Advancements in q-Hankel Transforms Based on Certain Approach of Big q-Bessel Functions and Applications

Abstract

This study presents a novel variant of a finite q-Hankel transform derived from big q-Bessel functions and investigates its analytical structure, with particular emphasis on the distribution and properties of its zeros. A key focus is placed on the inherent symmetry in the zero distribution of these transforms, which plays a central role in their analytical characterization. We establish rigorous conditions under which the finite q-Hankel transforms exhibit only real zeros and demonstrate their adherence to well-defined asymptotic and symmetric patterns. Moreover, we introduce a series of q-analogs to classical theorems, such as those of Pólya, further illustrating the symmetric nature of these results within the framework of q-calculus. The findings not only deepen the understanding of q-integral transforms and their symmetry properties but also underscore their relevance in the broader context of special functions and mathematical analysis.

1. Introduction

The common q-analogs of the classical Bessel function

$$J_{\nu }\left(z\right)=\sum _{j=0}^{\infty }{\displaystyle \frac{{(-1)}^j{\left(\frac{z}{2}\right)}^{2j+\nu }}{k!\Gamma (j+\nu +1)}},$$

were defined by Jackson in the early 20th century [1]. He introduced three primary q-versions of the classical Bessel function, commonly represented as $J_{\nu }^{\left(k\right)}(z;q),k=1,2,3$. These functions have attracted considerable interest due to their applications in quantum theory, harmonic analysis, and special function theory, cf [2,3]. As research has progressed, the analytical and geometric properties of these q-Bessel functions have been extensively examined, along with their integration, cf [4], into q-calculus-based transforms such as the q-Hankel transform. Recent advances have expanded their role in various fields, focusing on their analytical properties, geometric behavior, and integral transforms. Based on these Jackson q-Bessel functions, a critical area of exploration involves the q-Hankel transforms, cf [2,3,5,6], defined by

$${\mathcal{H}}_{\nu ,\varphi }^{\left(i\right)}(z;q):=z^{-\nu }{\int }_0^1{u}^{-\nu }\varphi \left(u\right)J_{\nu }^{\left(i\right)}(uz;q^2)d_{q}u,z\in \mathbb{C},i=2,3.$$

These transforms serve as q-analogs of classical Hankel transforms and have been the subject of several recent investigations due to their connections with q-calculus, harmonic analysis, and special functions. There are several studies in different techniques on zeros of the associated finite q-Hankel transforms ${\mathcal{H}}_{\nu ,f}^{\left(i\right)}(z;q)$. These techniques fall into four main categories: the Rouché and Hurwitz theorems approach, cf e.g., [6], the $\theta$-function approach, cf e.g., [5], the Hurwitz–Biehler approach, cf e.g., [2,7], and the Direct approach, cf e.g., [8,9]. Studies have explored the asymptotic behavior and location of zeros of q-Hankel transforms, utilizing techniques like the Rouché and Hurwitz theorems. These efforts enhance the understanding of their analytic continuation and stability, cf e.g., [10]. In this paper, we introduce a slightly different q-Hankel transform. We use different techniques to study the reality and asymptotics of their zeros. For applications, cf e.g., [10,11,12,13,14,15].

Our main contributions are as follows:

• We define a new finite q-Hankel transform utilizing big q-Bessel functions and derive its convergence and analytical structure.
• We establish sufficient conditions for the reality and simplicity of the zeros of this transform.
• We derive precise asymptotic estimates for the location of the zeros.
• We provide q-analogs to classical results such as the Pólya theorem.
• We explore potential applications of our results in mathematical physics and special functions.

This paper is structured as follows: In the remaining part, we have five sections. In Section 2, we introduce the preliminaries, definitions and results that will be used later in the conclusion of our main result. Section 3 introduces the latest definitions and properties of the big q-Bessel functions. Section 4 discusses the behavior and zeros of our introduced q-analog of the Hankel transform based on the big q-Bessel functions. We also give sufficient conditions to guarantee that all zeros of our q-Hankel transform are real and simple. Section 5 presents applications, including a q-analog of classical results like Pólya’s theorem and geometric insights into the big q-Bessel functions. Finally, Section 6 summarizes the advancements and identifies directions for future research.

2. Preliminaries

Throughout this paper, unless otherwise stated, q is a positive number with $0<q<1$. We follow [16,17] in the definitions of the q-shifted factorial, the q-hypergeometric series, and the q-gamma function.

• The q-shifted factorial, see [16], for $a\in \mathbb{C}$, is defined by

$${(a;q)}_n:=\left\{\begin{array}{cc}1,& n=0,\hfill \\ {\displaystyle \prod _{i=0}^{n-1}(1-a{q}^i),}& n=1,2,\dots .\hfill \end{array}\right.$$

(1)

The limit of ${(a;q)}_n$ as n tends to infinity exists and will be denoted by ${(a;q)}_{\infty }$. Moreover, ${(a;q)}_{\infty }$ has the following series representation, cf, e.g., [16], p. 11:

$${(a;q)}_{\infty }=\sum _{n=0}^{\infty }{(-1)}^n{q}^{{\displaystyle \frac{n(n-1)}{2}}}{\displaystyle \frac{a^n}{{(q;q)}_n}}.$$

(2)

The multiple q-shifted factorial for complex numbers $a_1,\dots ,a_k$ is defined by

$${(a_1,a_2,\dots ,a_k;q)}_n:=\prod _{j=1}^k{(a_j;q)}_n.$$

(3)

The $q\text{-}$Hypergeometric function ${}_r\varphi _s$ is defined by, cf [16]

$${}_r\varphi _s\left(\begin{array}{c}{a}_1,\dots ,a_r\\ b_1,\dots ,b_s\end{array}q,z\right):=\sum _{n=0}^{\infty }{\displaystyle \frac{{(a_1,\dots ,a_r;q)}_n}{{(q,b_1,\dots ,b_s;q)}_n}}{z}^n{(-q^{(n-1)/2})}^{n(s+1-r)}.$$

(4)

The q-Gamma function is defined by, cf [16]

$${\Gamma }_q\left(z\right):={\displaystyle \frac{{(q;q)}_{\infty }}{{(q^z;q)}_{\infty }}}{(1-q)}^{1-z},z\in \mathbb{C},\left|q\right|<1,$$

(5)

where we take the principal values of $q^z$ and ${(1-q)}^{1-z}$ and

$${\left[z\right]}_q:={\displaystyle \frac{1-q^z}{1-q}}.$$

(6)

To support the main results, several mathematical preliminaries are established, including definitions related to entire functions, asymptotic equivalence, and growth characteristics. We adopt the following notation:

• For $\varphi ,\psi$ be entire functions, we say that

$$\varphi \left(z\right)=O\left(\psi \left(z\right)\right),asz\to \infty .$$

If ${lim}_{z\to \infty }{\displaystyle \frac{\varphi \left(z\right)}{\psi \left(z\right)}}=1$, we write

$$\varphi \left(z\right)\sim \psi \left(z\right),z\to \infty .$$

(7)

If $\varphi \left(z\right)={\sum }_{n=0}^{\infty }{e}_n{z}^n$ is an entire function, then the maximum modulus is defined for $r>0$ by

$$M(r;\varphi ):=sup\left\{\left|\varphi \right(z\left)\right|:\left|z\right|=r\right\}.$$

(8)

The order of $\varphi$, $\rho \left(f\right)$, is, cf [18],

$$\rho \left(\varphi \right):=\underset{r\to \infty }{lim\; sup}{\displaystyle \frac{loglogM(r,\varphi )}{logr}}=\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{nlogn}{log|e_n{|}^{-1}}}.$$

(9)

Theorem 1

([18]). An entire function ψ with finite $\rho \left(\varphi \right)$ and $\rho \left(\varphi \right)\notin Z_{+}$ is either a polynomial or has infinite number of zeros.

Theorem 2

(Rouché and Hurwitz theorem, [19]). Let ${\left\{{\psi }_n\left(z\right)\right\}}_{n\in \mathbb{N}}$ be a sequence of entire functions that have real zeros only. If

$$\underset{n\to \infty }{lim}{\psi }_n\left(z\right)=\psi \left(z\right),$$

uniformly in any finite domain, then the entire function $\psi \left(z\right)$ can have only real zeros.

• Regarding entire functions, we have the following useful theorem which will be used in Section 6.

Theorem 3

([14]). If ϱ is an entire function which has order zero or one with only real zeros with R positive zeros. Then,

$$\sum _{j=0}^{\infty }{(-1)}^j{\displaystyle \frac{\varrho \left(2j\right)}{j!}}{z}^{2j}$$

has at most $2R$ zeros which are not real.

• The following version of the Hurwitz–Biehler theorem for entire functions of order zero is a useful tool.

Theorem 4

(Hurwitz–Biehler theorem, [20]). Let

$$\Psi \left(z\right)=f\left(z\right)+ig\left(z\right),$$

where the entire function $\Psi \left(z\right)$ is of zero order and with the coefficients of $f\left(z\right)$ and $g\left(z\right)$ being real. Then, roots of $\Psi \left(z\right)$ lie in the upper half plane if $f\left(z\right)$ and $g\left(z\right)$ have simple real interlacing zeros.

• The theorem is vital because the roots of $\Psi \left(z\right)$ lie in the upper half plane if $\Psi (-iz)$ has roots with negative real parts. Thus, the results of Katkova et al. [7] are useful in the present work.

Theorem 5.

Assume that $\Phi \left(z\right):={\sum }_{j=1}^{\infty }{e}_j{z}^j$, $e_j>0$ and that $x_0$ is the unique positive root of the polynomial $x^3-x^2-2x-1$, i.e., $x_0\approx 2.1479$. Then, $x_0$ is the smallest possible constant such that if

$$e_j{e}_{j+1}⩾x_0{e}_{j-1}{e}_{j+2},j⩾1,$$

(10)

then every zero of $\Phi \left(z\right)$ has a negative real part.

• From the above theorem, we get the following:

Corollary 1.

If condition (10) is satisfied by the sequence ${\left\{e_j\right\}}_{j=0}^{\infty }$, then

$$\sum _{j=0}^{\infty }{(-1)}^j{e}_{2j}{z}^{2j}and\sum _{j=0}^{\infty }{(-1)}^j{e}_{2j+1}{z}^{2j+1}$$

have real simple interlacing zeros only.

• The space denoted by $L_q^1(0,1)$ is the space of all integrable functions satisfy ${\int }_0^1{t}^n\left|\varphi \left(t\right)\right|d_{q}t<\infty ,n\in \mathbb{N}$. Two functions are said to be equivalent if they are equal on the sequence $\{q^n:n\in {\mathbb{N}}_0\}$. The norm of Banach Space $L_q^1(0,1)$ is

  $$∥\varphi ∥:={\int }_0^1\left|\varphi \left(t\right)\right|d_{q}t,\varphi \in L_q^1(0,1).$$

  For $\varphi \in L_q^1(0,1)$, we denote by $A_i\left(\varphi \right)$ to the q-moments of $\varphi$, i.e.,

  $$A_i\left(\varphi \right):={\int }_0^1{t}^i\varphi \left(t\right)d_{q}t,i\in \mathbb{N}.$$

  (11)

  Let also $c_{i,\nu }^a\left(\varphi \right)$ denote the number

  $$c_{i,\nu }^a\left(\varphi \right):={\displaystyle \frac{A_{2i+2}\left(\varphi \right)(1+\frac{q^{2i}}{a^2})}{A_{2i}\left(\varphi \right)(1-q^{2i+2+2\nu })(1-q^{2i+2})}},i\in \mathbb{N},$$

  (12)

Proposition 1.

Let $\varphi \in L_q^1(0,1)$ be positive on $\left\{0,q^n,n\in \mathbb{N}\right\}$. Then, the numbers

$$c_{\nu ,\varphi }^a:=\underset{i\in \mathbb{N}}{inf}{c}_{i,\nu }^a\left(\varphi \right),C_{\nu ,\varphi }^a:=\underset{i\in \mathbb{N}}{sup}{c}_{i,\nu }^a\left(\varphi \right),$$

(13)

exist and they are finite positive numbers.

Proof. 

Consider the strictly decreasing ${\left\{A_i\left(\varphi \right)\right\}}_{i=0}^{\infty }$ with $i⩾2$, $i\in \mathbb{N}$. From cf [11], we have

$$\begin{array}{ccc}\hfill {\left(A_{2i}\left(\varphi \right)\right)}^2& =& {\left({\int }_0^1{t}^{2i}\varphi \left(t\right)d_{q}t\right)}^2={\left(\sum _{j=0}^{\infty }{q}^j(1-q)q^{\left(2i\right)j}\varphi \left(q^j\right)\right)}^2\hfill \\ & ⩽& \sum _{j=0}^{\infty }{q}^j(1-q)q^{(2i+2)j}\varphi \left(q^j\right)\sum _{j=0}^{\infty }{q}^j(1-q)q^{(2i-2)j}\varphi \left(q^j\right)\hfill \\ & =& A_{2i+2}\left(\varphi \right)A_{2i-2}\left(\varphi \right).\hfill \end{array}$$

So, ${\left\{A_{2i+2}\left(\varphi \right)/A_{2i}\left(\varphi \right)\right\}}_{k=0}^{\infty }$ is an increasing sequence of positive numbers. Since ${\left\{A_{2i}\left(\varphi \right)\right\}}_{i=0}^{\infty }$ is strictly decreasing, we have

$${\displaystyle \frac{A_2\left(\varphi \right)}{A_0\left(\varphi \right)}}⩽{\displaystyle \frac{A_{2i+2}\left(\varphi \right)}{A_{2i}\left(\varphi \right)}}<1,i=0,1,\dots .$$

(14)

Hence, for $i\in \mathbb{N}$, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{A_2\left(\varphi \right)}{A_0\left(\varphi \right)}}& ⩽{\displaystyle \frac{A_{2i+2}\left(\varphi \right)}{A_{2i}\left(\varphi \right)}}⩽{\displaystyle \frac{A_{2i+2}\left(\varphi \right)(1+\frac{q^{2i}}{a^2})}{A_{2i}\left(\varphi \right)(1-q^{2i+2+2\nu })(1-q^{2i+2})}}=c_{k,\nu }^a\left(\varphi \right)\hfill \\ \hfill & <{\displaystyle \frac{(1+\frac{q^{2i}}{a^2})}{(1-q^{2i+2+2\nu })(1-q^{2i+2})}}⩽{\displaystyle \frac{(1+\frac{1}{a^2})}{(1-q^{2\nu +2})(1-q^2)}}.\hfill \end{array}$$

(15)

Inequalities (15) and Bolzano Weierstrass’ theorem assure that positive numbers $c_{\nu ,\varphi }^a,C_{\nu ,\varphi }^a$ exist. □

3. The Big q-Bessel Functions

The big q-Bessel function was first defined by Ciccolo, Koelink and Koornwinder in [21,22], using a rigorous limit transition from the big q-Jacobi functions. The following definition of the big q-Bessel functions was given by Koelink et al. in [23], by

$$J_{\gamma }(x,b;q)={}_1\varphi _1\left(\begin{array}{c}-{\displaystyle \frac{1}{x}}\\ b\end{array}q,b\gamma x\right),$$

and then they showed that $J_{\gamma }(x,b;q)$ satisfies the q-difference equation

$$(1-q)({\displaystyle \frac{1}{b}}+{\displaystyle \frac{1}{bx}})D_{q^{-1},x}{J}_{\gamma }(x,b;q)-(1-q)(1+{\displaystyle \frac{1}{bx}})D_{q,x}{J}_{\gamma }(x,b;q)=-\gamma J_{\gamma }(x,b;q).$$

Later, Bouzeffour et al., cf [24], redefined the big q-Bessel functions by

$$\begin{array}{ccc}\hfill J_{\nu }(x,\lambda ;q^2)& :=& {}_1\varphi _1\left(\begin{array}{c}-{\displaystyle \frac{1}{x^2}}\\ q^{2\nu +2}\end{array}|q^2,{\lambda }^2{x}^2{q}^{2\nu +2}\right)\hfill \\ & =& \sum _{n=0}^{\infty }{(-1)}^n{\displaystyle \frac{q^{n(2\nu +n+1)}}{{(q^2,q^{2\nu +2};q^2)}_n}}{(-{\displaystyle \frac{1}{x^2}};q^2)}_n{\left(\lambda x\right)}^{2n}.\hfill \end{array}$$

(16)

Therefore, one can see that

$$\underset{q⟶1}{lim}{J}_{\nu }({\displaystyle \frac{x}{1-q^2}},{(1-q^2)}^2\lambda ;q^2)={\displaystyle \frac{\Gamma (\nu +1)}{{\left(\lambda x\right)}^{\nu }}}{J}_{\nu }\left(2\lambda x\right),\nu >-1.$$

Throughout the following investigated results, we are interested in the big q-Bessel function with $\lambda$ as a parameter. So, we only recall the most important results related to the behavior and asymptotic of the zeros of $J_{\nu }(a;\lambda ;q^2)$. In the following, we collect the properties of the zeros of $J_{\nu }(a,\lambda ;q^2)$ in a single proposition, see [24].

Proposition 2

([24]). The big q-Bessel function satisfies the following properties:

• For $\nu >-1$, $J_{\nu }(a,\lambda ;q^2)$ is entire in λ.
• For $a>0,\nu >-1$, the zeros of $J_{\nu }(a,\lambda ;q^2)$ are real.
• For $a>0,\nu >-1$, the non-zero real zeros of $J_{\nu }(a,\lambda ;q^2)$ are simple.
• For $\nu >{\displaystyle \frac{-1}{2}}$, $J_{\nu }(a,\lambda ;q^2)$ is entire of order zero.
• For $a\in \mathbb{R},\nu >{\displaystyle \frac{-1}{2}}$, $J_{\nu }(a,\lambda ;q^2)$ has infinitely many zeros.
• If $j_{n,\nu }$ are the positive zeros of $J_{\nu }(a,.;q^2),\nu >{\displaystyle \frac{-3}{2}}$, then for sufficiently large n

  $$\begin{array}{c}\hfill j_{n,\nu }=q^{-n-\nu }{a}^{-1}(1+O\left(q^n\right)),\left|a^{-1}\right|<1.\end{array}$$

  (17)
• For $\lambda \in B_{n,\nu }$, we have the following asymptotic relation for $j_{1,\nu }\le \left|q^{-1}\lambda \right|$, uniformly for sufficiently large n

  $$log|J_{\nu }(a,\lambda ;q^2)|=-2{\displaystyle \frac{{\left(log|\lambda q^{2\nu }{a}^2|\right)}^2}{logq}}-2log|\lambda q^{2\nu }{a}^2|+log|1-{\displaystyle \frac{{\lambda }^2}{{\left(j_{n,\nu }\right)}^2}}|+O\left(1\right),$$

  (18)

where $B_{n,\nu },n\ge 1$ is the annulus defined by

$$B_{n,\nu }:=\{\lambda \in \mathbb{C}:j_{n,\nu }{q}^{d_{n,\nu }}\le \left|\lambda \right|\le j_{n,\nu }{q}^{-e_{n,\nu }}\},$$

(19)

and, the sequences ${\left\{e_{n,\nu }\right\}}_{n=1}^{\infty }$ and ${\left\{d_{n,\nu }\right\}}_{n=1}^{\infty }$ are defined by

$$e_{n,\nu }:=\left\{\begin{array}{cc}{\displaystyle {\beta }_{n,\nu }+1,}\hfill & \mathrm{if}{\beta }_{n,\nu }\ne 1\hfill \\ \\ {\displaystyle {\beta }_{\nu },}\hfill & \mathrm{if}{\beta }_{n,\nu }=1\hfill \end{array}\right..$$

(20)

and

$$d_{1,\nu }:=1,d_{n+1,\nu }:=\left\{\begin{array}{cc}{\displaystyle {\beta }_{n,\nu }-1,}\hfill & \mathrm{if}{\beta }_{n,\nu }\ne 1,\hfill \\ \\ {\displaystyle 1,}\hfill & \mathrm{if}{\beta }_{n,\nu }=1\hfill \end{array}\right..$$

(21)

while

$${\beta }_{n,\nu }:=log\left(j_{n,\nu }/j_{n+1,\nu }\right)/log{q}^2=1+log\left(1+O\left(q^{2n}\right)\right)/log{q}^2,n\in {\mathbb{Z}}^{+}.$$

(22)

• Now, we use some results of Proposition 2 to prove the following useful proposition.

Proposition 3.

For $r:=|\lambda q^{2\nu }{a}^2|⟶\infty$, we have

$$M\left(r;J_{\nu }(a,\lambda ;q^2)\right)=O\left(exp\left(-2{\displaystyle \frac{{(logr)}^2}{logq}}-2logr\right)\right).$$

(23)

Proof. 

From (18), it is clear that ∃$n_0\in \mathbb{N}$ and a constant $\zeta >0$, such that for all $n⩾n_0$ and $\lambda \in B_{n,\nu }$,

$$\begin{array}{cc}\hfill -2{\displaystyle \frac{{\left(log|\lambda q^{2\nu }{a}^2|\right)}^2}{logq}}& -2log|\lambda q^{2\nu }{a}^2|+log|1-{\displaystyle \frac{{\lambda }^2}{{\left(j_{n,\nu }\right)}^2}}|-\zeta ⩽log|J_{\nu }(a,\lambda ;q^2)|\\ & \le -2{\displaystyle \frac{{\left(log|\lambda q^{2\nu }{a}^2|\right)}^2}{logq}}-2log|\lambda q^{2\nu }{a}^2|+log|1-{\displaystyle \frac{{\lambda }^2}{{\left(j_{n,\nu }\right)}^2}}|+\zeta .\end{array}$$

(24)

From the definition of $B_{n,\nu }$ we have

$$q^{2d_{n,\nu }}⩽{\displaystyle \frac{{\left|\lambda \right|}^2}{{\left(j_{n,\nu }\right)}^2}}⩽q^{-2e_{n,\nu }}.$$

Hence, for all $n⩾1$

$$log|1-{\displaystyle \frac{{\lambda }^2}{{\left(j_{n,\nu }\right)}^2}}|⩽log\left(1+{\displaystyle \frac{{\left|\lambda \right|}^2}{{\left(j_{n,\nu }\right)}^2}}\right)⩽{\displaystyle \frac{{\left|\lambda \right|}^2}{{\left(j_{n,\nu }\right)}^2}}⩽q^{-2e_{n,\nu }}.$$

(25)

We have a bounded sequence ${\left\{e_{n,\nu }\right\}}_{n=1}^{\infty }$, so there exists a constant $\mathcal{R}>$ 0 such that for $n⩾n_0$

$$log|J_{\nu }(a,\lambda ;q^2)|\le -2{\displaystyle \frac{{\left(log|\lambda q^{2\nu }{a}^2|\right)}^2}{logq}}-2log\left|\lambda q^{2\nu }{a}^2\right|+\mathcal{R}>0,\lambda \in B_{n,\nu }.$$

(26)

Therefore,

$$|J_{\nu }(a,\lambda ;q^2)|⩽{\displaystyle \frac{e^{\mathcal{R}}}{|\lambda q^{2\nu }{a}^2{|}^2}}exp\left(-4{\displaystyle \frac{log|\lambda q^{2\nu }{a}^2|}{logq}}\right),\lambda \in B_{n,\nu },n⩾n_0,$$

(27)

which completes the proof. □

• The big q-trigonometric functions $cos(z,\lambda ;q)$ and $sin(z,\lambda ;q)$ are defined on $\mathbb{C}$ by

  $$\begin{array}{ccc}\hfill cos(a,\lambda ;q^2)& :=& J_{-1/2}(a;\lambda ;q^2)\hfill \\ & =& \sum _{n=0}^{\infty }{(-1)}^n{\displaystyle \frac{q^{n^2}}{{(q;q)}_{2n}}}{(-{\displaystyle \frac{1}{a^2}};q^2)}_n{\left(a\lambda \right)}^{2n},\hfill \end{array}$$

  (28)

  $$\begin{array}{ccc}\hfill sin(a,\lambda ;q^2)& :=& {\displaystyle \frac{a\lambda }{1-q}}{J}_{1/2}(a;\lambda ;q^2)\hfill \\ & =& \sum _{n=0}^{\infty }{(-1)}^n{\displaystyle \frac{q^{n(n+2)}}{{(q;q)}_{2n+1}}}{(-{\displaystyle \frac{1}{a^2}};q^2)}_n{\left(a\lambda \right)}^{2n+1}.\hfill \end{array}$$

  (29)

  From the above definition, it follows that the functions $cos(a,\lambda ;q^2)$ and $sin(a,\lambda ;q^2)$ have only real simple zeros.

4. The Big q-Hankel Transforms

We present the following main results: Let us begin by defining big q-Hankel transform by

$${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2):={\int }_0^1\varphi \left(t\right)J_{\nu }(a,\lambda t;q^2)d_{q}t.$$

(30)

Now, we introduce some results concerning the zeros of the big q-Hankel transform ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$. In general, we prove that all zeros of ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ are real and simple except at most a finite number of non-real zeros, where $\varphi \in L_q^1(0,1)$. We add more restrictions on q to guarantee that all zeros are real.

Proposition 4.

For $\varphi \in L_q^1(0,1)$, which is positive on $\left\{0,q^n,n\in {\mathbb{N}}_0\right\},$ ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ has infinitely many zeros.

Proof. 

It is sufficient to show that ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ is entire of order zero. One can see that

$${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)=\sum _{k=0}^{\infty }{(-1)}^k{\displaystyle \frac{q^{k(2\nu +k+1)}{A}_{2k}\left(\varphi \right)}{{(q^2;q^{2\nu +2};q^2)}_k}}{(-{\displaystyle \frac{1}{a^2}};q^2)}_k{\left(a\lambda \right)}^{2k}$$

(31)

Thus, by Proposition 1, one can see that the radius of convergence of the series in (31) is ∞. Hence, ${\mathcal{U}}_{\nu ,\varphi }(z,\lambda ;q^2)$ is entire. The order of ${\mathcal{U}}_{\nu ,\varphi }(z,\lambda ;q^2)$ is ${\displaystyle lim\; sup\frac{klogk}{log{e}_k^{-1}}}$ as $k⟶\infty$, where

$$e_k={\displaystyle \frac{q^{k(2\nu +k+1)}{A}_{2k}\left(\varphi \right)}{{(q^2;q^{2\nu +2};q^2)}_k}}{(-{\displaystyle \frac{1}{a^2}};q^2)}_k{a}^{2k}$$

Direct calculations lead us to

$${\displaystyle \underset{k\to \infty }{lim}\frac{log\frac{1}{e_k}}{klogk}=\infty .}$$

By Theorem 1, the proof is completed. □

• Now, we prove that all zeros of ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ are infinite real and simple zeros, where $\varphi \in L_q^1(0,1)$.

Theorem 6.

Let $\varphi \in L_q^1(0,1)$. The zeros of ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ are infinite real simple zeros ${\left\{\pm {\zeta }_{n,\nu }\right\}}_{n=1}^{\infty }$. Moreover, the positive zeros ${\zeta }_{n,\nu }>0$, satisfy that ${\zeta }_{n,\nu }\sim j_{n,\nu }$ as $n\to \infty$. More precisely,

$${\zeta }_{n,\nu }=j_{n,\nu }(1+O\left(q^n\right))asn\to \infty .$$

(32)

Proof. 

The function ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ can be written as

$${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)=H_{\nu ,\varphi }(a,\lambda )+R_{\nu ,\varphi }(a,\lambda ),$$

(33)

where

$$\begin{array}{cc}\hfill H_{\nu ,\varphi }(a,\lambda )& :=(1-q)\varphi \left(1\right)J_{\nu }(a,\lambda ;q^2),\hfill \\ \hfill R_{\nu ,\varphi }(a,\lambda )& :=\sum _{k=1}^{\infty }{q}^k(1-q)\varphi \left(q^k\right)J_{\nu }(a,q^k\lambda ;q^2).\hfill \end{array}$$

(34)

• We apply the Rouché theorem to conclude that $H_{\nu ,\varphi }(a,\lambda )$ and ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ have the same number of zeros; then, we study the asymptotic behavior of the zeros of ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$. From (23), there exists constants $r_0,\mathcal{C}>0$ such that

  $$\begin{array}{cc}\hfill \underset{|a^2{q}^{2\nu }\lambda |=r}{max}\left|J_{\nu }(a;\lambda ;q^2)\right|& ⩽{\displaystyle \frac{\mathcal{C}}{r^2}}exp\left(-2{(logr)}^2/logq\right),\mathrm{for}\mathrm{all} r>r_0\hfill \\ \hfill & ={\displaystyle \frac{{\mathcal{C}}_{\mathcal{0}}}{{\left|\lambda \right|}^2}}exp\left(-2{(log|\lambda \left|\right)}^2/logq\right){,\left|\lambda \right|>\left|a\right|}^{-2}{q}^{-4\nu }{r}_0.\hfill \end{array}$$

  (35)

  Thus,

  $$\begin{array}{cc}\hfill |R_{\nu ,\varphi }(a;\lambda )|& ⩽{\displaystyle \frac{{\mathcal{C}}_{\mathcal{0}}}{{\left|q\lambda \right|}^2}}exp\left(-2{(log|q\lambda \left|\right)}^2/logq\right)\sum _{k=1}^{\infty }{q}^k(1-q)\left|\varphi \left(q^k\right)\right|\hfill \\ \hfill & ={\displaystyle \frac{{\mathcal{C}}_{\mathcal{0}}}{q^4{\left|\lambda \right|}^6}}exp\left(-2{(log|\lambda \left|\right)}^2/logq\right){\int }_0^q\left|\varphi \left(t\right)\right|d_{q}t.\hfill \end{array}$$

  (36)

  Therefore,

  $$log|R_{\nu ,\varphi }(a;\lambda )|⩽{\mathcal{C}}_1-6log\left|\lambda \right|-2{\displaystyle \frac{{(log|\lambda \left|\right)}^2}{logq}},$$

  (37)

  where ${\mathcal{C}}_1:=log\left({\displaystyle q^{-4}{\mathcal{C}}_0{\int }_0^q\left|\varphi \left(t\right)\right|d_{q}t}\right)$.

Consequently, by (24), Equation (37) would now be

$$\begin{array}{ccc}\hfill log|R_{\nu ,\varphi }(a;\lambda )|& ⩽& {\mathcal{C}}_1-4log\left|\lambda \right|+{\zeta }_1+log|J_{\nu }(a;\lambda ;q^2)|-log|1-{\displaystyle \frac{{\lambda }^2}{{\left(j_n^{\nu }\right)}^2}}|\hfill \\ & =& log|H_{\nu ,\varphi }(a;\lambda )|-4log|\lambda |-log|1-{\displaystyle \frac{{\lambda }^2}{{\left(j_n^{\nu }\right)}^2}}|+{\mathcal{C}}_2,\hfill \end{array}$$

(38)

and ${\mathcal{C}}_2:=C_1+{\zeta }_1-log(1-q)\left|f\left(1\right)\right|$. Consider $D_{n_0}$,

$${\mathcal{D}}_{n_0}:=\left\{\lambda \in \mathbb{C}:\left|\lambda \right|<j_{n_0}^{\nu }{q}^{d_{n_0,\nu }}\right\}.$$

(39)

Hence,

$${\displaystyle \mathbb{C}:={\mathcal{D}}_{n_0}\bigcup \bigcup _{n⩾n_0}{B}_{n,\nu }.}$$

For $\lambda \in \partial {\mathcal{D}}_{n_0}$, where $\partial {\mathcal{D}}_{n_0}$ is the boundary of ${\mathcal{D}}_{n_0}$ then $\left|\lambda \right|=j_{n_0}^{\nu }{q}^{d_{n_0,\nu }}$; moreover,

$$log|R_{\nu ,\varphi }(a;\lambda )|⩽log|H_{\nu ,\varphi }(a;\lambda )|-4log|\lambda |-log|1-q^{2d_{n_0,\nu }}|+{\mathcal{C}}_2.$$

(40)

If we choose large $n_0$ such that $-4log\left|\lambda \right|-log|1-q^{2d_{n_0,\nu }}|+{\mathcal{C}}_2<0$, we get

$$log|R_{f,\nu }(a;\lambda )|⩽log|H_{f,\nu }(a;\lambda )|,|\lambda |=j_{n_0}^{\nu }{q}^{d_{n_0,\nu }}.$$

(41)

The theorem of Rouché leads us to conclude that $H_{\nu ,\varphi }(a;\lambda )$ and ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ have the same number of zeros inside $D_{n_0}$. For the zeros of ${\mathcal{U}}_{\nu ,f}(a,\lambda ;q^2)$ in ${\cup }_{n⩾n_0}{B}_{n,\nu }$, if $\lambda \in \partial B_{n,\nu }$, $n⩾n_0$, then

$$\begin{array}{cc}\hfill |1-{\displaystyle \frac{{\lambda }^2}{{\left(j_n^{\nu }\right)}^2}}|& ⩾1-q^{2d_{n,\nu }},\left|\lambda \right|=j_{n_0}^{\nu }{q}^{d_{n,\nu }},\hfill \\ \hfill |1-{\displaystyle \frac{{\lambda }^2}{{\left(j_n^{\nu }\right)}^2}}|& ⩾q^{-2e_{n,\nu }}-1,\left|\lambda \right|=j_{n_0}^{\nu }{q}^{e_{n,\nu }}.\hfill \end{array}$$

(42)

Thus, when $\lambda$ lies in the boundary of $B_{n,\nu }$ we have

$$|\lambda |=j_{n_0}^{\nu }{q}^{d_{n,\nu }}|⩾\mathcal{L},\mathcal{L}:=\underset{n\in \mathbb{N}}{inf}\left\{1-q^{2d_{n,\nu }},q^{-2e_{n,\nu }}-1\right\}>0.$$

(43)

Therefore,

$$log|R_{\varphi ,\nu }(a;\lambda )|⩽log|H_{\varphi ,\nu }(a;\lambda )|-4log|\lambda |+{\mathcal{C}}_2-{\mathcal{C}}_3.$$

(44)

Once more, choosing a large $n_0$, $r=\left|\lambda \right|\in B_{n,\nu }$, $n⩾n_0$, then $-4log\left|\lambda \right|+{\mathcal{C}}_3-{\mathcal{C}}_2<0$. Therefore,

$$|R_{\varphi ,\nu }(a;\lambda )|⩽|H_{\varphi ,\nu }(a;\lambda )|,\lambda \in \partial B_{n,\nu },n⩾n_0.$$

Again by applying Rouché theorem on $B_{n,\nu }$, $n⩾n_0$, one can conclude that $H_{\varphi ,\nu }(a;\lambda )$ and $U_{\varphi }(a;\lambda )$ have the same number of zeros inside $B_{n,\nu }$. It remains to give the asymptotic behavior of such zeros. Let ${\zeta }_{n,\nu }$ be non-negative zero of ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ in $B_{n,\nu }$. Then, $log\left|H_{\varphi ,\nu }(a,{\zeta }_{n,\nu })\right|=log\left|R_{\varphi ,\nu }(a,{\zeta }_{n,\nu })\right|$. Therefore, by (38)

$$log\left|1-{\displaystyle \frac{{\zeta }_{n,\nu }^2}{{\left(j_n^{\nu }\right)}^2}}\right|⩽-4log\left|{\zeta }_{n,\nu }\right|+{\mathcal{C}}_2,n⩾n_0.$$

(45)

Thus, after some simplifications, one can see that ${\zeta }_{n,\nu }\sim j_n^{\nu }$ for sufficiently large n. More precisely,

$${\zeta }_{n,\nu }=j_n^{\nu }(1+O\left(q^n\right)),asn\to \infty .$$

(46)

□

• Now, we introduce another theorem concerning the real zeros of ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$.

Theorem 7.

Let $\varphi \in L_q^1(0,1)$ be positive on $\left\{0,q^n,n\in \mathbb{N}\right\}$. If

$$q^{-1}(1-q){\displaystyle \frac{c_{\nu ,\varphi }^a}{C_{\nu ,\varphi }^a}}>1,$$

(47)

then all the zeros of ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ are real simple zeros and the positive zeros lie in the intervals

$$\left({\displaystyle \frac{q^{-r-\nu +\frac{1}{2}}}{\sqrt{C_{\nu ,\varphi }^a}}},{\displaystyle \frac{q^{-r-\nu -\frac{1}{2}}}{\sqrt{C_{\nu ,\varphi }^a}}}\right),r=1,2,3,\dots ,$$

(48)

with one zero in each interval. Moreover, ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ is an even function with no zeros in $\left[0,{\displaystyle \frac{q^{-\nu -1}}{\sqrt{C_{\nu ,\varphi }^a}}}\right)$.

Proof. 

Similarly to the proof of theorem 2.2 in [2], we get the results, so we only give the outlines. We first show that ${\mathcal{U}}_{\nu ,f}(a,\lambda ;q^2)$ has no zeros in $B_R\left(0\right)$, where

$$B_R\left(0\right):=\left\{\lambda \in \mathbb{C}:\left|\lambda \right|⩽R:=q^{-\nu -1}/\sqrt{C_{\nu ,\varphi }^a},z\ne \pm R\right\}.$$

Then, we prove that ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ has a zero in the interval $(z_{r-1},z_r)$, where $z_r,r\in {\mathbb{N}}_0$ denote the sequence

$$z_r:=q^{-r-\nu -{\displaystyle \frac{1}{2}}}/\sqrt{C_{\nu ,\varphi }^a}.$$

This can be done by considering the truncated series ${\mathcal{U}}_{2n,\nu ,\varphi }(a,\lambda ;q^2)$, where

$$\begin{array}{c}\hfill {\mathcal{U}}_{2n,\nu ,\varphi }(a,\lambda ;q^2)=\sum _{k=0}^{2n}{(-1)}^k{\displaystyle \frac{q^{k(2\nu +k+1)}{A}_{2k}\left(\varphi \right)}{{(q^2,q^{2\nu +2};q^2)}_k}}{(-{\displaystyle \frac{1}{a^2}};q^2)}_k{\left(a\lambda \right)}^{2k}.\end{array}$$

The sequence of partial sums ${\mathcal{U}}_{2n,\nu ,\varphi }(a,\lambda ;q^2)$ approaches ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ uniformly as $n\to \infty$ on compact subsets of $\mathbb{C}$. Since ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ is even, it suffices to prove that ${\mathcal{U}}_{2n,\nu ,\varphi }(a,\lambda ;q^2)$ has at least one zero in each interval $(z_{r-1},z_r)$, $r=1,2,\dots ,2n$. This means it has at least $2n$ positive zeros. Finally, we show that

$$\mathrm{sgn}{\mathcal{U}}_{\nu ,\varphi }(a,z_r;q^2)={(-1)}^r,r\in {\mathbb{N}}_0,$$

proving that ${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)$ has an infinite number of real zeros $\pm {\xi }_{r,\nu }$, ${\xi }_{r,\nu }\in (z_{r-1},z_r)$, $r\in \mathbb{N}$. □

5. Applications

Now we present certain applications of the proven results. The zeroes of $J_{\nu }(a,\lambda ;q^2)$ are investigated. Then, another q-version of a theorem of Pólya with regard to the q-trigonometric functions $cos(a,\lambda ;q^2)$ and $sin(a,\lambda ;q^2)$ is derived.

Proposition 5.

If

$$q^{-1}(1-q)(1-q^2)(1-q^{2\nu +2})>1+{\displaystyle \frac{1}{a^2}},$$

(49)

then the positive real simple zeros of $J_{\nu }(a,\lambda ;q^2)$ lie in the intervals

$$\left(q^{-r-\nu +{\displaystyle \frac{1}{2}}}\sqrt{{\displaystyle \frac{(1-q^2)(1-q^{2\nu +2})}{1+\frac{1}{a^2}}}},q^{-2r-\nu -{\displaystyle \frac{1}{2}}}\sqrt{{\displaystyle \frac{(1-q^2)(1-q^{2\nu +2})}{1+\frac{1}{a^2}}}}\right),r=1,2,3,\dots ,$$

(50)

with one zero in each interval. Moreover, $J_{\nu }(a,\lambda ;q^2)$ is an even function with no zeros in $\left[0,q^{-\nu -1}\sqrt{{\displaystyle \frac{(1-q^2)(1-q^{2\nu +2})}{1+\frac{1}{a^2}}}}\right)$.

Proof. 

Applying Theorem 7 with

$$\varphi \left(t\right):={(1-q)}^{-1},t=1and\varphi \left(t\right)=0otherwise.$$

${\mathcal{U}}_{\nu ,\varphi }(a,\lambda ;q^2)=J_{\nu }(a,\lambda ;q^2)$. We have $C_{\nu ,\varphi }^a={\displaystyle \frac{1+\frac{1}{a^2}}{(1-q^2)(1-q^{2\nu +2})}},c_{\nu ,\varphi }^a=1$. Therefore, condition (47) is nothing but (49), and the proposition is proved. □

• We now present a q-analog of the following theorems of Pólya [19].

Theorem 8.

If the function $\varphi \left(t\right)\in L^1(0,1)$ is positive and increasing, then the zeros of the entire functions of exponential type

$$U\left(z\right)={\int }_0^1\varphi \left(t\right)cos\left(zt\right)dt,V\left(z\right)={\int }_0^1\varphi \left(t\right)sin\left(zt\right)dt$$

(51)

are real, infinite and simple. Moreover, $U\left(z\right)$ is an even function having no zeros in $[0,{\displaystyle \frac{\pi }{2}})$, and its positive zeros are situated in the intervals $(\pi k-\pi /2,\pi k+\pi /2)$, $1⩽k<\infty$, with one zero in each interval. The odd function $V\left(z\right)$ has only one zero $z=0$ in $[0,\pi )$, and its positive zeros are situated in the intervals $(\pi k,\pi (k+1\left)\right)$, $1⩽k<\infty$, with one zero in each interval.

Theorem 9.

Let $\varphi \in L_q^1(0,1)$ be positive on $\left\{0,q^n,n\in \mathbb{N}\right\}$. If

$$q^{-1}(1-q){\displaystyle \frac{c_{-1/2,\varphi }^a}{C_{-1/2,\varphi }^a}}>1,$$

(52)

then the positive real simple zeros of ${\mathcal{U}}_f(a,\lambda ;q^2)$ lie in the intervals

$$\left({\displaystyle \frac{q^{-r+1}}{\sqrt{C_{-1/2,\varphi }^a}}},{\displaystyle \frac{q^{-r}}{\sqrt{C_{-1/2,\varphi }^a}}}\right),r=1,2,3,\dots ,$$

(53)

with one zero in each interval. Moreover, $U_{\varphi }(a,\lambda ;q^2)$ is an even function with no zeros in $\left[0,\sqrt{{\displaystyle \frac{q}{C_{-1/2,f}^a}}}\right)$, where

$$U_{\varphi }(a,\lambda ;q^2):={\int }_0^1\varphi \left(t\right)cos(a,\lambda t;q^2)d_{q}t.$$

Proof. 

The result is a consequence of Theorem 7 with $\nu =-1/2$ and (28), where $U_{\varphi }(a,\lambda ;q^2)={\mathcal{U}}_{-1/2,f}(a,\lambda ;q^2).$ □

Theorem 10.

Let $\varphi \in L_q^1(0,1)$ be positive on $\left\{0,q^n,n\in {\mathbb{N}}_0\right\}$ and let $\psi \left(t\right)=t^{-1}\varphi \left(t\right),0⩽t⩽1.$ If

$$q^{-1}(1-q){\displaystyle \frac{c_{1/2,\varphi }^a}{C_{1/2,\varphi }^a}}>1,$$

(54)

then the zeros of the entire function of order zero

$$V_{\varphi }(a,\lambda ;q^2):={\int }_0^1\psi \left(t\right)sin(a,\lambda t;q^2)d_{q}t,$$

lie in the intervals

$$\left({\displaystyle \frac{q^{-r}}{\sqrt{C_{1/2,\varphi }^a}}},{\displaystyle \frac{q^{-r-1}}{\sqrt{C_{1/2,\varphi }^a}}}\right),r=1,2,3,\dots ,$$

(55)

with one zero in each interval. Moreover, $V_{\varphi }(a,\lambda ;q^2)$ is an odd function with no zeros in $\left(0,1/\sqrt{q^3{C}_{1/2,\varphi }^a}\right)$.

Proof. 

The proof comes directly by applying Theorem 7 with $\nu =1/2$ and (29), where $V_{\varphi }(a,\lambda ;q^2)={\displaystyle \frac{a\lambda }{1-q}}{\mathcal{U}}_{1/2,\varphi }(a,\lambda ;q^2)$. □

Example 1.

Consider the function

$$U_{1-t}(a,\lambda ;q^2):={\int }_0^1(1-t)cos(a,\lambda t;q^2)d_{q}t.$$

$$c_{k,{\displaystyle \frac{-1}{2}}}^a(1-t)={\displaystyle \frac{q^2(1+\frac{q^{2k}}{a^2})}{(1-q^{2k+3})(1-q^{2k+4})}},C_{{\displaystyle \frac{-1}{2}},1-t}^a={\displaystyle \frac{q^2(1+\frac{1}{a^2})}{(1-q^3)(1-q^4)}},c_{{\displaystyle \frac{-1}{2}},1-t}^a=q^2.$$

Then, condition (52) reduces to be

$$q^{-1}(1-q)(1-q^3)(1-q^4)>(1+{\displaystyle \frac{1}{a^2}}),$$

and the zeros ${\left\{\pm u_r\right\}}_{r=0}^{\infty }$ of the $q\text{-}$function $U_{1-t}(a,\lambda ;q^2)$ are real, infinite and simple. Moreover,

$$u_r\in (q^{-r}\sqrt{{\displaystyle \frac{(1-q^3)(1-q^4)}{1+\frac{1}{a^2}}}},q^{-r-1}\sqrt{{\displaystyle \frac{(1-q^3)(1-q^4)}{1+\frac{1}{a^2}}}}),r\in \mathbb{N}.$$

6. Conclusions

We have introduced big q-Bessel function-based finite q-Hankel transforms and analysed the behaviour of their zeroes. We have also provided the criteria for finite q-Hankel transforms to have only real zeros satisfying certain asymptotic relations. Our results have been supported with suitable applications.