A $q$-Type $k$-Lidstone series for entire functions

1.   Introduction and preliminaries

A Lidstone series provides a generalization of Taylor's theorem that approximates an entire function $f$ of exponential type less than $\pi$ in a neighborhood of two points instead of one:

where the polynomials $(A_n(z))_n$ are called Lidstone polynomials (see ^[14]).

Several authors including Boas ^[5,6], Poritsky ^[19], Schoenberg ^[22], Whittaker ^[23], and Widder ^[24] gave necessary and sufficient conditions for representation of functions by Lidstone series (1.1).

In ^[13], Leeming and Sharma introduced an extension of Lidstone series. They proved that for a given integer $k\geq 2$, the following representation holds for a certain class of entire functions:

where $(C_{kn}(z))_n$ and $(A_{kn+v}(z))_n$ are certain polynomials which they called the fundamental polynomials of the series defined on the right-hand side of (1.2).

Recently, Ismail and Mansour ^[11] introduced a $q$-analog of the Lidstone expansion theorem for a certain class of entire functions as in the following formula:

where $(A_n)_n$ and $(B_n)_n$ are the $q$-Lidstone polynomials defined by the generating functions

respectively, and $E_q(\cdot)$ is one of Jackson's $q$-exponential function defined by

On the other hand, Al-Towailb ^[3] has constructed another $q$-type Lidstone theorem by expanding a class of entire functions in terms of $q$-derivatives of even orders at 0 and $q$-derivatives of odd orders at $1$. Also, in ^[3], we proved that

where $f$ is an entire function satisfying some prescribed conditions, the sequences $(r_n)_n$ and $(s_n)_n$ are two sequences of non-negative integers, and $\{\pi_n(z; q), \zeta_n(z; q)\}_n$ are the set of polynomials (called a $q^{-1}$-standard set) that satisfies the following conditions:

where $\delta_{n, k}$ is the Kronecker delta $(k\in \mathbb{N})$. In particular, the set of polynomials $\{A_n(z), B_n(z) \}_n$ which defined in (1.4) and (1.5) form a $q^{-1}$-standard set of polynomials in relation to the pair of sequences $(r_n; s_n) = (2n;2n)_{n\in \mathbb{N}_0}$.

For details and more results to the $q$-Lidstone's theorem, we also refer the reader to ^[2,4,15,16].

Our aim here is to introduce another extension of $q$-Lidstone series, which will be called $q$-type $k$-Lidstone series, and determine the class of functions for which this series is valid, to obtain a $q$-analog of Leeming and Sharma's result. Furthermore, we consider the problem of expanding an entire function in the $q$-type $3$-Lidstone series. These results will be derived by using Cauchy's integral formula and complex contour integration.

Throughout this paper, we assume that $q$ is a positive number less than one and $\mathbb{N}$ is the set of positive integers. We follow Gasper and Rahman ^[9] for the definitions, notations and properties of the $q$-shifted factorials $(a; q)_n$, $q$-gamma function $\Gamma_q(n)$, $q$-numbers $[n]_q$ and $q$-factorial $[n]_q!$.

The Jackson's $q$-derivative of a function $f$ is defined by

and $D_qf(0)$ is usually defined as $f'(0)$ if $f$ is differentiable at zero (see ^[9]).

We start by stating some definitions in Section 2 and introduce a $q$ analog of the generalized circular functions of order $k \; (k\in \mathbb{N})$, which we need in our investigation. In Section 3, we state and prove the principle theorem, and define the fundamental polynomials of a $q$-type $k$-Lidstone series. Then, we present some properties of these polynomials. Section 4 studies the problem of expanding an entire function in the $q$-type $3$-Lidstone series. Also, we give six tables that deal with the generating functions of the fundamental polynomials associated with the six kinds of $q$-type $3$-Lidstone series.

2.   $q$-analogs of circular functions of high orders

$q$-analogs of the trigonometric functions $\sin z$ and $\cos z$ are defined by

respectively, where $E_q(z)$ is defined as in (1.6). $q$-analogs of the hyperbolic functions $\text{Sinh}_q z$ and $\text{Cosh}_q z$ are defined by

In 1948, Mikusinski ^[17] introduced the generalized circular functions of order $k \; (k\in \mathbb{N})$ by

Note that there exists a relationship between these functions and the Mittag-Leffler function

(see [8, Section 18.1]), that is

We consider the following $q$-special functions $M_{k, j}(z; q)$ and $N_{k, j}(z; q) \; (k\in \mathbb{N})$, which are $q$-analogs of the functions (2.3) and (2.4), respectively.

Observe that $M_{1, 0}(z; q) = E_q(-z)$, $M_{2, 0}(z; q) = \text{Cos}_q z$, and $M_{2, 1}(z; q) = \text{Sin}_q z$. Also, it is easy to conclude that

Remark 2.1. The function $N_{k, j}(z; q)$ is a special case of the big $q$-Mittag-Leffler function which is introduced in ^[21], and defined by

where $q, z, c, \alpha, \beta \in \mathbb{C}; \, \Re(\alpha) > 0, \, \Re(\beta) > 0$ and $|q| < 1$. More precisely,

Proposition 2.2. Let $k\in \mathbb{N}$, $j = 0, 1, \ldots, k-1$, and $\omega = \exp(2\pi i/k)$. Then, the following results hold:

Proof. To prove Eq (2.8), we use (1.6) to obtain

Since $\omega = \exp(2\pi i/k)$, then $\omega^k = 1$ and $1+\omega+\omega^2+\ldots+\omega^{k-1} = 0$. Therefore,

We obtain the required result by substituting from (2.12) into (2.11) and then multiplying (2.11) with $\omega^{-(j/2)}$.

Formula (2.9) follows immediately from definitions (2.5) and (2.6). Finally, we get (2.10) and complete the proof from the results (2.8) and (2.9).

Now, we consider the following boundary value problems:

and the adjoint problem:

Then, the real eigenvalues $(\lambda_m)_{m = 1}^{\infty}$ are zeros of the $q$-circular function $M_{k, k-1}(x; q)$ (defined in Eq (2.5)). The eigenfunctions of Problem (2.13) are

and the eigenfunctions of Problem (2.14) are $\{\widetilde{M}_{k, k-1}(x, \lambda_k; q)\}_{m = 1}^{\infty}$, where in general

Notice, Ismail in ^[10] defined a $q$-translation operator by

and acts on polynomials as a linear operator. Therefore, one can verify that

We use $\widetilde{M}_{k, j}(x; q)$ to denote $\widetilde{M}_{k, j}(x, 1;q)$. One can also verify that

In the following, we construct the addition formula of the $q$-circular functions. For this, we define the function

One can verify that $K_{k, j}(x, \lambda, 1;q) = M_{k, j}(\lambda x), \, \, K_{k, j}(0, \lambda, y; q) = \widetilde{M}_{k, j}(y, \lambda; q)$. Moreover,

Theorem 2.3. The following result hold for $j = 0, 1, \ldots, k-1$.

Proof. For fixed $y$, the functions $\{y_j = K_{k, j}(x, \lambda, y)\}_{j = 0}^{k-1}$ form a fundamental set of solutions of the initial value problem

Therefore, there exist some constants $c_n \; (0\leq n\leq k-1)$ such that

Hence, for $r\in\{0, 1, \ldots, k-1\}$

If we set $x = 0$ on the previous identity, we get

Theorem 2.4. The following biorthogonal property holds:

where $\delta_{j, m}$ is the Kronecker's delta, and $(\lambda_m)_{m = 1}^{\infty}$ are the set of the real zeros of the function $M_{k, k-1}(x; q)$.

Proof. We set $y(x) = M_{k, k-1}(\lambda x; q)$, $z(x) = \widetilde{M}_{k, k-1}(y, \lambda; q)$. Then, we have

Consequently,

Applying the $q$-integration by parts on Eq (2.17) $j$ times, we obtain

That is,

But

Set $(x*y)^n: = \sum_{k = 0}^{n}{{n} \brack {k}}_qq^{k\choose 2} x^k (-y)^{n-k}(1/y; q)_{n-k}$. Then

Hence, $K_{k, k-2}(x, \lambda, x;q) = M_{k, k-2}(\lambda; q)$, and then we obtain

3.   $q$-type $k$-Lidstone expansion theorem

Recall that $\Psi$ is a comparison function if $\Psi(t) = \sum_{n = 0}^{\infty}\Psi_n t^n$ and $\Psi_n > 0\; (n\in\mathbb{N}_0)$ such that $\Big(\Psi_{n+1}/\Psi_n\Big)$ is a decreasing sequence that converges to zero (see ^[6]). We denote by $\mathcal{R}_{\Psi}$, the class of all entire functions $f$ such that, for some numbers $\tau$,

as $r\rightarrow \infty$. The infimum of the numbers $\tau$ for which (3.1) holds is the $\Psi$-type of the function $f$. This type can be computed by applying Nachbin's theorem ^[18] which states that a function $f(z) = \sum_{n = 0}^{\infty} f_n z^n$ is of $\Psi$-type $\tau$ if and only if

We will use the following result from [6, Theorem 2.9].

Theorem 3.1. Let $\Psi$ be a comparison function and $f$ is a function in the class $\mathcal{R}_{\Psi}$. Suppose that

If $D(f)$ is a closed set consisting of the union of all singular points of $F$ and all points exterior to the domain of $F$, then

where $\Gamma$ encloses $D(f)$ and

Ramis ^[20] defined an entire function $f$ to have a $q$-exponential growth $(q > 1)$ of order $\gamma$ and a finite type if there exist positive numbers $\alpha$ and $K > 0$ such that

Also, from [20, Lemma 2.2], if the series $f(z): = \sum_{n = 0}^{\infty}a_n z^n$ satisfies (3.2), then

Proposition 3.2. Let $\mu_1$ be the zero with the smallest positive absolute magnitude of $N_{k, k-1}(w; q)$, defined in (2.5), and let $w$ be a complex number such that $|w| < |\mu_1|$. Assume that

where $E_q$ is the $q$-exponential function defined in (1.6). Then, for $j\in\{0, 1, \ldots, k-2\}$:

where $N_{k, j}(z; q)$ are the functions defined in (2.6).

Proof. Replace $w$ in Eq (3.4) by $\omega w, \omega^2 w, \ldots, \omega^{k-1} w$, with $\omega^{k} = 1\, (\omega\neq 1)$, and note that the functions $\varphi$ and $\psi_j$ remain unchanged. Then, we obtain the following system of $k$ equations:

where $n\in\{0, 1, \ldots, k-1\}$. If we multiply Eq (3.6) by $\omega^{-nj}$ and then adding these equations for $n\in\{0, 1, \ldots, k-1\}$, we obtain

Therefore, from (2.10), we get

Thus, the result follows at once from (2.12).

Obviously, $\varphi(z, w^k)$ and $w^j \psi_j(z, w^k)$ are analytic functions for $|w| < |\mu_1|$.

According to the above results, we can prove the following main theorem.

Theorem 3.3. Let $\mu_1$ be the zero with the smallest positive absolute magnitude of $N_{k, k-1}(w; q)$, defined in (2.5). If $f$ is an entire function of $q^{-1}$-exponential growth of order less than $1$, or of order $1$ and a finite type $\alpha$ such that

then, for all $z\in \mathbb{C}$, the following representation holds

where $A_{kn}(z)$ and $B_{kn+j}(z)$ are the polynomials defined by the following generating functions:

and $w$ is a complex number such that $|w| < \rvert\mu_1\rvert$. Furthermore, the series on right-hand side of (3.8) converges to $f(z)$ uniformly on all compact subsets of the plane.

Proof. We apply Theorem 3.1. Set $\Psi(z) = E_q(z)$ and $f(z): = \sum_{n = 0}^{\infty}a_n z^n$. Then

is decreasing and vanishes at $\infty$. Since $\Psi(z)$ has a $q^{-1}$-exponential growth of order $1$, then Eq (3.1) holds if $f$ is a function of $q^{-1}$-exponential growth $\gamma$, $\gamma\leq 1$. Then from (3.3), the type $\tau$ of the function $f$, $\tau\geq 0$ of the function $f$ has the upper bound

Consequently, if $\gamma < 1$, then $\tau = 0$, if $\gamma = 1$, then $\tau\leq \frac{q^{\frac{1}{2}- \alpha}}{1-q}$. So, $D(f)$ lies in the closed disk $|w|\leq \tau \leq \dfrac{q^{\frac{1}{2}- \alpha}}{1-q} < \mu_1$ and we choose $\Gamma$ to be the circle $|w| = \tau+\epsilon < \mu_1$, $\epsilon > 0$ which encloses $D(f)$. Note that the inequality $\frac{q^{\frac{1}{2}- \alpha}}{1-q} < |\mu_1|$ satisfies the condition (3.7) on the type of $f(z)$. Then, we obtain

Therefore,

By setting

and using Proposition 3.2, we have

Finally, from the definitions of $\phi$ and $\psi_j$, we have that the right-hand side of (3.4) is analytic in the disk $|w| < |\mu_1|$. Therefore, the series defined by $\phi$ and $\psi_j$ converges uniformly in every compact subset of the disk $|w| < |\mu_1|$.

We will say that the formula (3.8) is $q$-type of $k$-Lidstone series of the function $f$, and the functions $A_{kn}(z)$ and $B_{kn+j}(z) \, (j = 0, 1, \ldots, k-2)$ are the fundamental polynomials of this series.

Remark 3.4. If we set $k = 2$ in Eq (3.8), we have the $q$-Lidstone series expansion (1.3).

The following result gives some properties of the fundamental polynomials of the $q$-type $k$-Lidstone series.

Proposition 3.5. For $n\in\mathbb{N}$, $k\geq 2$, and $j\in\{0, 1, 2, \ldots, k-2\}$, the fundamental polynomials of the $q$-type of $k$-Lidstone series, $A_{kn}(x)$ and $B_{kn+j}(x)$, satisfy the following properties:

(i) $A_0(x) = x^{k-1}$ and $B_0(x) = 1- x^{k-1};$

(ii) $D^k_{q^{-1}}A_{kn}(x) = A_{k(n-1)}(x)$ and $D^k_{q^{-1}}B_{kn+j}(x) = B_{k(n-1)+j}(x);$

(iii) $A_{kn}(1) = 0$ and $B_{kn+j}(1) = 0;$

(iv) $D_{q^{-1}}^rA_{kn}(0) = 0$ and $D^{r}_{q^{-1}}B_{kn+j}(0) = 0$ for $r\in \{0, 1, \ldots, k-2\}$.

Proof. The proof of (i) follows from the substitution with $w = 0$ in the generating functions in (3.9). To prove (ii), we act on the two sides of (3.9) by the operator $D_{q^{-1}, x}^k$ and use (2.7). To prove the first identity in (iii), we substitute with $z = 1$ in the first equation in (3.9) and using that $A_0(1) = 1$, consequently, $\sum_{n = 1}^{\infty} w^{kn} A_{kn}(1) = 0$, This yields $A_{kn}(1) = 0$. The substitution with $z = 1$ in the second identity in (3.9) yields $\sum_{n = 0}^{\infty} w^{kn+j} B_{kn+j}(1) = 0$. Hence, $B_{kn+j}(1) = 0$ for all $n\in\mathbb{N}_0$. Finally, the proof of (iv) follows at once from (ii) and (iii).

4.   Special case: A $q$-type $3$-Lidstone series

In this section, we give several problems that illustrate the $q$-type of $3$-Lidstone series. One of them can be derived immediately from Theorem 3.3 (see Table 1).

In the following, we discuss another problem of expanding an entire function in a $q$-type $3$-Lidstone series.

Theorem 4.1. Let $\mu_{1, 3}$ be the zero with the smallest positive absolute magnitude of $M_{3, 0}(z; q)$ which defined in (2.5). Then, for every entire function $f(z)$ of $q^{-1}$-exponential growth of order less than $1$, or of order $1$ and a finite type $\alpha$ such that $\alpha < \Big(\frac{1}{2}- \frac{\log |\mu_{1, 3}(1-q)|}{\log q}\Big)$ the following representation holds:

where $\tilde{A}_{3n}(z)$, $\tilde{B}_{3n+1}(z)$ and $\tilde{B}_{3n+2}(z)$ are polynomials defined by the following generating functions:

The series on (4.1) converges to $f(z)$ for all $z$ and the convergence is uniform on all compact subsets of the plane.

Proof. First, we consider the functions $\sum_{n = 0}^{\infty} w^{3n} \tilde{A}_{3n}(z) = \tilde{\varphi} (z, w^3) \equiv \tilde{\varphi}$,

and assume that the function $E_q(wz)$ has the following representation:

Let $\omega^3 = 1$ with $\omega \neq 1$ and replacing $w$ by $\omega w$ and $\omega^2 w$ in Eq (4.3), we obtain

If we add the Eqs (4.3)–(4.5), we obtain $\tilde{\varphi}$. In order to get the function $w\tilde{\psi}_1$, we multiply the Eqs (4.3)–(4.5) by $1, \omega^{-1}$ and $\omega^{-2}$, respectively and add. $w^2\tilde{\psi}_2$ obtained by multiplying the same equations by $1, \omega^{-2}$ and $\omega^{-1}$, respectively. The proof is then completed similar to the proof of Theorem 3.3.

Corollary 4.2. For $n\in\mathbb{N}$, the polynomials $\tilde{A}_{3n}$ and $\tilde{B}_{3n+j} \; (j = 1, 2)$ satisfy the $q$-difference equations

with the boundary conditions

Proof. The proof follows by using the generating functions in (4.2).

Remark 4.3. According to formula (4.1), if $P$ any polynomial of degree less than or equals to $6$, then we have

So, by setting $P(z) = 1, z, \ldots, z^6$, successively, in (4.6) we get

Example 4.4. We apply Theorem 4.1 on the function $f(z) = (z; q)_{3n}$ ($n\in\mathbb{N}$) and using that for any $m\in\mathbb{N}_0$,

This gives

Example 4.5. Consider the Al-Salam-Carlitz II polynomials (see ^[12]) defined by

Since, $D_{q^{-1}}^k V_m^{(a)}(x; q) = q^{-{m\choose 2}+{m-k\choose 2}} \frac{[m]_q!}{[m-k]_q!} V_{m-k}^{(a)}(x; q)$ if $0\leq k\leq m$, we obtain

for $k\leq m$. Consequently, applying Theorem 4.1 yields

Remark 4.6. For the $q$-type of $3$-Lidstone series, we can consider six problems:

where $f$ is assumed to be entire function and satisfies the conditions on Theorem 4.1.

The tables below give the generating functions of the $q$-type fundamental polynomials and the values at $n = 0$ in these problems, respectively.

5.   Conclusions

In this paper, we introduced an extension of $q$-Lidstone series which was called a $q$-type $k$-Lidstone series:

where $(A_{kn}(z))_n$ and $(B_{kn+j}(z))_n$ are polynomials defined by the following generating functions:

and determined the class of functions for which (5.1) is valid.

Notice, by following the same manner as a proof of (5.1), we can conclude that the function $f$ can be given also by the convergent another $q$-type $k$-Lidstone series expansion with different $q$-polynomials. More precisely, we can obtain the following result.

Theorem 5.1. Let $\mu_1$ be the zero with the smallest positive absolute magnitude of $N_{k, 0}(w; q)$ which defined in (2.5). If the function $f(z)$ is an entire function of $q^{-1}$-exponential growth of order less than $1$, or of order $1$ and a finite type $\alpha$ such that

then, for all $z\in \mathbb{C}$ the following representation holds

where $(C_{kn}(z))_n$ and $(P_{kn+j}(z))_n$ are polynomials defined by the following generating functions:

As a special case, we considered six problems of expanding an entire function in the $q$-type $3$-Lidstone series. The Lidstone polynomials are used in many interpolation and boundary value problems. See for example, ^[1,7]. We studied boundary value problems includes the 2 type $q$-Lidstone polynomials in ^[15], and we aim to study boundary value problems associated with the $k$ type $q$-Lidstone polynomials, $k > 2$.

Acknowledgments

The authors would like to express appreciation to the editor and the referees for their helpful comments and suggestions that improved this article.

Conflict of interest

The authors declare no conflicts of interest.