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

: 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.


Introduction
In 1929, Lidstone [1] introduced a generalization of Taylor's series that approximates an entire function f (z) of exponential type less than π in a neighborhood of two points instead of one: where the set {A n (z)} n called Lidstone polynomials. In [2], Whittaker proved that where B n (x) is the Bernoulli polynomial of degree n, which may be defined by the generating function te xt e t − 1 = ∞ ∑ n=0 B n (x) 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 δ q (see Section 2), that is where A n (z) and B n (z) are the q-Lidstone polynomials defined by the generating functions exp q (zw) − exp q (−zw) exp q (w) − exp q (−w) = ∞ ∑ n=0 A n (z) w 2n , Moreover, it turns out that where B n (z; q) are q-Bernoulli polynomials generated by and the function exp q (.) is the q-exponential function which has the series representation In this paper, we assume that q is a positive number less than one and the set A * q is defined by where N 0 := {0, 1, 2, . . .}. We present the q-Lidstone polynomials A n (z) and B n (z) based on the Green's function of a q-boundary value problem where f and φ 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 A n (z) and B n (z) for large n.

Preliminaries
Recall that the q-derivative D q of the function f is defined by and the q-derivative at zero is defined to be f (0) if it exists, see [11]. The q-shifted fractional (a; q) n of a ∈ C is defined by (a; q) 0 := 1 and (a; q) n := n ∏ j=0 (1 − aq j ), for n ∈ N, and the q-number factorial [n] q ! is defined for q = 1 by Jackson [12] introduced the following integral, as a right inverse of the q-derivative provided that the series converges at z = a and z = b. We can interchange the order of double q-integral by The symmetric q-difference operator δ q which is acting on a function f defined by (see [11,13]). From (7) and (9), it follows A function f defined on A * q is called q-regular at zero if it satisfies The q-integration by parts rule on A * q (see [13]) is where f and g are complex valued q-regular functions at zero.
We will use a q-exponential function exp q (.) defined in (5) and the q-linear sine and cosine, S q (z) and C q (z), which defined by They can be written in terms of the third Jackson q-Bessel function J These functions satisfy see [11,13]. We denote to the derivative of S q (z) by S q (z) and we assume that {w k : k ∈ N with w 1 < w 2 < w 3 < . . .} is the set of positive zeroes of S q (z).

Existence Solutions of q-Differential System
In this section, we construct the solution of the q-differential system (51). Let C n q (A * q ) denote the space of all continues functions with continuous q-derivatives up to order n − 1 on A * q with values in R.
Then, the solution of the q-differential equation subject to the boundary conditions f (0) = f (1) = 0 is equivalent to the basic Fredholm qintegral equation where G(z, t) is the Green's function defined on A * q by Proof. The q-differential Equation (15) can be written as By taking double q-integral for (18) and using (8), we obtain where c 0 and c 1 are arbitrary constant. Using the boundary conditions, we get c 0 = 0 and Substituting in (19), we have and then we obtain (16).

Remark 1. By induction on n, one can verify that
, then the function is the solution of the q-boundary value problem where G 1 (z, t) is the Green's function defined as in (17) and Theorem 1. If f (z) and φ(z) are functions of class C 2n q (A * q ), then any solution of the system is given by where the functions G n (z, qt) (n ∈ N) defined as in (17) and (22).

Proof.
From (17), (22) and Equation (23) we get Using the rule (11), after some simplifications, we obtain Repeating the q-integration by parts on the last q-integral of Equation (26) (n − 1) times, we get Computing the last integral of (27), we get Now, by substituting (28) in (27), we obtain the required result.

Certain q-Fourier Expansions
In this section, we consider the q-trigonometric functions C q (z) and S q (z) which are defined in (12). Our aim is to obtain the q-Fourier expansions of certain q-integral transforms involving the Green's functions G n (z, qt) defined in Section 3.
Recall that the q-Fourier series expansion for f (x) = 1 and g(x) = x are given [13,16] by where {w k : k ∈ N} is the set of positive zeroes of S q (z).

Lemma 3.
For z ∈ A * q , the following q-Fourier series expansion holds: where Proof. According to (29), we have Multiplying (34) by G(z, qt), and integrating with respect to t from zero to unity, we get and using Lemma 2, we obtain the result.

Theorem 2.
For z ∈ A * q , the following q-Fourier series expansion holds: 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 ≥ 2. Then

Lemma 4.
For z ∈ A * q , the following q-Fourier series expansion holds: Proof. Consider the function g(t) = t. From (29), we have Hence, the proof can be performed by using (38) similar to the proof of Lemma 3.

Theorem 3.
For z ∈ A * q , the following q-Fourier series expansion holds: Proof. The proof can be performed by induction similar to the proof of Theorem 2. So, we omit it.

Fourier Series Expansions of the 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 A n (z) and B n (z) defined in (2). We define these polynomials by using the Green's functions 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 ∈ N, the q-polynomials A n (z) and B n (z) satisfy the q-difference equations with the boundary conditions A n (0) = A n (1) = 0 = B n (0) = B n (1) = 0. Moreover, We have the following: Proposition 1. The q-Lidstone polynomials A n and B n can be expressed as A 0 (z) = z, B 0 (z) = z − 1, and for n ∈ N A n (z) = q where G n (z, qt) = 1 0 G(z, qw) G n−1 (qw, qt) d q w (n = 2, 3, . . .). (42) Proof. We use the induction on n. By Lemma 5, we have So, if n = 1 we get the q-boundary value problem According to Lemma 1, we have the result. Next, assume that (40) is true for n ≥ 1. According to Remark (1), the solution A n+1 (z) of the q-boundary value problem is given by Similarly, we can prove Equation (41). Finally, by induction on n (n ≥ 2) again, it is easy to see that G n (z, qt) = 1 0 G n−1 (z, qw) 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 (z). Theorem 4. Let 0 < q < 1 and f ∈ C 2 q (A * q ). Then where Proof. The proof follows immediately from Theorem 1 and Proposition 1, if we replace a k , b k and φ(z) in Equation (24) by their values in terms of f (z) as given by the system (23).

Proposition 2.
For z ∈ A * q and n ∈ N, the Fourier series for q-Lidstone polynomials A n (z) and B n (z) are given by where {w k : k ∈ N with w 1 < w 2 < w 3 < . . .} is the set of positive zeroes of S q (z).
Proof. By using Equation (40)  We end this section by determining the asymptotic behavior of A n (z) and B n (z) for large n. Proposition 3. Let z ∈ A * q . Then, there exist some constants K q and L q such that where w 1 is the smallest positive zero of S q (z).
Proof. From Equation (47), we get S q (w k z) .
Since the function S q (.) is bounded on A * q , there exists a constant M > 0 such that Note that w 1 < w 2 < . . ., this implies the series in brackets tends to unity when n → ∞. Set K q = M w 1 S q (w 2 ) , we get (49). Inequality (50) can be proved in the same manner by using Equation (48).

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 δ 2k q f (0) δ q z 2k = a k , δ 2k q f (1) δ q z 2k = b k (k = 0, 1, . . . , n − 1).
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.