A q-Difference Equation and Fourier Series Expansions of q-Lidstone Polynomials

Abstract

In this paper, we present the q-Lidstone polynomials which are q-Bernoulli polynomials generated by the third Jackson q-Bessel function, based on the Green’s function of a certain q-difference equation. Also, we provide the q-Fourier series expansions of these polynomials and derive some results related to them.

1. Introduction

In 1929, Lidstone [1] introduced a generalization of Taylor’s series that approximates an entire function $f\left(z\right)$ of exponential type less than $\pi$ in a neighborhood of two points instead of one:

$$f\left(z\right)=\sum _{n=0}^{\infty }{A}_n\left(z\right)f^{2n}\left(1\right)+\sum _{n=0}^{\infty }{A}_n(1-z)f^{2n}\left(0\right),$$

where the set ${\left\{A_n\left(z\right)\right\}}_n$ called Lidstone polynomials. In [2], Whittaker proved that

$$A_n\left(z\right)=\frac{2^{2n+1}}{(2n+1)!}{B}_{2n+1}\left(\frac{z+1}{2}\right),$$

where $B_n\left(x\right)$ is the Bernoulli polynomial of degree n, which may be defined by the generating function

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

Recently, Ismail and Mansour [3] introduced a q analog of the Lidstone expansion theorem where they expand a class of entire functions of q-exponential growth in terms of Jackson q-derivatives of even degree at 0 and 1. See also [4,5,6] for some results and applications to the q-Lidstone theorem.

In [7], the authors constructed another formula of q-Lidstone expansion theorem by using the symmetric q-difference operator ${\delta }_q$ (see Section 2), that is

$$f\left(z\right)=\sum _{n=0}^{\infty }\left[\frac{{\delta }_q^{2n}f\left(1\right)}{{\delta }_q{z}^{2n}}{\tilde{A}}_n\left(z\right)-\frac{{\delta }_q^{2n}f\left(0\right)}{{\delta }_q{z}^{2n}}{\tilde{B}}_n\left(z\right)\right],$$

(1)

where ${\tilde{A}}_n\left(z\right)$ and ${\tilde{B}}_n\left(z\right)$ are the q-Lidstone polynomials defined by the generating functions

$$\begin{array}{c}{\displaystyle \frac{ex{p}_q\left(zw\right)-ex{p}_q(-zw)}{ex{p}_q\left(w\right)-ex{p}_q(-w)}}={\displaystyle \sum _{n=0}^{\infty }}{\tilde{A}}_n\left(z\right)w^{2n},\\ {\displaystyle \frac{ex{p}_q\left(zw\right)ex{p}_q(-w)-ex{p}_q(-zw)ex{p}_q\left(w\right)}{ex{p}_q\left(w\right)-ex{p}_q(-w)}}={\displaystyle \sum _{n=0}^{\infty }}{\tilde{B}}_n\left(z\right)w^{2n}.\end{array}$$

(2)

Moreover, it turns out that

$${\tilde{B}}_n\left(z\right)=\frac{2^{2n+1}}{{[2n+1]}_q!}{\tilde{B}}_{2n+1}(z/2;q),$$

(3)

where ${\tilde{B}}_n(z;q)$ are q-Bernoulli polynomials generated by

$${\displaystyle \frac{wex{p}_q\left(zw\right)ex{p}_q\left(\frac{-w}{2}\right)}{ex{p}_q\left(\frac{w}{2}\right)-ex{p}_q\left(\frac{-w}{2}\right)}}=\sum _{n=0}^{\infty }{\tilde{B}}_n(z;q)\frac{w^n}{{\left[n\right]}_q!},$$

(4)

and the function $ex{p}_q(.)$ is the q-exponential function which has the series representation

$$ex{p}_q\left(z\right)=\sum _{n=0}^{\infty }\frac{q^{\frac{n(n-1)}{4}}}{{\left[n\right]}_q!}{z}^n;z\in \mathbb{C}.$$

(5)

In this paper, we assume that q is a positive number less than one and the set $A_q^{*}$ is defined by

$$A_q^{*}:=\{q^n:n\in {\mathbb{N}}_0\}\cup \left\{0\right\},$$

where ${\mathbb{N}}_0:=\{0,1,2,\dots \}$. We present the q-Lidstone polynomials ${\tilde{A}}_n\left(z\right)$ and ${\tilde{B}}_n\left(z\right)$ based on the Green’s function of a q-boundary value problem

$$\left\{\begin{array}{c}\frac{{\delta }_q^{2n}f\left(z\right)}{{\delta }_q{z}^{2n}}=\varphi \left(z\right),\hfill \\ \\ \frac{{\delta }_q^{2k}f\left(0\right)}{{\delta }_q{z}^{2k}}=a_k,\frac{{\delta }_q^{2k}f\left(1\right)}{{\delta }_q{z}^{2k}}=b_k(k=0,1,\dots ,n-1),\hfill \end{array}\right.$$

(6)

where f and $\varphi$ are assumed to be continuous functions on $A_q^{*}$. Also, we introduce the q-Fourier series expansions of these functions and derive some results related to them. For other recent contributions on this area, one may refer to [8,9,10].

This article is organized as follows: In the next section, we present some background on q-analysis which we need in our investigations. In Section 3, we establish the existence of a solution for the system (51). In Section 4, we introduce the q-Fourier series expansions of some functions. As an application, in Section 5, we define q-Lidstone polynomials based on the Green’s function of the system (51), and we provide the q-Fourier series expansions of these polynomials. Moreover, relying on the obtained q-Fourier series, we derive a close approximation to ${\tilde{A}}_n\left(z\right)$ and ${\tilde{B}}_n\left(z\right)$ for large n.

2. Preliminaries

Recall that the q-derivative $D_q$ of the function f is defined by

$$D_{q}f\left(z\right):=\frac{f\left(z\right)-f\left(qz\right)}{z-qz},\mathrm{for}z\ne 0,$$

(7)

and the q-derivative at zero is defined to be $f^{\prime }\left(0\right)$ if it exists, see [11]. The q-shifted fractional ${(a;q)}_n$ of $a\in \mathbb{C}$ is defined by

$${(a;q)}_0:=1\mathrm{and}{(a;q)}_n:=\prod _{j=0}^n(1-a{q}^j),\mathrm{for}n\in \mathbb{N},$$

and the q-number factorial ${\left[n\right]}_q!$ is defined for $q\ne 1$ by

$${\left[n\right]}_q!=\prod _{j=0}^n{\left[j\right]}_q,{\left[j\right]}_q=\frac{1-q^j}{1-q}.$$

Jackson [12] introduced the following integral, as a right inverse of the q-derivative (7), by

$${\int }_a^{b}f\left(t\right)d_{q}t:={\int }_0^{b}f\left(t\right)d_{q}t-{\int }_0^{a}f\left(t\right)d_{q}t(a,b\in \mathbb{C}),$$

where

$${\int }_0^{z}f\left(t\right)d_{q}t:=(1-q)\sum _{n=0}^{\infty }z{q}^{n}f\left(z{q}^n\right),$$

provided that the series converges at $z=a$ and $z=b$. We can interchange the order of double q-integral by

$${\int }_0^z{\int }_0^{v}f\left(t\right)d_{q}t{d}_{q}v={\int }_0^z{\int }_{qt}^{z}f\left(t\right)d_{q}v{d}_{q}t={\int }_0^z(z-qt)f\left(t\right)d_{q}t.$$

(8)

The symmetric q-difference operator ${\delta }_q$ which is acting on a function f defined by

$$\frac{{\delta }_{q}f\left(z\right)}{{\delta }_{q}z}:={\displaystyle \frac{f\left(q^{\frac{1}{2}}z\right)-f\left(q^{\frac{-1}{2}}z\right)}{z(q^{\frac{1}{2}}-q^{\frac{-1}{2}})}},\mathrm{for}z\ne 0.$$

(9)

(see [11,13]). From (7) and (9), it follows

$$\frac{{\delta }_{q}f\left(z\right)}{{\delta }_{q}z}:=D_{q}f\left(\frac{z}{\sqrt{q}}\right).$$

Therefore, we have

$${\int }_0^a\frac{{\delta }_{q}f\left(z\right)}{{\delta }_{q}z}{d}_{q}z=q^{\frac{1}{2}}[f\left(q^{-\frac{1}{2}}a\right)-f\left(0\right)].$$

(10)

A function f defined on $A_q^{*}$ is called q-regular at zero if it satisfies

$$\underset{n\to \infty }{\lim }f\left(x{q}^n\right)=f\left(0\right),\mathrm{for}\mathrm{all}x\in A_{q,t}^{*}.$$

The q-integration by parts rule on $A_q^{*}$ (see [13]) is

$${\int }_0^{a}g\left(q^{-\frac{1}{2}}t\right)\frac{{\delta }_{q}f\left(t\right)}{{\delta }_{q}t}{d}_{q}t=q^{\frac{1}{2}}\left(gf\right)\left(q^{-\frac{1}{2}}t\right){|}_0^a-{\int }_0^{a}f\left(q^{\frac{1}{2}}t\right)\frac{{\delta }_{q}g\left(t\right)}{{\delta }_{q}t}{d}_{q}t,$$

(11)

where f and g are complex valued q-regular functions at zero.

We will use a q-exponential function $ex{p}_q(.)$ defined in (5) and the q-linear sine and cosine, $S_q\left(z\right)$ and $C_q\left(z\right)$, which defined by

$$\begin{array}{cc}\hfill S_q\left(z\right)& :=\frac{ex{p}_q\left(iz\right)-ex{p}_q(-iz)}{2i}=\sum _{n=0}^{\infty }{(-1)}^n\frac{q^{n(n+\frac{1}{2})}}{{[2n+1]}_q!}{z}^{2n+1},\hfill \\ \hfill C_q\left(z\right)& :=\frac{ex{p}_q\left(iz\right)+ex{p}_q(-iz)}{2}=\sum _{n=0}^{\infty }{(-1)}^n\frac{q^{n(n-\frac{1}{2})}}{{\left[2n\right]}_q!}{z}^{2n}.\hfill \end{array}$$

(12)

They can be written in terms of the third Jackson q-Bessel function $J_{\nu }^{\left(3\right)}(z;q)$ [14,15] as follows

$$\begin{array}{cc}\hfill S_q\left(z\right)& :=q^{1/8}\frac{{(q^2;q^2)}_{\infty }}{{(q;q^2)}_{\infty }}{z}^{1/2}{J}_{1/2}^{\left(3\right)}(q^{-1/4}z;q^2),\hfill \\ \hfill C_q\left(z\right)& :=q^{-3/8}\frac{{(q^2;q^2)}_{\infty }}{{(q;q^2)}_{\infty }}{z}^{1/2}{J}_{-1/2}^{\left(3\right)}(q^{-3/4}z;q^2).\hfill \end{array}$$

(13)

These functions satisfy

$$\frac{{\delta }_q{C}_q\left(wz\right)}{{\delta }_{q}z}=-w{S}_q\left(wz\right),\frac{{\delta }_q{S}_q\left(wz\right)}{{\delta }_{q}z}=w{C}_q\left(wz\right),$$

(14)

see [11,13]. We denote to the derivative of $S_q\left(z\right)$ by $S_q^{\prime }\left(z\right)$ and we assume that $\{w_k:k\in \mathbb{N} \mathrm{with} w_1<w_2<w_3<\dots \}$ is the set of positive zeroes of $S_q\left(z\right)$.

3. Existence Solutions of $\mathit{q}$-Differential System

In this section, we construct the solution of the q-differential system (51). Let $C_q^n\left(A_q^{*}\right)$ denote the space of all continues functions with continuous q-derivatives up to order $n-1$ on $A_q^{*}$ with values in $\mathbb{R}$.

Lemma 1.

Let $f,\varphi \in C_q^2\left(A_q^{*}\right)$. Then, the solution of the q-differential equation

$$\frac{{\delta }_q^{2}f\left(z\right)}{{\delta }_q{z}^2}-\varphi \left(z\right)=0,$$

(15)

subject to the boundary conditions $f\left(0\right)=f\left(1\right)=0$ is equivalent to the basic Fredholm q-integral equation

$$f\left(z\right)={\int }_0^1\tilde{G}(z,qt)\varphi \left(qt\right)d_{q}t,$$

(16)

where $\tilde{G}(z,t)$ is the Green’s function defined on $A_q^{*}$ by

$$\tilde{G}(z,t):=\left\{\begin{array}{cc}\sqrt{q}z(t-1),\hfill & z<t;\hfill \\ \\ \sqrt{q}t(z-1),\hfill & t<z.\hfill \end{array}\right.$$

(17)

Proof. 

The q-differential Equation (15) can be written as

$$D_q^{2}f\left(z\right)-\sqrt{q}\varphi \left(qz\right)=0(z\in A_q^{*}).$$

(18)

By taking double q-integral for (18) and using (8), we obtain

$$f\left(z\right)=c_0+c_{1}z+\sqrt{q}{\int }_0^z(z-qt)\varphi \left(qt\right)d_{q}t,$$

(19)

where $c_0$ and $c_1$ are arbitrary constant. Using the boundary conditions, we get $c_0=0$ and

$$c_1=-\sqrt{q}{\int }_0^1(1-qt)\varphi \left(qt\right)d_{q}t.$$

Substituting in (19), we have

$$f\left(z\right)=-\sqrt{q}z{\int }_0^1(1-qt)\varphi \left(qt\right)d_{q}t+\sqrt{q}{\int }_0^z(z-qt)\varphi \left(qt\right)d_{q}t,$$

and then we obtain (16).  ☐

Remark 1.

By induction on n, one can verify that if $f,\varphi \in C_q^{2n}\left(A_q^{*}\right)$, then the function

$$f\left(z\right)={\int }_0^1{\tilde{G}}_n(z,qt)\varphi \left(qt\right)d_{q}t,$$

(20)

is the solution of the q-boundary value problem

$$\left\{\begin{array}{c}\frac{{\delta }_q^{2n}f\left(z\right)}{{\delta }_q{z}^{2n}}=\varphi \left(z\right),\hfill \\ \\ \frac{{\delta }_q^{2k}f\left(1\right)}{{\delta }_q{z}^{2k}}=\frac{{\delta }_q^{2k}f\left(0\right)}{{\delta }_q{z}^{2k}}=0(k=0,1,\dots ,n-1),\hfill \end{array}\right.$$

(21)

where ${\tilde{G}}_1(z,t)$ is the Green’s function defined as in (17) and

$$\begin{array}{cc}\hfill {\tilde{G}}_n(z,qt)=& {\int }_0^1\tilde{G}(z,qw){\tilde{G}}_{n-1}(qw,qt)d_{q}w\hfill \\ \hfill =& {\int }_0^1{\tilde{G}}_{n-1}(z,qw)\tilde{G}(qw,qt)d_{q}w(n=2,3,\dots ).\hfill \end{array}$$

(22)

Theorem 1.

If $f\left(z\right)$ and $\varphi \left(z\right)$ are functions of class $C_q^{2n}\left(A_q^{*}\right)$, then any solution of the system

$$\left\{\begin{array}{c}\frac{{\delta }_q^{2n}f\left(z\right)}{{\delta }_q{z}^{2n}}=\varphi \left(z\right),\hfill \\ \\ \frac{{\delta }_q^{2k}f\left(0\right)}{{\delta }_q{z}^{2k}}=a_k,\frac{{\delta }_q^{2k}f\left(1\right)}{{\delta }_q{z}^{2k}}=b_k(k=0,1,\dots ,n-1)\hfill \end{array}\right.$$

(23)

is given by

$$\begin{array}{c}f\left(z\right)=a_0(z-1)+{\displaystyle \sum _{k=1}^{n-1}}{a}_k{\int }_0^1(qt-1){\tilde{G}}_k(z,qt)d_{q}t+b_{0}z\\ +{\displaystyle \sum _{k=1}^{n-1}}{b}_k{\int }_0^1\left(qt\right){\tilde{G}}_k(z,qt)d_{q}t+{\int }_0^1{\tilde{G}}_n(z,qt)\varphi \left(qt\right)d_{q}t,\end{array}$$

(24)

where the functions ${\tilde{G}}_n(z,qt)$ $(n\in \mathbb{N})$ defined as in (17) and (22).

Proof. 

From (17), (22) and Equation (23) we get

$$\begin{array}{cc}\hfill R_n\left(z\right)=& {\int }_0^1{\tilde{G}}_n(z,qt)\varphi \left(qt\right)d_{q}t\hfill \\ \hfill =& {\int }_0^1{\tilde{G}}_{n-1}(z,qw){\int }_0^1\tilde{G}(qw,qt)\frac{{\delta }_q^{2n}f\left(qt\right)}{{\delta }_q{z}^{2n}}{d}_{q}t{d}_{q}w\hfill \\ \hfill =& {\int }_0^1{\tilde{G}}_{n-1}(z,qw)[\sqrt{q}(qw-1){\int }_0^{qw}\left(qt\right)\frac{{\delta }_q^{2n}f\left(qt\right)}{{\delta }_q{z}^{2n}}{d}_{q}t\hfill \\ \hfill +& q\sqrt{q}w{\int }_{qw}^1(qt-1)\frac{{\delta }_q^{2n}f\left(qt\right)}{{\delta }_q{z}^{2n}}{d}_{q}t]d_{q}w.\hfill \end{array}$$

(25)

Using the rule (11), after some simplifications, we obtain

$$\begin{array}{c}{R}_n\left(z\right)=\frac{{\delta }_q^{2n-2}f\left(0\right)}{{\delta }_q{z}^{2n-2}}{\int }_0^1(qw-1){\tilde{G}}_{n-1}(z,qw)d_{q}w-\\ \frac{{\delta }_q^{2n-2}f\left(1\right)}{{\delta }_q{z}^{2n-2}}{\int }_0^1\left(qw\right){\tilde{G}}_{n-1}(z,qw)d_{q}w+{\int }_0^1{\tilde{G}}_{n-1}(z,qw)\frac{{\delta }_q^{2n-2}f\left(qw\right)}{{\delta }_q{z}^{2n-2}}{d}_{q}w.\end{array}$$

(26)

Repeating the q-integration by parts on the last q-integral of Equation (26) $(n-1)$ times, we get

$$\begin{array}{c}{R}_n\left(z\right)={\displaystyle \sum _{k=1}^{n-1}}\frac{{\delta }_q^{2k}f\left(0\right)}{{\delta }_q{z}^{2k}}{\int }_0^1(qw-1){\tilde{G}}_k(z,qt)d_{q}w-\\ {\displaystyle \sum _{k=1}^{n-1}}\frac{{\delta }_q^{2k}f\left(1\right)}{{\delta }_q{z}^{2k}}{\int }_0^1\left(qw\right){\tilde{G}}_k(z,qw)d_{q}w+{\int }_0^1\tilde{G}(z,qw)\frac{{\delta }_q^{2}f\left(qw\right)}{{\delta }_q{z}^2}{d}_{q}w.\end{array}$$

(27)

Computing the last integral of (27), we get

$$\begin{array}{cc}\hfill & {\int }_0^1\tilde{G}(z,qw)\frac{{\delta }_q^{2}f\left(qw\right)}{{\delta }_q{z}^2}{d}_{q}w\hfill \\ \hfill =& \sqrt{q}(1-z){\int }_0^z(-qw)\frac{{\delta }_q^{2}f\left(qw\right)}{{\delta }_q{z}^2}{d}_{q}w-\sqrt{q}z{\int }_z^1(1-qw)\frac{{\delta }_q^{2}f\left(qw\right)}{{\delta }_q{z}^2}{d}_{q}w\hfill \\ \hfill =& a_0(1-z)+b_{0}z-f\left(z\right).\hfill \end{array}$$

(28)

Now, by substituting (28) in (27), we obtain the required result.  ☐

4. Certain $\mathit{q}$-Fourier Expansions

In this section, we consider the q-trigonometric functions $C_q\left(z\right)$ and $S_q\left(z\right)$ which are defined in (12). Our aim is to obtain the q-Fourier expansions of certain q-integral transforms involving the Green’s functions ${\tilde{G}}_n(z,qt)$ defined in Section 3.

Recall that the q-Fourier series expansion for $f\left(x\right)=1$ and $g\left(x\right)=x$ are given [13,16] by

$$\begin{array}{cc}\hfill 1=& 2\sum _{k=1}^{\infty }\frac{1-C_q\left(q^{1/2}{w}_k\right)}{w_k{C}_q\left(q^{1/2}{w}_k\right)S_q^{\prime }\left(w_k\right)}{S}_q\left(q{w}_{k}x\right),x\in A_q^{*},\hfill \\ \hfill x=& -\frac{1}{q}\sum _{k=1}^{\infty }\frac{2}{w_k{S}_q^{\prime }\left(w_k\right)}{S}_q\left(q{w}_{k}x\right),\hfill \end{array}$$

(29)

where $\{w_k:k\in \mathbb{N}\}$ is the set of positive zeroes of $S_q\left(z\right)$.

Lemma 2.

Let $z\in A_q^{*}$. Then

$${\int }_0^1\tilde{G}(z,qt)S_q\left(q{w}_{k}t\right)d_{q}t=-\frac{1}{w_k^2}{S}_q\left(w_{k}z\right).$$

Proof. 

From (17), we get

$$\begin{array}{c}{\int }_0^1\tilde{G}(z,qt)S_q\left(q{w}_{k}t\right)d_{q}t=\sqrt{q}(1-z){\int }_0^z(-qt)S_q\left(q{w}_{k}t\right)d_{q}t\\ -\sqrt{q}z{\int }_z^1(1-qt)S_q\left(q{w}_{k}t\right)d_{q}t.\end{array}$$

(30)

Using q-integration by parts (11), we obtain

$$\begin{array}{c}\hfill {\int }_0^z(-qt)S_q\left(q{w}_{k}t\right)d_{q}t=\frac{z}{\sqrt{q}{w}_k}{C}_q\left(\frac{w_{k}z}{\sqrt{q}}\right)-\frac{1}{\sqrt{q}{w}_k^2}{S}_q\left(w_{k}z\right),\end{array}$$

(31)

$$\begin{array}{c}\hfill {\int }_z^1(1-qt)S_q\left(q{w}_{k}t\right)d_{q}t=\frac{(1-z)}{\sqrt{q}{w}_k}{C}_q\left(\frac{w_{k}z}{\sqrt{q}}\right)+\frac{1}{\sqrt{q}{w}_k^2}{S}_q\left(w_{k}z\right).\end{array}$$

(32)

Substituting from (31) and (32) into (30), we have the required result.  ☐

Lemma 3.

For $z\in A_q^{*}$, the following q-Fourier series expansion holds:

$${\int }_0^1\tilde{G}(z,qt)d_{q}t=-\sum _{k=1}^{\infty }\frac{L_k}{w_k^2}{S}_q\left(w_{k}z\right),$$

(33)

where

$$L_k:=\frac{2-2C_q\left(q^{1/2}{w}_k\right)}{w_k{C}_q\left(q^{1/2}{w}_k\right)S_q^{\prime }\left(w_k\right)}.$$

Proof. 

According to (29), we have

$$1=2\sum _{k=1}^{\infty }\frac{1-C_q\left(q^{1/2}{w}_k\right)}{w_k{C}_q\left(q^{1/2}{w}_k\right)S_q^{\prime }\left(w_k\right)}{S}_q\left(q{w}_{k}t\right),t\in A_q^{*}.$$

(34)

Multiplying (34) by $\tilde{G}(z,qt)$, and integrating with respect to t from zero to unity, we get

$$\begin{array}{c}{\int }_0^1\tilde{G}(z,qt)d_{q}t=\\ 2{\displaystyle \sum _{k=1}^{\infty }}\frac{1-C_q\left(q^{1/2}{w}_k\right)}{w_k{C}_q\left(q^{1/2}{w}_k\right)S_q^{\prime }\left(w_k\right)}{\int }_0^1\tilde{G}(z,qt)S_q\left(w_{k}qt\right)d_{q}t.\end{array}$$

(35)

By setting ${\displaystyle L_k:=\frac{2-2C_q\left(q^{1/2}{w}_k\right)}{w_k{C}_q\left(q^{1/2}{w}_k\right)S_q^{\prime }\left(w_k\right)}}$ and using Lemma 2, we obtain the result.  ☐

Theorem 2.

For $z\in A_q^{*}$, the following q-Fourier series expansion holds:

$${\int }_0^1{\tilde{G}}_n(z,qt)d_{q}t={(-1)}^n\sum _{k=1}^{\infty }\frac{L_k}{w_k^{2n}}{S}_q\left(w_{k}z\right).$$

(36)

Proof. 

We prove the result by mathematical induction with respect to n. We first observe that for $n=1$, the Formula (36) reduces to the formula in Lemma 3; that is, Equation (36) is true for $n=1$.

Next, assume that (36) is true for some $n\ge 2$. Then

$$\begin{array}{cc}\hfill {\int }_0^1{\tilde{G}}_{n+1}(z,qt)d_{q}t=& {\int }_0^1{\int }_0^1\tilde{G}(z,qy){\tilde{G}}_n(qy,qt)d_{q}y{d}_{q}t\hfill \\ \hfill =& {\int }_0^1\tilde{G}(z,qy)\left[{\int }_0^1{\tilde{G}}_n(qy,qt)d_{q}t\right]d_{q}y\hfill \\ \hfill =& {(-1)}^n\sum _{k=1}^{\infty }\frac{L_k}{w_k^{2n}}{\int }_0^1\tilde{G}(z,qy)S_q\left(w_{k}qy\right)d_{q}y\hfill \\ \hfill =& {(-1)}^n\sum _{k=1}^{\infty }\frac{L_k}{w_k^{2n}}\left[\frac{-1}{w_k^2}{S}_q\left(w_{k}z\right)\right]\hfill \\ \hfill =& {(-1)}^{n+1}\sum _{k=1}^{\infty }\frac{L_k}{w_k^{2(n+1)}}{S}_q\left(w_{k}z\right).\hfill \end{array}$$

 ☐

Lemma 4.

For $z\in A_q^{*}$, the following q-Fourier series expansion holds:

$${\int }_0^1\left(qt\right)\tilde{G}(z,qt)d_{q}t=2\sum _{k=1}^{\infty }\frac{1}{w_k^3{S}_q^{\prime }\left(w_k\right)}{S}_q\left(w_{k}z\right),k\in \mathbb{N}.$$

(37)

Proof. 

Consider the function $g\left(t\right)=t$. From (29), we have

$$t=-\frac{1}{q}\sum _{k=1}^{\infty }\frac{2}{w_k{S}_q^{\prime }\left(w_k\right)}{S}_q\left(q{w}_{k}t\right),0<t<1.$$

(38)

Hence, the proof can be performed by using (38) similar to the proof of Lemma 3.  ☐

Theorem 3.

For $z\in A_q^{*}$, the following q-Fourier series expansion holds:

$${\int }_0^1\left(qt\right){\tilde{G}}_n(z,qt)d_{q}t={(-1)}^n\sum _{k=1}^{\infty }\frac{2}{w_k^{2n+1}{S}_q^{\prime }\left(w_k\right)}{S}_q\left(w_{k}z\right).$$

(39)

Proof. 

The proof can be performed by induction similar to the proof of Theorem 2. So, we omit it.  ☐

5. Fourier Series Expansions of the $\mathit{q}$-Lidstone Polynomials

The Fourier expansion of special polynomials has been studied by some mathematicians; see [17,18,19,20]. In this section, we consider the q-Lidstone polynomials ${\tilde{A}}_n\left(z\right)$ and ${\tilde{B}}_n\left(z\right)$ defined in (2). We define these polynomials by using the Green’s functions ${\tilde{G}}_n(z,qt)$ defined in (17) and (22) and then, we introduce the q-Fourier Series Expansions for them.

We begin with the following result from [7]:

Lemma 5.

For $n\in \mathbb{N}$, the q-polynomials ${\tilde{A}}_n\left(z\right)$ and ${\tilde{B}}_n\left(z\right)$ satisfy the q-difference equations

$$\frac{{\delta }_q^2{\tilde{A}}_n\left(z\right)}{{\delta }_q{z}^2}={\tilde{A}}_{n-1}\left(z\right)and\frac{{\delta }_q^2{\tilde{B}}_n\left(z\right)}{{\delta }_q{z}^2}={\tilde{B}}_{n-1}\left(z\right),$$

with the boundary conditions ${\tilde{A}}_n\left(0\right)={\tilde{A}}_n\left(1\right)=0={\tilde{B}}_n\left(0\right)={\tilde{B}}_n\left(1\right)=0$. Moreover,

$${\tilde{A}}_0\left(z\right)=z,{\tilde{B}}_0\left(z\right)=z-1.$$

We have the following:

Proposition 1.

The q-Lidstone polynomials ${\tilde{A}}_n$ and ${\tilde{B}}_n$ can be expressed as ${\tilde{A}}_0\left(z\right)=z$, ${\tilde{B}}_0\left(z\right)=z-1$, and for $n\in \mathbb{N}$

$$\begin{array}{cc}\hfill {\tilde{A}}_n\left(z\right)& =q{\int }_0^{1}t{\tilde{G}}_n(z,qt)d_{q}t,\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\tilde{B}}_n\left(z\right)& ={\int }_0^1(qt-1){\tilde{G}}_n(z,qt)d_{q}t,\hfill \end{array}$$

(41)

where

$$\begin{array}{c}\tilde{G}(z,t):={\tilde{G}}_1(z,t)=\left\{\begin{array}{cc}\sqrt{q}z(t-1),\hfill & 0\le z<t\le 1;\hfill \\ \\ \sqrt{q}t(z-1),\hfill & 0\le t<z\le 1.\hfill \end{array}\right.\\ {\tilde{G}}_n(z,qt)={\int }_0^1\tilde{G}(z,qw){\tilde{G}}_{n-1}(qw,qt)d_{q}w(n=2,3,\dots ).\end{array}$$

(42)

Proof. 

We use the induction on n. By Lemma 5, we have

$$\left\{\begin{array}{c}\frac{{\delta }_q^2{\tilde{A}}_n\left(z\right)}{{\delta }_q{z}^2}={\tilde{A}}_{n-1}\left(z\right)(n\in \mathbb{N}),\hfill \\ \\ {\tilde{A}}_n\left(0\right)={\tilde{A}}_n\left(1\right)=0.\hfill \end{array}\right.$$

(43)

So, if $n=1$ we get the q-boundary value problem

$$\left\{\begin{array}{c}\frac{{\delta }_q^2{\tilde{A}}_1\left(z\right)}{{\delta }_q{z}^2}=z(z\in A_q^{*}),\hfill \\ \\ {\tilde{A}}_1\left(0\right)={\tilde{A}}_1\left(1\right)=0.\hfill \end{array}\right.$$

(44)

According to Lemma 1, we have the result.

Next, assume that (40) is true for $n\ge 1$. According to Remark (1), the solution ${\tilde{A}}_{n+1}\left(z\right)$ of the q-boundary value problem

$$\left\{\begin{array}{c}\frac{{\delta }_q^2{\tilde{A}}_{n+1}\left(z\right)}{{\delta }_q{z}^2}={\tilde{A}}_n\left(z\right),\hfill \\ \\ {\tilde{A}}_{n+1}\left(0\right)={\tilde{A}}_{n+1}\left(1\right)=0,\hfill \end{array}\right.$$

(45)

is given by

$$\begin{array}{cc}\hfill {\tilde{A}}_{n+1}\left(z\right)& ={\int }_0^1\tilde{G}(z,qw){\tilde{A}}_n\left(qw\right)d_{q}w\hfill \\ & ={\int }_0^1\tilde{G}(z,qw)\left[{\int }_0^{1}qt{\tilde{G}}_n(qw,qt)d_{q}t\right]d_{q}w\hfill \\ & ={\int }_0^1\left[{\int }_0^{1}qt\tilde{G}(z,qw){\tilde{G}}_n(qw,qt)d_{q}w\right]d_{q}t\hfill \\ & ={\int }_0^{1}qt{\tilde{G}}_{n+1}(z,qt)d_{q}t.\hfill \end{array}$$

Similarly, we can prove Equation (41). Finally, by induction on n$(n\ge 2)$ again, it is easy to see that

$${\tilde{G}}_n(z,qt)={\int }_0^1{\tilde{G}}_{n-1}(z,qw)\tilde{G}(qw,qt)d_{q}w.$$

 ☐

The following result offers the explicit representation of the interpolating q-Lidstone polynomials and the associated error function $R_n\left(z\right)$.

Theorem 4.

Let $0<q<1$ and $f\in C_q^2\left(A_q^{*}\right)$. Then

$$f\left(z\right)=\sum _{k=0}^{n-1}\left[\frac{{\delta }_q^{2k}f\left(1\right)}{{\delta }_q{z}^{2k}}{\tilde{A}}_k\left(z\right)+\frac{{\delta }_q^{2k}f\left(0\right)}{{\delta }_q{z}^{2k}}{\tilde{B}}_k\left(z\right)\right]+R_n\left(z\right),$$

(46)

where

$$R_n\left(z\right)={\int }_0^1{\tilde{G}}_n(z,qt)\frac{{\delta }_q^{2n}f\left(qz\right)}{{\delta }_q{z}^{2n}}{d}_{q}t.$$

Proof. 

The proof follows immediately from Theorem 1 and Proposition 1, if we replace $a_k$, $b_k$ and $\varphi \left(z\right)$ in Equation (24) by their values in terms of $f\left(z\right)$ as given by the system (23).  ☐

Proposition 2.

For $z\in A_q^{*}$ and $n\in \mathbb{N}$, the Fourier series for q-Lidstone polynomials ${\tilde{A}}_n\left(z\right)$ and ${\tilde{B}}_n\left(z\right)$ are given by

$$\begin{array}{ccc}\hfill {\tilde{A}}_n\left(z\right)& =& {(-1)}^n\sum _{k=1}^{\infty }\frac{2}{w_k^{2n+1}{S}_q^{\prime }\left(w_k\right)}{S}_q\left(w_{k}z\right),\hfill \end{array}$$

(47)

$$\begin{array}{ccc}\hfill {\tilde{B}}_n\left(z\right)& =& {(-1)}^n\sum _{k=1}^{\infty }\frac{2}{w_k^{2n+1}{S}_q^{\prime }\left(w_k\right)C_q\left(q^{1/2}{w}_k\right)}{S}_q\left(w_{k}z\right),\hfill \end{array}$$

(48)

where $\{w_k:k\in \mathbb{N} with w_1<w_2<w_3<\dots \}$ is the set of positive zeroes of $S_q\left(z\right)$.

Proof. 

By using Equation (40) and Theorem 3 we get (47). Similarly, Equation (48) follows immediately from (41), (36) and (37).  ☐

We end this section by determining the asymptotic behavior of ${\tilde{A}}_n\left(z\right)$ and ${\tilde{B}}_n\left(z\right)$ for large n.

Proposition 3.

Let $z\in A_q^{*}$. Then, there exist some constants $K_q$ and $L_q$ such that

$$\begin{array}{ccc}\hfill & & |{(-1)}^n{\tilde{A}}_n\left(z\right)-\frac{2S_q\left(w_{1}z\right)}{w_1^{2n+1}{S}_q^{\prime }\left(w_1\right)}|<\frac{K_q}{w_1^{2n}},\hfill \end{array}$$

(49)

$$\begin{array}{ccc}\hfill & & |{(-1)}^n{B}_n\left(z\right)-\frac{2S_q\left(w_{1}z\right)}{w_1^{2n+1}{S}_q^{\prime }\left(w_1\right)C_q\left(\sqrt{q}{w}_1\right)}|<\frac{L_q}{w_1^{2n}},\hfill \end{array}$$

(50)

where $w_1$ is the smallest positive zero of $S_q\left(z\right)$.

Proof. 

From Equation (47), we get

$$|{(-1)}^n{\tilde{A}}_n\left(z\right)-\frac{2}{w_1^{2n+1}{S}_q^{\prime }\left(w_1\right)}{S}_q\left(w_{1}z\right)|=\left|\sum _{k=2}^{\infty }\frac{2}{w_k^{2n+1}{S}_q^{\prime }\left(w_k\right)}{S}_q\left(w_{k}z\right)\right|.$$

Since the function $S_q(.)$ is bounded on $A_q^{*}$, there exists a constant $M>0$ such that

$$\begin{array}{ccc}& |& \sum _{k=2}^{\infty }\frac{2}{w_k^{2n+1}{S}_q^{\prime }\left(w_k\right)}{S}_q\left(w_{k}z\right)|\hfill \\ & <& \frac{M}{w_2^{2n+1}{S}_q^{\prime }\left(w_2\right)}\left[1+{\left(\frac{w_2}{w_3}\right)}^{2n+1}\frac{S_q^{\prime }\left(w_2\right)}{S_q^{\prime }\left(w_3\right)}+{\left(\frac{w_2}{w_4}\right)}^{2n+1}\frac{S_q^{\prime }\left(w_2\right)}{S_q^{\prime }\left(w_4\right)}+\dots \dots \right].\hfill \end{array}$$

Note that $w_1<w_2<\dots$, this implies the series in brackets tends to unity when $n\to \infty$. Set $K_q=\frac{M}{w_1{S}_q^{\prime }\left(w_2\right)}$, we get (49). Inequality (50) can be proved in the same manner by using Equation (48).  ☐

6. Conclusions and Future Work

In this paper, we have introduced some definitions of the q-Lidstone polynomials which are q-Bernoulli polynomials generated by the third Jackson q-Bessel function, based on the Green’s function of the q-difference equation

$$\left\{\begin{array}{c}\frac{{\delta }_q^{2n}f\left(z\right)}{{\delta }_q{z}^{2n}}=\varphi \left(z\right),\hfill \\ \\ \frac{{\delta }_q^{2k}f\left(0\right)}{{\delta }_q{z}^{2k}}=a_k,\frac{{\delta }_q^{2k}f\left(1\right)}{{\delta }_q{z}^{2k}}=b_k(k=0,1,\dots ,n-1).\hfill \end{array}\right.$$

(51)

New results are obtained; particularly the q-Fourier series expansions of these functions.

Another study to give a characterization of those functions on the plane given by absolutely convergent of q-Lidstone series expansion (1), using the results in Section 5, is in progress.