Integral transforms involving a generalized k-Bessel function

1 Introduction

The theory of special functions plays a discernible role in a variety of appropriate fields of mathematical analysis, applied science, and engineering. Bessel functions and their extensions with their various generalizations are among the most important in the special functions. In recent years, the k -calculus started with Diaz and Parigun’s [1] definition (see also Diaz et al. [2], Diaz and Teruel [3]). Following Diaz and Parigaun’s [1] concept, several researchers later introduced different types of k -special functions; for example, the properties of k -gamma, k -beta, and k -zeta have been studied by Kokologiannaki [4]. Krasniqi [5] studied the limits to k -gamma and k -beta functions.

In addition, the hypergeometric function and its generalized extensions have been introduced and extensively investigated mainly due to their applications in diverse areas of mathematics [6–8]. Similarly, the hypergeometric k -function has become important in recent years; for example, in [9,11], Mubeen and Habibullah defined the integral notation of generalized confluent hypergeometric k -functions and hypergeometric k -functions and k -fractional integral with applications based on Pochhammer k -symbols, k -gamma, and k -beta functions. Mansour [12] defined the k -generalized gamma function using a functional equation. The power product constraints for the k -gamma function were provided by Merovci [13].

Suthar et al. [14] have studied and presented the interesting Pochhammer k -symbol (rising factorial), fractional integral representations of k -gamma with some other related functions, such as k -beta and k -digamma functions, and k -Bessel function (which is one of the valuable speculations of the Bessel function). Also, for more details regarding the developed and applied new class of k -Bessel function with their applied properties, several inequalities, and integrals in different fields have been studied by various researchers and mathematicians [15–21].

Recently, Ali et al. [22], Ghayasuddin and Khan [23], Khan et al. [24–26], and Nisar et al. [27,28] gave certain interesting new class of integral formulas involving some types of the generalized Bessel functions, which are expressed in terms of the generalized (right) hypergeometric function (for more details, see [6–8]). Motivated by these recent studies, we start by mentioning some notations with some results and definitions used in this study.

The k -Bessel function of the first kind has been established and discussed by Romero et al. [16] given as follows:

where k ∈ ℜ ; λ , γ , α , ϑ ∈ C ; ℜ ( λ ) > 0 ; and ℜ ( ϑ ) > 0 .

The generalized k -Bessel function of the first kind introduced by [27] has the form:

where k ∈ ℜ ; λ , γ , ν , b , c ∈ C ; ℜ ( ν ) > 0 , and ℜ ( λ ) > 0 , and ( γ ) r , k is the k -Pochhammer symbol [1] defined by:

The k -gamma functions Γ k ( x ) , where Γ k ( x ) ⟶ Γ ( x ) if k ⟶ 1 , are given by the following relationship:

To obtain our main results, we use the following definitions of integral transforms:

1. Euler transform [29] of the function f ( z ) is defined by:

   (1.5) B { f ( z ) ; λ , μ } = ∫ 0 1 z λ − 1 ( 1 − z ) μ − 1 f ( z ) d z , λ , μ ∈ C and ℜ ( λ ) , ℜ ( μ ) > 0 .

2. Laplace transform [29] of the function f ( z ) is defined by:

   (1.6) F ( s ) = L { f ( z ) ; s } = ∫ 0 ∞ e − s z f ( z ) d z , ℜ ( s ) > 0 .

3. We also recall the following known Whittaker transform results [30]:

   (1.7) ∫ 0 ∞ t ϑ − 1 e − t 2 W λ , μ ( t ) d t = Γ 1 2 + μ + ϑ Γ 1 2 − μ + ϑ Γ ( 1 − λ + ϑ ) , ℜ ( ϑ ± μ ) > − 1 2 ,

   where W λ , μ is the Whittaker function [31] (see, e.g., [32])

   W λ , μ ( t ) = Γ ( − 2 μ ) Γ 1 2 − λ − μ M λ , μ ( t ) + Γ ( 2 μ ) Γ ( 1 2 + λ + μ ) M λ , − μ ( t ) ,

   where M λ , μ ( t ) is the confluent hypergeometric function written as:

   M λ , μ ( t ) = z 1 2 + μ e − 1 2 t F 1 1 1 2 + μ − λ ; 2 μ + 1 ; t .

   Also, we have the transformation formula

   (1.8) ∫ 0 ∞ t ϑ − 1 e − t 2 M λ , μ ( t ) d t = Γ ( 2 μ + 1 ) Γ 1 2 + μ + ϑ Γ ( λ − ϑ ) Γ ( 1 2 + μ − ϑ ) Γ ( 1 2 + μ + λ ) .

4. The K -transform [31] with a complex parameter a is defined by:

   (1.9) R ϑ { f ( z ) ; a } = g ( a ; ϑ ) = ∫ 0 ∞ ( a z ) 1 2 K ϑ ( a z ) f ( z ) d z ,

   where ℜ ( a ) > 0 and K ϑ ( z ) is the Bessel function of the second kind defined in ([31], p. 32) as:

   K ϑ ( z ) = π 2 z 1 2 W 0 , ϑ ( 2 z ) ,

   where W 0 , ϑ ( . ) is the Whittaker function defined in (1.7).

The generalized hypergeometric function is defined in [33] as:

provided p ≤ q , p = q + 1 and ∣ x ∣ < 1 , where ( A ) r is known as the Pochhammer symbol.

Also, the Fox-Wright generalization Ω q p of the hypergeometric function F q p [34–36] is

where A i > 0 ( i = 1 , 2 , 3 , … , p ) ; B i > 0 ( i = 1 , 2 , 3 , … , q ) ; and 1 + ∑ i = 1 q B i − ∑ i = 1 p A i ≥ 0 for a suitably bounded value of ∣ x ∣ .

In [20], the generalized k -Wright function defined for k ∈ ℜ + ; z ∈ C , α i , β j ∈ ℜ ( α i , β j ≠ 0 ; i = 1 , 2 , 3 , … , p ; j = 1 , 2 , 3 , … , q ) and ( s i + α i r ) , ( d j + β j r ) ∈ C \ k Z − is written as:

In this article, we initiate a generalized k -Bessel function. We also establish some integral formulas for the generalized k -Bessel function by applying the Laplace transform, Euler transform, Whittaker transform, and k -transform. Moreover, as consequences, we present some special cases of our obtained results.

2 Main results

First, we introduce the generalized k -Bessel function of the first kind in the following form:

for k ∈ ℜ ; λ , γ , α , ϑ ∈ C ; ℜ ( λ ) > 0 ; and ℜ ( ϑ ) > 0 , where ( γ ) n , k and ( δ ) n , k are the k -Pochhammer symbol.

Note that

The integral transforms for generalized k -Bessel functions such as Euler, Laplace, and Whittaker are listed in the following theorems:

If we let k = 1 in Theorem 2.1, we state the following:

Setting γ = δ , b = 1 , and c = − 1 in Theorem 2.1, we obtain

If we take k = 1 in Theorem 2.2, then we have

Moreover, if we set γ = δ in the aforementioned corollary, we obtain the following result:

In Theorem 2.3, by setting k = 1 we obtain

If we set k = 1 in Theorem 2.4, we obtain the following result:

Now, we list another theorem that can be proved in steps similar to the proof of Theorem 2.4, using the integral transform given in equation (1.8).