displaystyle \frac{2}{3}}{G}_n\left(x\right)-{\displaystyle \frac{1}{3}}{H}_n\left(x\right)\right\}2\sqrt{p}\mathrm{s}\mathrm{i}\mathrm{n}\left(p\pi \right)$$

(90)

When n=1, $p={\displaystyle \frac{1}{3}}$ and (85), (86), (89) and (90) reduce to (12), (48), (49) and (50), respectively.

Upon using (18) and (73) in (85) and (86), the generalized Scorer functions are expressed in the following forms in terms of Bessel’s modified functions:

$$G_n\left(x\right)=p\sqrt{px} \left[I_p\left(\zeta \right)+I_{-p}\left(\zeta \right)\right]+{\displaystyle \frac{p\sqrt{x}}{\sin \left(p\pi \right)}} \left\{I_{- p}\left(\zeta \right){\int }_0^{\zeta }\left[I_p\left(t\right)\right]dt-I_p\left(\zeta \right){\int }_0^{\zeta }\left[I_{-p}\left(t\right)\right]dt \right\}$$

(91)

$$H_n\left(x\right)=2p\sqrt{px} \left[I_p\left(\zeta \right)+I_{-p}\left(\zeta \right)\right]-{\displaystyle \frac{p \sqrt{x}}{\mathrm{s}\mathrm{i}\mathrm{n}\left(p\pi \right)}}\left\{I_{-p}\left(\zeta \right){\int }_0^{\zeta }\left[I_p\left(t\right)\right]dt-I_p\left(\zeta \right){\int }_0^{\zeta }\left[I_{-p}\left(t\right)\right]dt \right\}$$

(92)

The above yields the following results.

Result 6: Particular solutions to the generalized inhomogeneous Airy’s equations: ${u\text{'}\text{'}}_n-x^n{u}_n=-{\displaystyle \frac{1}{\pi }}$ and ${u\text{'}\text{'}}_n-x^n{u}_n={\displaystyle \frac{1}{\pi }}$, are given, respectively, by either:

$G_n\left(x\right)or{\displaystyle \frac{1}{2\sqrt{p}\mathrm{s}\mathrm{i}\mathrm{n}\left(p\pi \right)}}{N}_n\left(x\right)$and

$H_n\left(x\right)$ $or-{\displaystyle \frac{1}{2\sqrt{p}\mathrm{s}\mathrm{i}\mathrm{n}\left(p\pi \right)}}{N}_n\left(x\right)$.

Result 7: The generalized Scorer functions $G_n\left(x\right)$ and $H_n\left(x\right)$ are defined in terms of the generalized Nield-Kuznetsov function of the first-kind by equations (85) and (86).

We parallel properties (30), (31) and (32) by stating the following conjecture, where when $n=1$, the integrals reduce to (30)-(32).

Conjecture 1: The following integrals follow from properties (30)-(32):

$${\int }_{-\infty }^0{B}_n\left(x\right)dx=0$$

(93)

$${\int }_{-\infty }^{\infty }{A}_n\left(x\right)dx=3p$$

(94)

$${\int }_0^{\infty }{A}_n\left(x\right)dx=p$$

(95)

We conclude this section by tabulating values at zero of $A_n\left(x\right)$, $B_n\left(x\right)$, $G_n\left(x\right){, H_n\left(x\right) and N}_n\left(x\right)$, and their derivatives:

Table 2. Values of the generalized integral functions and their. derivatives at x = 0.

$A_n\left(0\right)={\displaystyle \frac{{\left(p\right)}^{1-p}}{\mathsf{\Gamma }(1-p)}}$	${A^{\text{'}}}_n\left(0\right)=-{\displaystyle \frac{{\left(p\right)}^p}{\mathsf{\Gamma }\left(p\right)}}$
$B_n\left(0\right)={\displaystyle \frac{{\left(p\right)}^{1/2-p}}{\mathsf{\Gamma }(1-p)}}$	${B^{\text{'}}}_n\left(0\right)=-{\displaystyle \frac{{\left(p\right)}^{p-\frac{1}{2}}}{\mathsf{\Gamma }\left(p\right)}}$
$N_n\left(0\right)=0$	${N^{\text{'}}}_n\left(0\right)=0$
$G_n\left(0\right)=p{B}_n\left(0\right)={\displaystyle \frac{{\left(p\right)}^{3/2-p}}{\mathsf{\Gamma }(1-p)}}$	${G\text{'}}_n\left(0\right)=-{\displaystyle \frac{{\left(p\right)}^{p+\frac{1}{2}}}{\mathsf{\Gamma }\left(p\right)}}$
$H_n\left(0\right)=2G_n\left(0\right)=2{\displaystyle \frac{{\left(p\right)}^{\frac{3}{2}-p}}{\mathsf{\Gamma }\left(1-p\right)}}$	${H^{\text{'}}}_n\left(0\right)=2{G^{\text{'}}}_n\left(0\right)=-2{\displaystyle \frac{{\left(p\right)}^{p+\frac{1}{2}}}{\mathsf{\Gamma }\left(p\right)}}$

Table 2. Values of the generalized integral functions and their. derivatives at x = 0.

$A_n\left(0\right)={\displaystyle \frac{{\left(p\right)}^{1-p}}{\mathsf{\Gamma }(1-p)}}$	${A^{\text{'}}}_n\left(0\right)=-{\displaystyle \frac{{\left(p\right)}^p}{\mathsf{\Gamma }\left(p\right)}}$
$B_n\left(0\right)={\displaystyle \frac{{\left(p\right)}^{1/2-p}}{\mathsf{\Gamma }(1-p)}}$	${B^{\text{'}}}_n\left(0\right)=-{\displaystyle \frac{{\left(p\right)}^{p-\frac{1}{2}}}{\mathsf{\Gamma }\left(p\right)}}$
$N_n\left(0\right)=0$	${N^{\text{'}}}_n\left(0\right)=0$
$G_n\left(0\right)=p{B}_n\left(0\right)={\displaystyle \frac{{\left(p\right)}^{3/2-p}}{\mathsf{\Gamma }(1-p)}}$	${G\text{'}}_n\left(0\right)=-{\displaystyle \frac{{\left(p\right)}^{p+\frac{1}{2}}}{\mathsf{\Gamma }\left(p\right)}}$
$H_n\left(0\right)=2G_n\left(0\right)=2{\displaystyle \frac{{\left(p\right)}^{\frac{3}{2}-p}}{\mathsf{\Gamma }\left(1-p\right)}}$	${H^{\text{'}}}_n\left(0\right)=2{G^{\text{'}}}_n\left(0\right)=-2{\displaystyle \frac{{\left(p\right)}^{p+\frac{1}{2}}}{\mathsf{\Gamma }\left(p\right)}}$

4.3. Computational Algorithm of Generalized Functions

Following Swanson and Headley, [12], the generalized Airy’s functions are evaluated using the following relationships:

$${\rho }_n={\displaystyle \frac{({p)}^{1-p}}{\mathsf{\Gamma }(1-p)}} \mathrm{and} {\phi }_n={\displaystyle \frac{{\left(p\right)}^p}{\mathsf{\Gamma }\left(p\right)}}$$

(96)

$$g_{n1}\left(x\right)=1+{\sum }_{k=1}^{\infty }{p}^{2k}\prod _{j=1}^k{\displaystyle \frac{x^{\left(n+2\right)k}}{j\left(j-p\right)}}$$

(97)

$$g_{n2}\left(x\right)=x\left[1+{\sum }_{k=1}^{\infty }{p}^{2k}\prod _{j=1}^k{\displaystyle \frac{x^{\left(n+2\right)k}}{j\left(j+p\right)}}\right]$$

(98)

$$A_n\left(x\right)={\rho }_n{g}_{n1}\left(x\right)-{\phi }_n{g}_{n2}\left(x\right)$$

(99)

$$B_n\left(x\right)={{\displaystyle \frac{1}{\sqrt{p}}}[\rho }_n{g}_{n1}\left(x\right)+{\phi }_n{g}_{n2}\left(x\right)]$$

(100)

Based on the above algorithm, the generalized Nield-Kuznetsov function of the first-kind and the generalized Scorer functions can be evaluated using the following expression:

$$N_n\left(x\right)={\displaystyle \frac{2}{\sqrt{p}}}{\rho }_n{\phi }_n\left\{g_{n1}\left(x\right){\int }_0^x{g}_{n2}\left(t\right)dt-g_{n2}\left(y\right){\int }_0^x{g}_{n1}\left(t\right)dt\right\}$$

(101)

$$G_n\left(x\right)={\displaystyle \frac{{\rho }_n{\phi }_n}{p\sin \left(p\pi \right)}}\left\{g_{n1}\left(x\right){\int }_0^x{g}_{n2}\left(t\right)dt-g_{n2}\left(y\right){\int }_0^x{g}_{n1}\left(t\right)dt\right\}+{\sqrt{p}[\rho }_n{g}_{n1}\left(x\right)+{\phi }_n{g}_{n2}\left(x\right)]$$

(102)

$$H_n\left(x\right)=-{\displaystyle \frac{{\rho }_n{\phi }_n}{p\sin \left(p\pi \right)}}\left\{g_{n1}\left(x\right){\int }_0^x{g}_{n2}\left(t\right)dt-g_{n2}\left(y\right){\int }_0^x{g}_{n1}\left(t\right)dt\right\}+2{\sqrt{p}[\rho }_n{g}_{n1}\left(x\right)+{\phi }_n{g}_{n2}\left(x\right)]$$

(103)

When $n=1$, equations (96)-(103) reduce to the following:

$${\rho }_1={\displaystyle \frac{({\frac{1}{3})}^{\frac{2}{3}}}{\mathsf{\Gamma }\left(\frac{2}{3}\right)}} and {\phi }_1={\displaystyle \frac{{\left(\frac{1}{3}\right)}^{\frac{1}{3}}}{\mathsf{\Gamma }\left(\frac{1}{3}\right)}}$$

(104)

$$g_{11}\left(x\right)=1+{\sum }_{k=1}^{\infty }{\left({\displaystyle \frac{1}{3}}\right)}^{2k}\prod _{j=1}^k{\displaystyle \frac{x^{3k}}{j\left(j-\frac{1}{3}\right)}}$$

(105)

$$g_{12}\left(x\right)=x\left[1+{\sum }_{k=1}^{\infty }{\left({\displaystyle \frac{1}{3}}\right)}^{2k}\prod _{j=1}^k{\displaystyle \frac{x^{3k}}{j\left(j+\frac{1}{3}\right)}}\right]$$

(106)

$$A_i\left(x\right)={\rho }_1{g}_{11}\left(x\right)-{\phi }_1{g}_{12}\left(x\right)$$

(107)

$$B_1\left(x\right)=\sqrt{3}[{\rho }_1{g}_{11}\left(x\right)+{\phi }_1{g}_{12}\left(x\right)]$$

(108)

$$N_i\left(x\right)=2{\sqrt{3}\rho }_1{\phi }_1\left\{g_{11}\left(x\right){\int }_0^x{g}_{12}\left(t\right)dt-g_{12}\left(y\right){\int }_0^x{g}_{11}\left(t\right)dt\right\}$$

(109)

$$G_i\left(x\right)=2\sqrt{3}{\rho }_1{\phi }_1\left\{g_{11}\left(x\right){\int }_0^x{g}_{12}\left(t\right)dt-g_{12}\left(y\right){\int }_0^x{g}_{11}\left(t\right)dt\right\}+{{\displaystyle \frac{1}{\sqrt{3}}}[\rho }_1{g}_{11}\left(x\right)+{\phi }_1{g}_{12}\left(x\right)]$$

(110)

$$H_i\left(x\right)=-\sqrt{3}{\rho }_1{\phi }_1\left\{g_{11}\left(x\right){\int }_0^x{g}_{12}\left(t\right)dt-g_{12}\left(y\right){\int }_0^x{g}_{11}\left(t\right)dt\right\}+{\displaystyle \frac{2}{\sqrt{3}}}{ [\rho }_1{g}_{11}\left(x\right)+{\phi }_1{g}_{12}\left(x\right)]$$

(111)

5. Higher Derivatives of ${\mathit{N}}_{\mathit{i}}\left(\mathit{x}\right),{\mathit{A}}_{\mathit{n}}\left(\mathit{x}\right),{\mathit{B}}_{\mathit{n}}\left(\mathit{x}\right)\mathit{a}\mathit{n}\mathit{d}{\mathit{N}}_{\mathit{n}}\left(\mathit{x}\right)$

5.1. Higher Derivatives of $N_i\left(x\right)$

In a recent article Hamdan et.al. [16] discussed higher derivatives of the standard Nield-Kuznetsov function of the first kind, $N_i\left(x\right)$, and arrived at the following iterative definition of its k+1^st derivative, where $k\ge 1$:

$$N_i^{(k+1)}\left(x\right)=P_{k+1}\left(x\right)N_i\left(x\right)+Q_{k+1}\left(x\right){N^{\text{'}}}_i\left(x\right)-{\displaystyle \frac{R_{k+1}\left(x\right)}{\pi }}$$

(112)

where $P_{k+1}\left(x\right)$, $Q_{k+1}\left(x\right)$, and $-R_{k+1}\left(x\right)$ are the polynomial coefficients of $N_i\left(x\right)$, ${N^{\text{'}}}_i\left(x\right)$ and the Wronskian $W\left(A_i\left(x\right), B_i\left(x\right)\right)={\displaystyle \frac{1}{\pi }}$, respectively, in the k+1st derivative of $N_i\left(x\right)$. These polynomial coefficients are obtained from the known polynomial coefficients in the kth derivative of $N_i\left(x\right)$ using the following relationships:

$$P_{k+1}={P^{\text{'}}}_k\left(x\right)+x{Q}_k\left(x\right)$$

(113)

$$Q_{k+1}={Q^{\text{'}}}_k\left(x\right)+P_k\left(x\right)$$

(114)

$$R_{k+1}\left(x\right)={R^{\text{'}}}_k\left(x\right)+Q_k\left(x\right)$$

(115)

5.2. Higher Derivatives of $A_n\left(x\right) and B_n\left(x\right)$

In what follows, we derive expressions for higher derivatives of the generalized Airy’s functions, $A_n\left(x\right) \mathrm{a}\mathrm{n}\mathrm{d} B_n\left(x\right)$, and the generalized Nield-Kuznetsov function of the first-kind, $N_n\left(x\right)$. The Wronskian of $A_n\left(x\right)$ and $B_n\left(x\right)$ is given by equation (19) as: $W\left(A_n\left(x\right),B_n\left(x\right)\right)={\displaystyle \frac{2}{\pi }}{p}^{{\displaystyle \frac{1}{2}}}\sin \left(p\pi \right)$.

Consider the homogeneous generalized Airy’s equation, written in the form

$${u\text{'}\text{'}}_n=x^n{u}_n$$

(116)

The first few derivatives of (116) are:

$${u\text{'}\text{'}\text{'}}_n=x^n{u\text{'}}_n+n{x}^{n-1}{u}_n$$

(117)

$${u^{iv}}_n=2n{x}^{n-1}{u\text{'}}_n+{[x}^{2n}+n\left(n-1\right)x^{n-2}]u_n$$

(118)

$${u^v}_n={[x}^{2n}+3n\left(n-1\right)x^{n-2}]{u\text{'}}_n+{[4nx}^{2n-1}+n\left(n-1\right)(n-2\left)x^{n-3}\right]u_n$$

(119)

Each of the above derivatives of $u_n$ is expressed in terms of $u_n$ and ${u\text{'}}_n$. Their coefficients are polynomials for any given index $n>-2$. The generalized Airy’s functions, $A_n\left(x\right)$ and $B_n\left(x\right)$ satisfy the homogeneous generalized Airy’s equation and the derivatives above. We can thus express the kth derivatives of $A_n\left(x\right)$ and $B_n\left(x\right)$ in the following forms:

$$A_n^{\left(k\right)}\left(x\right)=P_k\left(x\right){A^{\text{'}}}_n\left(x\right)+Q_k\left(x\right)A_n\left(x\right)$$

(120)

$$B_n^{\left(k\right)}\left(x\right)=P_k\left(x\right){B^{\text{'}}}_n\left(x\right)+Q_k\left(x\right)B_n\left(x\right)$$

(121)

where $P_k\left(x\right)$ is the polynomial coefficient of ${A^{\text{'}}}_n\left(x\right)$ and ${B^{\text{'}}}_n\left(x\right)$, and $Q_k\left(x\right)$ is the polynomial coefficient of $A_n\left(x\right)$ and $B_n\left(x\right)$ in the kth derivatives of $A_n\left(x\right)$ and $B_n\left(x\right)$. A few of these polynomials are shown in the Table below.

Table 3. The Polynomials ${\mathit{P}}_{\mathit{n}}\left(\mathit{x}\right)$ and ${\mathit{Q}}_{\mathit{n}}\left(\mathit{x}\right).$.

$k$	$P_n\left(x\right)$	$Q_n\left(x\right)$
0	0	1
1	1	0
2	0	$x^n$
3	$x^n$	$n{x}^{n-1}$
4	$2n{x}^{n-1}$	$x^{2n}+n\left(n-1\right)x^{n-2}$
5	$x^{2n}+3n\left(n-1\right)x^{n-2}$	${4nx}^{2n-1}+n\left(n-1\right)(n-2)x^{n-3}$

Table 3. The Polynomials ${\mathit{P}}_{\mathit{n}}\left(\mathit{x}\right)$ and ${\mathit{Q}}_{\mathit{n}}\left(\mathit{x}\right).$.

$k$	$P_n\left(x\right)$	$Q_n\left(x\right)$
0	0	1
1	1	0
2	0	$x^n$
3	$x^n$	$n{x}^{n-1}$
4	$2n{x}^{n-1}$	$x^{2n}+n\left(n-1\right)x^{n-2}$
5	$x^{2n}+3n\left(n-1\right)x^{n-2}$	${4nx}^{2n-1}+n\left(n-1\right)(n-2)x^{n-3}$

Now, from (120) and (121), we obtain the k+1^st derivatives, and express them as:

$$A_n^{\left(k+1\right)}\left(x\right)=P_{k+1}\left(x\right){A^{\text{'}}}_n\left(x\right)+Q_{k+1}\left(x\right)A_n\left(x\right)$$

(122)

$$B_n^{\left(k+1\right)}\left(x\right)=P_{k+1}\left(x\right){B^{\text{'}}}_n\left(x\right)+Q_{k+1}\left(x\right)B_n\left(x\right)$$

(123)

where the polynomial coefficients in the k+1^st derivatives are obtained with the knowledge of the polynomial coefficients in the kth derivatives using the following relationships:

$$P_{k+1}\left(x\right)={P^{\text{'}}}_k\left(x\right)+Q_k\left(x\right)$$

(124)

$${{Q_{k+1}\left(x\right)=x}^{n}P}_k\left(x\right)+{Q^{\text{'}}}_k\left(x\right)$$

(125)

The above derivations furnish the following results.

Result 7: Higher derivatives, of all orders, of $A_n\left(x\right)$ are expressible in terms of $A_n\left(x\right)$ and ${A\text{'}}_n\left(x\right)$, and higher derivatives, of all orders, of $B_n\left(x\right)$ are expressible in terms of $B_n\left(x\right)$ and ${B\text{'}}_n\left(x\right)$. Furthermore, the k+1^st derivatives of the generalized Airy’s functions are defined iteratively by equations (122) and (123).

Result 8: The generalized Airy’s polynomials arising from higher derivatives of the generalized Airy’s functions are defined iteratively by equations (124) and (125).

5.3. Higher Derivatives of $N_n\left(x\right),G_n\left(x\right) and H_n\left(x\right)$

The inhomogeneous generalized Airy’s equation (1), when $f\left(x\right)=\kappa$, can be written as:

$${u\text{'}\text{'}}_n=x^n{u}_n+\kappa$$

(126)

The function $N_n\left(x\right)$ satisfies the particular solution (67) to the inhomogeneous generalized Airy’s equation (126). We thus have

$${N\text{'}\text{'}}_n\left(x\right)=x^n{N}_n\left(x\right)-W(A_n\left(x\right),B_n\left(x\right))$$

(127)

Repeated differentiation of (127) gives

$${N\text{'}\text{'}\text{'}}_n\left(x\right)=x^n{N\text{'}}_n\left(x\right)+n{x}^{n-1}{N}_n\left(x\right)$$

(128)

$${N^{iv}}_n\left(x\right)=2n{x}^{n-1}{N\text{'}}_n\left(x\right)+\left[x^{2n}+n\left(n-1\right)x^{n-2}\right]N_n\left(x\right)-x^{n}W(A_n\left(x\right),B_n\left(x\right))$$

(129)

$${N^v}_n\left(x\right)=[x^{2n}+3n(n-1)x^{n-2}{]N\text{'}}_n\left(x\right)+\left[{4nx}^{2n-1}+n\left(n-1\right)(n-2)x^{n-3}\right]N_n\left(x\right)-{3nx}^{n-1}W(A_n\left(x\right),B_n\left(x\right))$$

(130)

$$N_n^{\left(k\right)}\left(x\right)=P_k\left(x\right){N^{\text{'}}}_n\left(x\right)+Q_k\left(x\right)N_n\left(x\right)-R_k\left(x\right)W(A_n\left(x\right),B_n\left(x\right))$$

(131)

The k+1^st derivative of $N_n\left(x\right)$ is obtained from (131) and takes the form

$$N_n^{\left(k+1\right)}\left(x\right)=P_{k+1}\left(x\right){N^{\text{'}}}_n\left(x\right)+{Q_{k+1}\left(x\right)N}_n\left(x\right)-R_{k+1}\left(x\right)W(A_n\left(x\right),B_n\left(x\right))$$

(132)

where the polynomial coefficients $P_{k+1}\left(x\right)$, $Q_{k+1}\left(x\right)$ and $R_{k+1}\left(x\right)$, in the k+1^st derivative are obtained from $P_k\left(x\right)$, $Q_k\left(x\right)$ and $R_k\left(x\right)$, in the kth derivative using the relationships:

$$P_{k+1}\left(x\right)={P^{\text{'}}}_k\left(x\right)+Q_k\left(x\right)$$

(133)

$${{Q_{k+1}\left(x\right)={Q^{\text{'}}}_k\left(x\right)+x}^{n}P}_k\left(x\right)$$

(134)

$$R_{k+1}\left(x\right)={R^{\text{'}}}_k\left(x\right)+P_k\left(x\right)$$

(135)

The k+1^st derivatives of generalized Scorer functions can be obtained using (85), (86), (123) and (132), as:

$$G_n^{\left(k+1\right)}\left(x\right)={\displaystyle \frac{1}{2\sqrt{p}\sin \left(p\pi \right)}}{N}_n^{\left(k+1\right)}\left(x\right)+p{B}_n^{\left(k+1\right)}\left(x\right)={\displaystyle \frac{1}{2\sqrt{p}\sin \left(p\pi \right)}}{[P}_{k+1}\left(x\right){N^{\text{'}}}_n\left(x\right)+{Q_{k+1}\left(x\right)N}_n\left(x\right)-R_{k+1}\left(x\right)W(A_n\left(x\right),B_n\left(x\right))]+{p[P}_{k+1}\left(x\right){B^{\text{'}}}_n\left(x\right)+Q_{k+1}\left(x\right)B_n\left(x\right)]$$

(136)

$$H_n^{\left(k+1\right)}\left(x\right)=2p{B}_n^{\left(k+1\right)}\left(x\right)-{\displaystyle \frac{1}{2\sqrt{p}\mathrm{s}\mathrm{i}\mathrm{n}\left(p\pi \right)}}{N}_n^{\left(k+1\right)}\left(x\right)=-{\displaystyle \frac{1}{2\sqrt{p}\sin \left(p\pi \right)}}{[P}_{k+1}\left(x\right){N^{\text{'}}}_n\left(x\right)+{Q_{k+1}\left(x\right)N}_n\left(x\right)-R_{k+1}\left(x\right)W(A_n\left(x\right),B_n\left(x\right))]+2{p[P}_{k+1}\left(x\right){B^{\text{'}}}_n\left(x\right)+Q_{k+1}\left(x\right)B_n\left(x\right)]$$

(137)

The above derivations furnish the following results.

Result 9: Higher derivatives, of all orders, of $N_n\left(x\right)$ are expressible in terms of $N_n\left(x\right)$ and ${N\text{'}}_n\left(x\right)$. Furthermore, the k+1^st derivatives of the generalized Nield-Kuznetsov function of the first-kind is defined iteratively by equation (132), and those of the generalized Scorer functions by equations (136) and (137).

Result 10: The generalized polynomials arising from higher derivatives of the generalized Nield-Kuznetsov function of the first-kind are defined iteratively by equations (133), (134) and (135).

Using (17), (18), (26) and (27) in (122) and (123), and using (73) and (73) in (132) we obtain the following expressions for the k+1^st derivatives of $A_n\left(x\right)$, $B_n\left(x\right)$ $N_n\left(x\right)$ in terms of the modified Bessel functions:

$$A_n^{\left(k+1\right)}\left(x\right)={-{\displaystyle \frac{x}{3}}P}_{k+1}\left(x\right)\left[I_{- {\displaystyle \frac{2}{3}}}\left(\zeta \right)-I_{{\displaystyle \frac{2}{3}}}\left(\zeta \right)\right]+p {\left(x\right)}^{1/2}{Q}_{k+1}\left(x\right) [I_{- p}\left(\zeta \right)-I_p\left(\zeta \right)]$$

(138)

$$B_n^{\left(k+1\right)}\left(x\right)={{\displaystyle \frac{x}{\sqrt{3}}}P}_{k+1}\left(x\right) \left[I_{-{\displaystyle \frac{2}{3}}}\left(\zeta \right)+I_{{\displaystyle \frac{2}{3}}}\left(\zeta \right)\right]+{\left(px\right)}^{1/2}{Q}_{k+1}\left(x\right) [I_p\left(\zeta \right)+I_{-p}\left(\zeta \right)]$$

(139)

$$N_n^{\left(k+1\right)}\left(x\right)=2p\sqrt{p}{\left(x\right)}^{{\displaystyle \frac{n+1}{2}}}{P}_{k+1}\left(x\right)\left\{I_{p-1}\left(\zeta \right){\int }_0^{\zeta }{I}_p\left(t\right)dt-I_{1-p}\left(\zeta \right){\int }_0^{\zeta }{I}_{-p}\left(t\right)dt\right\}+2p\sqrt{p} {\left(x\right)}^{{\displaystyle \frac{1}{2}}}{Q}_{k+1}\left(x\right)\left\{I_{-p}\left(\zeta \right){\int }_0^{\zeta }\left[I_p\left(t\right)\right]dt-I_p\left(\zeta \right){\int }_0^{\zeta }\left[I_{-p}\left(t\right)\right]dt \right\}-R_{k+1}\left(x\right)W(A_n\left(x\right),B_n\left(x\right))$$

(140)

Using the above derivatives, we tabulate in Table 4 the values of the k+1^st derivative at $x=0$ for each of the integral functions.

6. Conclusions

The main theme of this work has been the study and analysis of Airy’s and generalized Airy’s differential equations, and the integral functions that define their particular solutions. We attempted to fill in gaps that exit in the knowledge-base of the Nield-Kuznetsov and the Scorer functions. This includes defining the generalized Scorer functions, studying some of their properties and their relations to special functions that exist in the literature. We also provided further analysis and properties of the Nield-Kuznetsov and the generalized Nield-Kuznetsov functions and their relationships to the generalized Airy’s functions. All functions have been expressed in terms of modified Bessel functions. Furthermore, as higher derivatives of the Nield-Kuznetsov and Scorer functions give rise to important Airy’s polynomials and generalized Airy’s polynomials, we provided in this work iterative definitions of the higher derivatives together with an iterative method of generating these polynomials. It is expected that this work will make its way into the many applications of the Airy’s family of differential equations and their emerging integral functions.