Certain Fractional Integral Formulas Involving the Product of Generalized Bessel Functions

Abstract

We apply generalized operators of fractional integration involving Appell’s function 𝐹₃(⋅) due to Marichev-Saigo-Maeda, to the product of the generalized Bessel function of the first kind due to Baricz. The results are expressed in terms of the multivariable generalized Lauricella functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals are also presented. Some interesting special cases of our two main results are presented. We also point out that the results presented here, being of general character, are easily reducible to yield many diverse new and known integral formulas involving simpler functions.

1. Introduction, Definitions, and Preliminaries

The fractional calculus is nowadays one of the most rapidly growing subject of mathematical analysis. It is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders. The fractional integral operator involving various special functions has found significant importance and applications in various subfields of applicable mathematical analysis. Many applications of fractional calculus can be found in turbulence and fluid dynamics, stochastic dynamical system, plasma physics and controlled thermonuclear fusion, nonlinear control theory, image processing, nonlinear biological systems, astrophysics, and in quantum mechanics. Since last four decades, a number of workers like Love [1], McBride [2], Kalla [3, 4], Kalla and Saxena [5], Saigo [6, 7], Kilbas [8], Kilbas and Sebastian [9], Kiryakova [10, 11], Baleanu [12], Baleanu and Mustafa [13], Baleanu et al. [14], Baleanu et al. [15], Purohit and Kalla [16], Purohit [17], Agarwal [18–20] and Agarwal and Jain [21], and so forth have studied, in depth, the properties, applications, and different extensions of various operators of fractional calculus. A detailed account of fractional calculus operators along with their properties and applications can be found in the research monographs by Miller and Ross [22], Kiryakova [10], and so forth.

The computation of fractional derivatives (and fractional integrals) of special functions of one and more variables is important from the point of view of the usefulness of these results in the evaluation of generalized integrals and the solution of differential and integral equations. Motivated by these avenues of applications, a remarkably large number of fractional integral formulas involving a variety of special functions have been developed by many authors (see, e.g., [23–27]). Fractional integration formulae for the Bessel function and generalized Bessel functions are given recently by Kilbas and Sebastian [9], Saxena et al. [28], Malik et al. [29] and Purohit et al. [27]. A useful generalization of the Bessel function 𝑤_𝜈(𝑧) has been introduced and studied in [30–34]. Here we aim at presenting composition formula of Marichev-Saigo-Maeda fractional integral operators and the product of generalized Bessel function, which are expressed in terms of the multivariable generalized Lauricella functions. Some interesting special cases of our main results are also considered.

For our purpose, we begin by recalling some known functions and earlier works. This paper deals with two integral transforms defined for 𝑥>0 and 𝛼,𝛼^′,𝛽,𝛽^′,𝛾∈ℂ by 𝐼^{𝛼,𝛼′,𝛽,𝛽′,𝛾}_0,+𝑓(𝑥)=𝑥^−𝛼Γ𝛾^𝑥₀(𝑥−𝑡)^𝛾−1𝑡^−𝛼′𝐹₃𝛼,𝛼^′,𝛽,𝛽^′𝑡;𝛾;1−𝑥𝑥,1−𝑡ℜ𝛾,𝐼𝑑𝑥,>0_{𝛼,𝛼′,𝛽,𝛽′,𝛾0,−}𝑓𝑥)=𝑥_−𝛼′(Γ𝛾^∞_𝑥(𝑡−𝑥)^𝛾−1𝑡^−𝛼𝐹₃𝛼,𝛼^′,𝛽,𝛽^′𝑥;𝛾;1−𝑡𝑡,1−𝑥ℜ𝛾.𝑑𝑥,>0 (1)(2) These operators (integral transforms) were introduced by Marichev [35] as Mellin type convolution operators with a special function 𝐹₃(⋅) in the kernel. These operators were rediscovered and studied by Saigo in [36] as generalization of the so-called Saigo fractional integral operators (see [37]). Such further properties as, for example, their relations with the Mellin transform and with the hypergeometric operators (or the Saigo fractional integral operators), together with their decompositional, operational and other properties in the McBride space 𝐹_𝑝;𝜇 (see [38]) were studied by Saigo and Maeda [39].

In (1) and (2) the symbol 𝐹₃(⋅) denotes so-called 3rd Appell function (known also as Horn function) (see [40, page 413]): 𝐹₃𝛼,𝛼^′;𝛽,𝛽^′=;𝜂;𝑥;𝑦_{∞𝑚,𝑛=0}(𝛼)_𝑚𝛼^′_𝑛𝛽_𝑚𝛽^′_𝑛𝜂_𝑚+𝑛𝑥𝑚!𝑛!^𝑚𝑦^𝑛×max|𝑥|,𝑦<1. (3) The properties of this function are discussed in [40, pages 412–415]. In particular, its relation to the Gauss hypergeometric function is presented as follows: 𝐹₃=𝛼,𝜂−𝛼;𝛽,𝜂−𝛽;𝜂;𝑥;𝑦₂𝐹₁𝛼,𝛽;𝜂;𝑥+𝑦−𝑥𝑦. (4) It is known that the 3rd Appell function cannot be expressed as a product of two ₂𝐹₁ functions that satisfy pairs of linear partial differential equations of the second order.

We investigate compositions of integral transforms (1) and (2) with the product of generalized Bessel function of the first kind, 𝑤_𝜈(𝑧), which is defined for 𝑧∈ℂ⧵{0} and 𝑏,𝑐,𝜈∈ℂ with ℜ(𝜈)>−1 by the following series (see, e.g., [30, page 10, equation (1.15)]; for a very recent work, see also [31–33], [34, page 182, equation (2.2)], and [29, page 2, equation (8)]): 𝑤_𝜈(𝑧)=^∞_𝑘=0(−1)^𝑘𝑐^𝑘(𝑧/2)^𝜈+2𝑘(()),𝑘!Γ𝜈+𝑘+1+𝑏/2 (5) where ℂ denotes the set of complex numbers and Γ(𝑧) is the familiar Gamma function (see [41, Section 1.1]).

Then we show that the composition is expressed in terms of the multivariable generalized Lauricella functions due to Srivastava and Daoust (see, [42, page 454]) is a generalization of the Wright function _𝑝Ψ_𝑞 in several variables and defined by (see e.g., [43, page 107]) 𝑆^{𝑝;𝑝1;…;𝑝𝑛}_{𝑞;𝑞1;…;𝑞𝑛}(𝑎)_𝑝∶𝛼^′_𝑝,…,𝛼^𝑛_𝑝∶𝑐^′_𝑝1∶𝛾^′_𝑝1;…;(𝑐)^𝑛_𝑝1∶𝛾^𝑛_𝑝1;(𝑏)_𝑞∶𝛽^′_𝑞,…,𝛽^𝑛_𝑞∶𝑑^′_𝑞1∶𝛿^′_𝑞1;…;(𝑑)^𝑛_𝑞1∶𝛿^𝑛_𝑞1;𝑧₁,…,𝑧_𝑛=^∞_{𝑚1,…,𝑚𝑛=0}𝐴𝑚₁,…,𝑚_𝑛^𝑛_𝑗=1𝑧^𝑚𝑗_𝑗𝑚_𝑗!, (6) where 𝐴𝑚₁,…,𝑚_𝑛=∏^𝑝_𝑗=1Γ𝑎_𝑗+∑^𝑚_𝑗=1𝑚_𝑘𝛼^𝑘_𝑗∏^𝑞_𝑗=1Γ𝑏_𝑗+∑^𝑚_𝑗=1𝑚_𝑘𝛽^𝑘_𝑗×^𝑛_𝑗=1∏^𝑝𝑘_𝑗=1Γ𝑐^𝑘_𝑗+𝑚_𝑘𝛾^𝑘_𝑗∏^𝑞𝑘_𝑗=1Γ𝑑^𝑘_𝑗+𝑚_𝑘𝛿^𝑘_𝑗. (7)

The coefficients 𝛼^𝑘_𝑗 (𝑗=1,…,𝑝), 𝛽^𝑘_𝑗 (𝑗=1,…,𝑞), 𝛾^𝑘_𝑗 (𝑗=1,…,𝑝_𝑘), and 𝛿^𝑘_𝑗 (𝑗=1,…,𝑞_𝑘), for all 𝑘=1,…,𝑛, are real and positive and (𝑎_𝑝) means the array of 𝑝-parameters 𝑎₁,…,𝑎_𝑝; with similar interpretations for (𝑏_𝑝), (𝛾¹_𝑝1), (𝛼¹_𝑝1), and so forth, and where (𝑎)_𝑛 is the Pochhammer symbol defined (for 𝑎∈ℂ) by (see [41, page 2 and pages 4–6]) (𝑎)_𝑛(),𝑎()⋯(),({=Γ()∶=1,𝑛=0𝑎+1𝑎+𝑛−1𝑛∈ℕ∶=1,2,3,…})𝑎+𝑛Γ(𝑎)𝑎∈ℂ⧵ℤ⁻₀, (8) and ℤ⁻₀ denotes the set of nonpositive integers.

The paper is organized as follows. The composition formula of Marichev-Saigo-Maeda fractional integral operators (1) and (2) with the product of generalized Bessel function of the first kind (5) is proved in terms of the multivariable generalized Lauricella functions (6) in Sections 2 and 3, respectively. The corresponding results for the corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville and Weyl type of fractional integrals are also presented in Sections 2 and 3. Special cases giving compositions of fractional integrals with the product of trigonometric functions and concluding remarks are considered in Section 4.

2. Left-Side Fractional Integration of Generalized Bessel Functions

Our results in this Section are based on the preliminary assertions giving composition formula of fractional integral (1) with a power function.

Lemma 1 (Saigo and Maeda [39, Lemma 1]). Let 𝛼,𝛼^′,𝛽,𝛽^′,𝛾∈ℂ and if ℜ(𝛾)>0, ℜ(𝜌)>max{0,ℜ(𝛼+𝛼^′+𝛽−𝛾),ℜ(𝛼^′−𝛽^′)}, then 𝐼^{𝛼,𝛼′,𝛽,𝛽′,𝛾}_0,+𝑥^𝜌−1(𝑥)=Γ𝜌Γ𝜌+𝛾−𝛼−𝛼^′−𝛽Γ𝜌+𝛽^′−𝛼^′Γ𝜌+𝛽^′Γ𝜌+𝛾−𝛼−𝛼^′Γ𝜌+𝛾−𝛼^′−𝛽×𝑥^{𝜌+𝛾−𝛼−𝛼′−1}. (9)

The Marichev-Saigo-Maeda fractional integration (1) of product of generalized Bessel functions (5) and Binomial function is given by the following result.

Theorem 2. Let 𝛼,𝛼^′,𝛽,𝛽^′,𝜎,𝜆,𝛾,𝜈_𝑗,𝜌_𝑗,𝑏,𝑐∈ℂ and satisfying the inequalities ℜ𝛾>0,ℜ(𝜎)>0,ℜ^𝑛_𝑗=1𝜈_𝑗+𝑏+12ℜ>0,𝜎+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗>max0,ℜ𝛼+𝛼^′+𝛽−𝛾,ℜ𝛼^′−𝛽^′, (10) then for 𝑥>0, |𝑎𝑥/𝑏|<1, one has 𝐼^{(𝛼,𝛼′,𝛽,𝛽′,𝛾)}_0,+𝑡^𝜎−1(𝑏−𝑎𝑡)^𝑛−𝜆_𝑗=1𝑤_𝜈𝑗𝑎_𝑗𝑡^𝜌𝑗𝑥)=𝑥_{𝜎+𝛾−𝛼−𝛼′−1}(Γ(𝜆)𝑏^𝜆𝑛_𝑗=1𝑎_𝑗𝑥^𝜌𝑗2^𝜈𝑗𝑆^3;0;…;0;1_3;1;…;1;0−𝐴,𝐶,𝐸∶𝐺;𝑎𝐵,𝐷,𝐹∶𝐻;^2𝑛𝑥^2𝜌𝑛4,𝑎…,−^2𝑛𝑥^2𝜌𝑛4,𝑎𝑥𝑏, (11) where 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹, 𝐺, and 𝐻 are given by the following 𝐴=𝜎+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗∶2𝜌₁,…,2𝜌_𝑛,(1),𝐵=𝜎+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+𝛽^′∶2𝜌₁,…,2𝜌_𝑛,(1),𝐶=𝜎+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+𝛾−𝛽−𝛼−𝛼^′∶2𝜌₁,…,2𝜌_𝑛,(1),𝐷=𝜎+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+𝛾−𝛼−𝛼^′∶2𝜌₁,…,2𝜌_𝑛,(1),𝐸=𝜎+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+𝛽^′−𝛼^′∶2𝜌₁,…,2𝜌_𝑛,(1),𝐹=𝜎+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+𝛾−𝛼^′−𝛽∶2𝜌₁,…,2𝜌_𝑛,(1),[],𝜈𝐺=−;…;−;𝜆∶1𝐻=₁+1+𝑏2∶𝑘₁;…;𝜈_𝑛+1+𝑏2∶𝑘_𝑛;0. (12)(13)(14)(15)(16)(17)(18)(19)

Proof. For sake of convenience, let the left-hand side of the (11) be denoted by ℐ. Using definition (5) and the binomial expansion, namely, (𝑏−𝑎𝑡)^−𝜆=𝑏^∞−𝜆_𝑘=0(𝜆)_𝑘𝑘!𝑎𝑡𝑏^𝑘𝑎𝑡𝑏<1, (20) we find ℐ=𝐼^{(𝛼,𝛼′,𝛽,𝛽′,𝛾)}_0,+𝑡^𝜎−1(𝑏−𝑎𝑡)^𝑛−𝜆_𝑗=1𝑤_𝜈𝑗𝑎_𝑗𝑡^𝜌𝑗𝑥)=𝐼_{(𝛼,𝛼′,𝛽,𝛽′,𝛾)0,+}(×𝑡^𝜎−1𝑏^∞−𝜆_𝑘=0(𝜆)_𝑘𝑘!𝑎𝑡𝑏^𝑘×^∞_𝑘1=0(−1)^𝑘1(𝑐)^𝑘1𝑎₁𝑡^𝜌1/2^𝜈1+2𝑘1𝑘₁!Γ𝜈₁+𝑘₁+(𝑏+1)/2×⋯×^∞_𝑘𝑛=0(−1)^𝑘𝑛(𝑐)^𝑘𝑛𝑎_𝑛𝑡^𝜌𝑛/2^{𝜈𝑛+2𝑘𝑛}𝑘_𝑛!Γ𝜈_𝑛+𝑘_𝑛+)(𝑥).((𝑏+1/2) (21)
Following the convergence condition of Theorem 2, for any 𝑘∈ℕ₀, ℜ𝜎+𝑘+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+2^𝑛_𝑗=1𝜈_𝑗𝑘_𝑗≥ℜ𝜎+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗>max0,ℜ𝛼+𝛼^′+𝛽−𝛾,ℜ𝛼^′−𝛽^′ (22) and then changing the order of integration and summation, we obtain ℐ=𝑏^∞−𝜆_{𝑘,𝑘1,…,𝑘𝑛=0}(𝜆)_𝑘(−1)^𝑘1(𝑐)^𝑘1𝑎₁/2^𝜈1+2𝑘1𝑘!𝑘₁!Γ𝜈₁+𝑘₁+()𝑎𝑏+1/2𝑏^𝑘()×⋯−1^𝑘𝑛(𝑐)^𝑘𝑛𝑎_𝑛/2^{𝜈𝑛+2𝑘𝑛}𝑘_𝑛!Γ𝜈_𝑛+𝑘_𝑛+()×𝐼𝑏+1/2_{(𝛼,𝛼′,𝛽,𝛽′,𝛾)0,+}×𝑡^{𝜎+𝑘+𝜈1𝜌1+⋯+𝜈𝑛𝜌𝑛+2𝜈1𝑘1+⋯+2𝜈𝑛𝑘𝑛−1}(𝑥). (23) Now on applying Lemma 1 and using (9) with 𝜌 replaced by (𝜎+𝑘+𝜈₁𝜌₁+⋯+𝜈_𝑛𝜌_𝑛+2𝜈₁𝑘₁+⋯+2𝜈_𝑛𝑘_𝑛), we obtain ℐ=𝑥^{𝜎+𝛾−𝛼−𝛼′−1}Γ(𝜆)𝑏^𝜆×^𝑛_𝑗=1𝑎_𝑗𝑥^𝜌𝑗2^𝜈𝑗×^∞_{𝑘,𝑘1,…,𝑘𝑛=0}Γ(𝜆+𝑘)Γ𝜎+𝑘+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+2^𝑛_𝑗=1𝜈_𝑗𝑘_𝑗×Γ𝜎+𝛽^′+𝑘+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+2^𝑛_𝑗=1𝜈_𝑗𝑘_𝑗⁻¹×Γ𝜎+𝛾−𝛼−𝛼^′−𝛽+𝑘+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+2^𝑛_𝑗=1𝜈_𝑗𝑘_𝑗×Γ𝜎+𝛽^′−𝛼^′+𝑘+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+2^𝑛_𝑗=1𝜈_𝑗𝑘_𝑗×Γ𝜎+𝛾−𝛼−𝛼^′+𝑘+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+2^𝑛_𝑗=1𝜈_𝑗𝑘_𝑗×Γ𝜎+𝛾−𝛼^′−𝛽+𝑘+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+2^𝑛_𝑗=1𝜈_𝑗𝑘_𝑗⁻¹×1Γ𝜈₁+𝑘₁+(𝑏+1)/2⋯Γ𝜈_𝑛+𝑘_𝑛+()𝑎𝑏+1/2×−¹𝑥^2𝜌1/4^𝑘1𝑘₁!𝑎×⋯×−^𝑛𝑥^2𝜌𝑛/4^𝑘𝑛𝑘_𝑛!×(𝑎𝑥/𝑏)^𝑘.𝑘! (24)
This, in accordance with (6), gives the required result (11). This completed the proof of the Theorem 2.

On setting 𝛼^′=0 in Theorem 2, we get the following Saigo hypergeometric fractional image of the product of generalized Bessel function of the first formula kind 𝑤_𝜈(𝑧).

Corollary 3. If 𝛼,𝛽,𝛽^′,𝜎,𝜆,𝛾,𝜈_𝑗,𝜌_𝑗,𝑏,𝑐∈ℂ and 𝑥>0, |𝑎𝑥/𝑏|<1ℜ𝛾>0,ℜ(𝜎)>0,ℜ^𝑛_𝑗=1𝜈_𝑗+𝑏+12ℜ>0,𝜎+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗>0, (25) then there hold the results: 𝐼^{(𝛾,𝛼−𝛾,−𝛽)}_0,+𝑡^𝜎−1(𝑏−𝑎𝑡)^𝑛−𝜆_𝑗=1𝑤_𝜈𝑗𝑎_𝑗𝑡^𝜌𝑗𝑥)=𝑥_{𝜎+𝛾−𝛼−1}(Γ(𝜆)𝑏^𝜆𝑛_𝑗=1𝑎_𝑗𝑥^𝜌𝑗2^𝜈𝑗𝑆^2;0;…;0;1_2;1;…;1;0−𝐴,𝐶∶𝐺;𝑎𝐷,𝐹∶𝐻;²¹𝑥^2𝜌14,𝑎…,−^2𝑛𝑥^2𝜌𝑛4,𝑎𝑥𝑏, (26) where 𝐴, 𝐶, 𝐷, 𝐹, 𝐺, and 𝐻 are given by (12), (14), (15), (17), (18), and (19), respectively.

Again, on letting 𝛼^′=0 and 𝛼=0 in Theorem 2, we get the Riemann-Liouville fractional image of the product of generalized Bessel function of the first formula kind; asserted by the following corollary.

Corollary 4. Let 𝛽,𝛽^′,𝜎,𝜆,𝛾,𝜈_𝑗,𝜌_𝑗,𝑏,𝑐∈ℂ and 𝑥>0, |𝑎𝑥/𝑏|<1, and ℜ𝛾>0,ℜ(𝜎)>0,ℜ^𝑛_𝑗=1𝜈_𝑗+𝑏+12ℜ>0,𝜎+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗>0. (27) Then, one has 𝐼^(𝛾)_0,+𝑡^𝜎−1(𝑏−𝑎𝑡)^𝑛−𝜆_𝑗=1𝑤_𝜈𝑗𝑎_𝑗𝑡^𝜌𝑗𝑥)=𝑥_{𝜎+𝛾−1}(Γ(𝜆)𝑏^𝜆𝑛_𝑗=1𝑎_𝑗𝑥^𝜌𝑗2^𝜈𝑗𝑆^1;0;…;0;1_1;1;…;1;0−𝐴∶𝐺;𝑎𝐷∶𝐻;²¹𝑥^2𝜌14,𝑎…,−^2𝑛𝑥^2𝜌𝑛4,𝑎𝑥𝑏, (28) where 𝐴, 𝐷, 𝐺, and 𝐻 are given by (12), (15), (18), and (19), respectively.

3. Right-Side Fractional Integration of Generalized Bessel Functions

In the sequel, we use the following results.

Lemma 5 (Saigo and Maeda [39, Lemma 2]). Let 𝛼,𝛼^′,𝛽,𝛽^′,𝛾∈ℂ and if ℜ(𝛾)>0, ℜ(𝜌)<1+min{ℜ(−𝛽),ℜ(𝛼+𝛼^′−𝛾),ℜ(𝛼+𝛽^′−𝛾)}, then 𝐼^{𝛼,𝛼′,𝛽,𝛽′,𝛾}_0,−𝑥^𝜌−1(𝑥)=Γ1−𝜌−𝛽Γ1−𝜌−𝛾+𝛼+𝛼^′×Γ1−𝜌+𝛼+𝛽^′×ΓΓ−𝛾1−𝜌1−𝜌+𝛼+𝛼^′+𝛽^′−𝛾×Γ1−𝜌+𝛼−𝛽⁻¹𝑥^{𝜌+𝛾−𝛼−𝛼′−1}. (29)

The Marichev-Saigo-Maeda fractional integration (2) of product of generalized Bessel functions (5) and Binomial function is given by the following result.

Theorem 6. Let 𝛼,𝛼^′,𝛽,𝛽^′,𝜎,𝜆,𝛾,𝜈_𝑗,𝜌_𝑗,𝑏,𝑐∈ℂ and 𝑥>0, |𝑎/𝑏𝑥|<1 such that ℜ𝛾>0,ℜ(𝜎)>0,ℜ^𝑛_𝑗=1𝜈_𝑗+𝑏+12ℜ>0,𝜎+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗<1+minℜ−𝛽,ℜ𝛼+𝛼^′−𝛾,ℜ𝛼+𝛽^′−𝛾. (30) Then, one has 𝐼^{(𝛼,𝛼′,𝛽,𝛽′,𝛾)}_0,−𝑡^𝜎−1𝑎𝑏−𝑡^𝑛−𝜆_𝑗=1𝑤_𝜈𝑗𝑎_𝑗𝑡^𝜌𝑗𝑥)=𝑥_{𝜎+𝛾−𝛼−𝛼′−1}(Γ(𝜆)𝑏^𝜆𝑛_𝑗=1𝑎_𝑗2𝑥^{𝜌𝑗𝜈𝑗}𝑆^3;0;…;0;1_3;1;…;1;0𝐴^′,𝐶^′,𝐸^′𝐵∶𝐺;_′,𝐷^′,𝐹^′−𝑎∶𝐻;²¹4𝑥^2𝜌1,𝑎…,−^2𝑛4𝑥^2𝜌𝑛,𝑎,𝑏𝑥 (31) where 𝐴^′, 𝐵^′, 𝐶^′, 𝐷^′, 𝐸^′, and 𝐹^′ are given by the following: 𝐴^′=1−𝜎−𝛽+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗∶2𝜌₁,…,2𝜌_𝑛,(1)𝐵_′,=1−𝜎+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗∶2𝜌₁,…,2𝜌_𝑛,(1)𝐶_′,=1−𝜎−𝛾+𝛼+𝛼^′+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗∶2𝜌₁,…,2𝜌_𝑛,(1),𝐷^′=1−𝜎+𝛼−𝛽+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗∶2𝜌₁,…,2𝜌_𝑛,(1)𝐸_′,=1−𝜎−𝛾+𝛼+𝛽^′+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗∶2𝜌₁,…,2𝜌_𝑛,(1),𝐹^′=1−𝜎−𝛾+𝛼+𝛼^′+𝛽^′+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗∶2𝜌₁,…,2𝜌_𝑛,(1), (32)(33)(34)(35)(36)(37) and 𝐺,𝐻 are given by (18) and (19), respectively.

Proof. For convenience, let the left-hand side of the (31) be denoted by 𝒥. Applying definition (5) and the binomial expansion for (𝑏−𝑎𝑡)^−𝜆 and then changing the order of integration and summation, we find 𝒥=𝑏^∞−𝜆_{𝑘,𝑘1,…,𝑘𝑛=0}(𝜆)_𝑘(−1)^𝑘1(𝑐)^𝑘1𝑎₁/2^𝜈1+2𝑘1𝑘!𝑘₁!Γ𝜈₁+𝑘₁+()×𝑎𝑏+1/2𝑏^𝑘⋯(−1)^𝑘𝑛(𝑐)^𝑘𝑛𝑎_𝑛/2^{𝜈𝑛+2𝑘𝑛}𝑘_𝑛!Γ𝜈_𝑛+𝑘_𝑛+()×𝐼𝑏+1/2_{(𝛼,𝛼′,𝛽,𝛽′,𝛾)0,+}×𝑡^{𝜎−𝑘−𝜈1𝜌1−⋯−𝜈𝑛𝜌𝑛−2𝜈1𝑘1−⋯−2𝜈𝑛𝑘𝑛−1}(𝑥). (38) Following the convergence condition of Theorem 6, for any 𝑘∈ℕ₀ℜ𝜎−𝑘−^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗−2^𝑛_𝑗=1𝜈_𝑗𝑘_𝑗≤ℜ𝜎−^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗<1+minℜ−𝛽,ℜ𝛼+𝛼^′−𝛾,ℜ𝛼+𝛽^′−𝛾. (39)
Now, on making use of Lemma 5 and (9), we obtain 𝒥=𝑥^{𝜎+𝛾−𝛼−𝛼′−1}Γ(𝜆)𝑏^𝜆×^𝑛_𝑗=1𝑎_𝑗2𝑥^{𝜌𝑗𝜈𝑗}×^∞_{𝑘,𝑘1,…,𝑘𝑛=0}Γ(𝜆+𝑘)Γ1−𝜎−𝛽+𝑘+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+2^𝑛_𝑗=1𝜈_𝑗𝑘_𝑗×Γ1−𝜎−𝛾+𝛼+𝛼^′+𝛽^′++𝑘_𝑛𝑗=1𝜈_𝑗𝜌_𝑗+2^𝑛_𝑗=1𝜈_𝑗𝑘_𝑗⁻¹×Γ1−𝜎−𝛾+𝛼+𝛼^′++𝑘_𝑛𝑗=1𝜈_𝑗𝜌_𝑗+2^𝑛_𝑗=1𝜈_𝑗𝑘_𝑗×Γ1−𝜎−𝛾+𝛼+𝛽^′++𝑘_𝑛𝑗=1𝜈_𝑗𝜌_𝑗+2^𝑛_𝑗=1𝜈_𝑗𝑘_𝑗×Γ1−𝜎+𝑘+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+2^𝑛_𝑗=1𝜈_𝑗𝑘_𝑗×Γ1−𝜎+𝛼−𝛽+𝑘+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗+2^𝑛_𝑗=1𝜈_𝑗𝑘_𝑗⁻¹×1Γ𝜈₁+𝑘₁+(𝑏+1)/2⋯Γ𝜈_𝑛+𝑘_𝑛+()−𝑎𝑏+1/2₁/4𝑥^2𝜌1𝑘1𝑘₁!𝑎⋯−^𝑛/4𝑥^{2𝜌𝑛𝑘𝑛}𝑘_𝑛!(𝑎/𝑏𝑥)^𝑘.𝑘! (40)
This, in accordance with (6), gives the required result (31). This completed the proof of Theorem 6.

On taking 𝛼^′=0 in Theorem 6, we get the right-sided Saigo hypergeometric fractional image of the product of generalized Bessel function of the first formula kind, 𝑤_𝜈(𝑧) as follows.

Corollary 7. Let 𝛼,𝛽,𝛽^′,𝜎,𝜆,𝛾,𝜈_𝑗,𝜌_𝑗,𝑏,𝑐∈ℂ and 𝑥>0, |𝑎/𝑏𝑥|<1, and ℜ𝛾>0,ℜ(𝜎)>0,ℜ^𝑛_𝑗=1𝜈_𝑗+𝑏+12ℜ>0,𝜎+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗>0. (41) Then 𝐼^{(𝛾,𝛼−𝛾,−𝛽)}_𝑥,∞𝑡^𝜎−1𝑎𝑏−𝑡^𝑛−𝜆_𝑗=1𝑤_𝜈𝑗𝑎_𝑗𝑡^𝜌𝑗𝑥)=𝑥_{𝜎+𝛾−𝛼−𝛼′−1}(Γ(𝜆)𝑏^𝜆𝑛_𝑗=1𝑎_𝑗2𝑥^{𝜌𝑗𝜈𝑗}𝑆^2;0;…;0;1_2;1;…;1;0𝐴^′,𝐶^′𝐵∶𝐺;_′,𝐷^′−𝑎∶𝐻;²¹4𝑥^2𝜌1,𝑎…,−^2𝑛4𝑥^2𝜌𝑛,𝑎,𝑏𝑥 (42) where 𝐴^′, 𝐵^′, 𝐶^′, 𝐷^′, 𝐺 and 𝐻 are given by (32), (33), (34), (35), (18), and (19), respectively.

Again, on setting 𝛼^′=0 and 𝛼=0 in Theorem 6, we get the following image formula for the product of generalized Bessel function of the first kind involving Riemann-Liouville type fractional integral operator.

Corollary 8. Let 𝛽,𝛽^′,𝜎,𝜆,𝛾,𝜈_𝑗,𝜌_𝑗,𝑏,𝑐∈ℂ and 𝑥>0, |𝑎/𝑏𝑥|<1, and ℜ𝛾>0,ℜ(𝜎)>0,ℜ^𝑛_𝑗=1𝜈_𝑗+𝑏+12ℜ>0,𝜎+^𝑛_𝑗=1𝜈_𝑗𝜌_𝑗>0. (43) Then 𝐼^(𝛾)_𝑥,∞𝑡^𝜎−1𝑎𝑏−𝑡^𝑛−𝜆_𝑗=1𝑤_𝜈𝑗𝑎_𝑗𝑡^𝜌𝑗𝑥)=𝑥_{𝜎+𝛾−𝛼−𝛼′−1}(Γ(𝜆)𝑏^𝜆𝑛_𝑗=1𝑎_𝑗2𝑥^{𝜌𝑗𝜈𝑗}𝑆^1;0;…;0;1_1;1;…;1;0𝐶^′𝐵∶𝐺;_′−𝑎∶𝐻;²¹4𝑥^2𝜌1,𝑎…,−^2𝑛4𝑥^2𝜌𝑛,𝑎,𝑏𝑥 (44) where 𝐵^′, 𝐶^′, 𝐺, and 𝐻 are given by (33), (34), (18), and (19), respectively.

4. Consequence Results and Concluding Remarks

In this section, we briefly consider another variation of the results derived in the preceding sections. Bessel functions are important special functions that appear widely in science and engineering. Bessel functions of the first kind 𝐽_𝜈(𝑧) are oscillatory and may be regarded as generalizations of trigonometric functions. Indeed, for large argument (𝑧≥1) the function 𝜋𝑧/2𝐽_𝜈(𝑧) is well approximated by the trigonometric function cos(𝑧−𝜋𝜈/2−𝜋/4). Similarly, modified Bessel functions of the first kind 𝐼_𝜈(𝑧), which are Bessel functions of imaginary argument, may be regarded as generalization of exponentials. Exponential functions have the unique and special property that they are particularly easy to multiply and to raise to powers: 𝑒^𝑎𝑧𝑒^𝑏𝑧=𝑒^{(𝑎+𝑏)𝑧} and (𝑒^𝑧)^𝑟=𝑒^𝑟𝑧. Further, it can be easily seen that for 𝑐=1 and 𝑏=1, the Generalized Bessel function of the first kind (5) reduces to 𝐽_𝜈(𝑧), and when 𝑐=−1 and 𝑏=1 the function 𝑤_𝜈(𝑧) becomes 𝐼_𝜈(𝑧). Similarly, when 𝑐=1 and 𝑏=2, the function 𝑤_𝜈(𝑧) reduces to 2𝑗_𝜈/√𝜋, while if 𝑐=−1 and 𝑏=2, then 𝑤_𝜈(𝑧) becomes 2𝑖_𝜈/√𝜋. In the sequel from (5), we have 𝑤_𝜈(0)=0. Therefore, the results presented in this paper are easily converted in terms of the various special Bessel functions after some suitable parametric replacement.

For example, if we set 𝜈=−𝑏/2, then the generalized Bessel function 𝑤_𝜈(𝑧) in (5) have following relation with cosine function when 𝑐 is replaced by 𝑐² (see, e.g., [29]):𝑤_{−𝑏/2,𝑐2}(𝑧)=2𝑧^𝑏/2cos𝑐𝑧√𝜋. (45)

While if 𝜈=−𝑏/2 and 𝑐=−𝑐², 𝑤_𝜈(𝑧) in (5) have relation with hyperbolic cosine function as (see, e.g., [29]) 𝑤_{−𝑏/2,−𝑐2}(𝑧)=2𝑧^𝑏/2cosh𝑐𝑧√𝜋. (46) In the sequel, if we set 𝜈=1−𝑏/2, then the generalized Bessel function 𝑤_𝜈(𝑧) in (5) have following relation with sine function when 𝑐 is replaced by 𝑐² (see, e.g., [29]): 𝑤_{1−𝑏/2,𝑐2}(𝑧)=2𝑧^𝑏/2sin𝑐𝑧√𝜋. (47)

Further, if we put 𝜈=1−𝑏/2 and 𝑐=−𝑐² in (5), we have the following relation with 𝑤_𝜈(𝑧) and hyperbolic sine function as (see, e.g., [29]) 𝑤_{1−𝑏/2,−𝑐2}(𝑧)=2𝑧^𝑏/2sinh𝑐𝑧√𝜋. (48)

Now, by virtue of the relations (45) to (48), one can easily derive the Marichev-Saigo-Maeda type fractional integrals involving product of trigonometric functions. Therefore, we omit the details of these results.

Further, it is interesting to observe that, if we set at 𝛼^′=0, the results given by Malik et al. [29, pages 5 and 8, Theorems 4.1 and 5.1] follow as special cases of our main results. Again, if we set 𝜆=0 and 𝑏=𝑐=1, then our main results (11) and (31) follow the kown results due to [28]. Finally, if we set 𝜆=0, 𝑛=𝑏=𝑐=1 and give some suitable parametric replacement in the main results (11) and (31), we can arrive at the known results due to Purohit et al. [27] in a slightly different notation.

Fractional integral formulas involving products of Bessel functions have been developed and play an important role in several physical problems. In fact, Bessel functions are playing the important role in studying solutions of differential equations, integral equations [44], and they are associated with a wide range of problems in important areas of mathematical physics, like problems of acoustics, radio-physics, hydrodynamics, and atomic and nuclear physics. These considerations have led various workers in the field of special functions for exploring the possible extensions and applications for the Bessel functions. Among many properties of Bessel functions, they also have investigated some possible extensions of the Bessel functions. The generalized Bessel function defined by (5), possess the advantage that a number of Bessel functions, trigonometric functions and hyperbolic functions happen to be the particular cases of this function. Therefore, we conclude this paper with the remark that, the results deduced above are significant and can lead to yield numerous other fractional integrals involving various Bessel functions and trigonometric functions by the suitable specializations of arbitrary parameters in the theorems. More importantly, they are expected to find some applications to the solutions of fractional differential and integral equations. The results thus derived in this paper are general in character and likely to find certain applications in the theory of special functions.

Acknowledgments

The authors take this opportunity to express their deepest thanks to the members of the editorial and reviewer board for their valuable comments and essential suggestions to improve this paper as in the present form.