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Integral Transforms Involving the Product of Humbert and Bessel Functions and Its Application

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Integral Transforms Involving the Product of Humbert and Bessel Functions and Its Application AIMS Mathematics, 5(2): 1260–1274. DOI:10.3934/math.2020086 Received: 10 October 2019 Accepted: 06 January 2020 http://www.aimspress.com/journal/Math Published: 19 January 2020 Research article Integral transforms involving the product of Humbert and Bessel functions and its application A. Belafhal1, N. Nossir1, L. Dalil-Essakali1 and T. Usman2;∗ 1 LPNAMME, Laser Physics Group, Department of Physics, Faculty of Sciences, Choua¨ıb Doukkali University, P. B 20, 24000 El Jadida, Morocco 2 Department of Mathematics, School of Basic and Applied Sciences, Lingaya’s Vidyapeeth, Faridabad-121002, Haryana, India * Correspondence: Email: [email protected]; Tel: +917417803673. Abstract: In this paper, we develop some integral transforms involving a product of Humbert and Bessel functions with a weight e−γx2 . These integral transforms will be evaluated in terms of hypergeometric functions. Various transformation formulae are also evaluated in terms of Appell functions to complete this study. Some special cases of the evaluated integrals yield some infinite series of generalized hypergeometric and Appell functions. As application, one of our main results is investigated to give an expression of the Generalized Humbert-Gaussian beams (GHGBs) propagating through a paraxial ABCD optical system. Keywords: integral transforms; Bessel functions; Humbert functions; Appell functions; Hypergeometric functions Mathematics Subject Classification: 33B15, 33C10, 33C15 1. Introduction Integral transforms involving special functions have gained considerable attention in the bibliography. In the last decade, many papers are investigated to study the integral transforms involving the product of Bessel and other special functions (see [4,8,9]). In view of this, it is worth investigating a general integral transform involving Humbert and Bessel functions with a weight e−γx2 . This integral transform is important in evaluating the expression of the generalized Humbert-Gaussian beams propagating in the space. A closed form of the considered integral will be derived. To the best of our knowledge, the results of the present contribution have not been previously published. The following definitions are essential to recall for the present investigation: 1261 The four two-variables hypergeometric functions, called confluent Humbert functions [7], are defined by X1 X1 (a) (b) xr ys Ψ (a; b; c; d; x; y) = r+s r ; (1.1) 1 (c) (d) r! s! r=0 s=0 r s X1 X1 (a) (b) xr ys Φ (a; b; c; x; y) = r+s r ; (1.2) 1 (c) r! s! r=0 s=0 r+s X1 X1 (a) (b) (c) xr ys Ξ (a; b; c; d; x; y) = r s r ; (1.3) 1 (d) r! s! r=0 s=0 r+s and X1 X1 (a) (b) xr ys Ξ (a; b; c; x; y) = r r ; (1.4) 2 (c) r! s! r=0 s=0 r+s with jxj < 1; jyj < 1 and c; d , 0; −1; −2;::: The hypergeometric function 2F1 is defined by (see [1]) X1 (α) (β) zn F (α, β; γ; z) = n n ; (1.5) 2 1 (γ) n! n=0 n where (α)n is the Pochhammer symbol defined by Γ(α + n) (α) = ; n Γ(α) with Γ is the gamma function (see [1]). The Kummer function is defined by the series X1 (α) zk F (α; β; z) = k : (1.6) 1 1 (β) k! k=0 k In terms of the hypergeometric function (1.5), it’s easy to obtain Humbert functions (1.1)–(1.4) (see [13]). Therefore, we can write X1 (a) ys Ψ (a; b; c; d; x; y) = s F (a + s; b; c; x) ; (1.7) 1 (d) s! 2 1 s=0 s X1 (a) ys Φ (a; b; c; x; y) = s F (a + s; b; c + s; x) ; (1.8) 1 (c) s! 2 1 s=0 s X1 (b) ys Ξ (a; b; c; d; x; y) = s F (a; c; d + s; x) ; (1.9) 1 (d) s! 2 1 s=0 s and X1 1 ys Ξ (a; b; c; x; y) = F (a; b; c + s; x) ; (1.10) 2 (c) s! 2 1 s=0 s AIMS Mathematics Volume 5, Issue 2, 1260–1274. 1262 with jxj < 1; jyj < 1 and c; d , 0; −1; −2;::: The four Appell series [2], which are double hypergeometric series, are defined by X1 X1 (a) (b) (b0) xm yn F a; b; b0; c; x; y = m+n m n ; (1.11) 1 (c) m! n! m=0 n=0 m+n with maxfjxj; jyjg < 1; X1 X1 (a) (b) (b0) xm yn F a; b; b0; c; c0; x; y = m+n m n ; (1.12) 2 (c) (c0) m! n! m=0 n=0 m n with jxj + jyj < 1; X1 X1 (a) (a0) (b) (b0) xm yn F a; a0; b; b0; c; x; y = m n m n ; (1.13) 3 (c) m! n! m=0 n=0 m+n with maxfjxj; jyjg < 1; and X1 X1 (a) (b) xm yn F a; b; c; c0; x; y = m+n m+n ; (1.14) 4 (c) (c0) m! n! m=0 n=0 m n p p with jxj + jyj < 1. These expressions, with c and c0 are neither zero nor a negative integer, can be expressed in terms of 2F1 as follows (see [13]) X1 (a) (b) xm F a; b; b0; c; x; y = m m F a + m; b0; c + m; y ; (1.15) 1 (c) m! 2 1 m=0 m X1 (a) (b) xm F a; b; b0; c; c0; x; y = m m F a + m; b0; c0; y ; (1.16) 2 (c) m! 2 1 m=0 m X1 (a) (b) xm F a; a0; b; b0; c; x; y = m m F a0; b0; c + m; y ; (1.17) 3 (c) m! 2 1 m=0 m and X1 (a) (b) xm F a; b; c; c0; x; y = m m F a + m; b + m; c0; y : (1.18) 4 (c) m! 2 1 m=0 m 2. Main results 2.1. Evaluation of the integral Im In this section, we now evaluate the following integral transform containing Humbert functions F (= Ψ1; Φ1; Ξ1 or Ξ2): Z 1 2m+d −γt2 2 Im(F ) = t e Jd−1(βt)F (y; xt )dt; (2.1) 0 AIMS Mathematics Volume 5, Issue 2, 1260–1274. 1263 <(2m + 2d) > 0 and <(γ) > 0: In (2.1), we replaced x and y with y and xt2 respectively in (1.1), (1.2), (1.3) and (1.4). Theorem 2.1. The following transformations hold true: 1 x q 2 ! X (m + d)q ( γ ) −β I (Ψ ) = A (a) F (a + q; b; d; y) F m + d + q; d; ; (2.2) m 1 q (c0) q! 2 1 1 1 4γ q=0 q 1 x q 2 ! X (a)q(m + d)q ( γ ) −β I (Φ ) = A F (a + q; b; d + q; y) F m + d + q; d; ; (2.3) m 1 (d) q! 2 1 1 1 4γ q=0 q 1 x q 2 ! X (b)q(m + d)q ( γ ) −β I (Ξ ) = A F a; d; c0 + q; y F m + d + q; d; ; (2.4) m 1 (c0) q! 2 1 1 1 4γ q=0 q and 1 x q 2 ! X (m + d)q ( γ ) −β I (Ξ ) = A F (a; b; d + q; y) F m + d + q; d; ; (2.5) m 2 (d) q! 2 1 1 1 4γ q=0 q d A 1 β where, = βγm 2γ (d)m. Proof. The proof of integral transform in (2.2) is as follows: By substituting (1.7) in (2.1), we obtain 1 q X (a)q x I (Ψ ) = A F (a + q; b; d; y) I ; (2.6) m 1 (c0) q! 2 1 q q=0 q where Z 1 µ −γt2 Iq = t e Jd−1(βt)dt; (2.7) 0 with µ = 2m + 2q + d: (2.8) By means of the identity [6] Z 1 ν ν+µ+1 2 ! µ −γt2 1 β Γ( 2 ) ν + µ + 1 −β t e Jν(βt)dt = ν+µ+1 1F1 ; ν + 1; (2.9) 0 2 2 ν!γ 2 2 4γ (<(µ + ν) > −1 ; <(γ) > 0); and using (1.6), (2.7) can be written as − ! 1 βd 1 1 Γ(m + q + d) −β2 I = F m + q + d; d; : (2.10) q 2 2 (d − 1)! γm+q+d 1 1 4γ Substituting (2.10) into (2.6), the assertion (2.1) of Theorem 2.1 directly follows. AIMS Mathematics Volume 5, Issue 2, 1260–1274. 1264 Note that if m = 0 and c = d, one finds the undermentioned integral: Z 1 c −γt2 0 2 Im=0(Ψ1) = t e Jc−1(βt)Ψ1 a; b; c; c ; y; xt dt 0 !c 1 x q 2 ! (2.11) 1 β X (a)q(c)q ( γ ) −β = F (a + q; b; c; y) F c + q; c; : β 2γ (c0) q! 2 1 1 1 4γ q=0 q Proof. The proof of integral transform (2.3) is as follows: 2 2 Considering F (y; xt ) = Φ1 a; b; d; y; xt given by (1.8), (2.1) becomes in this case 1 q X (a)q x I (Φ ) = F (a + q; b; d + q; y) I ; (2.12) m 1 (d) q! 2 1 q q=0 q where Iq is given by (2.7). By using the identity (2.9), (2.12) can be written as !d ! 1 β Γ(m + q + d) −β2 I (Φ ) = F m + q + d; d; : (2.13) m 1 β 2γ Γ(d)γm+q 1 1 4γ Substituting this last equation in the expression of Im(Φ1), yields the assertion (2.3) of Theorem 2.1.
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