Theorems connecting Stieltjes transform and Hankel transform
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Theorems connecting Stieltjes transform and Hankel transform Virendra Kumar1
© Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract The aim of the present paper is to establish four theorems connecting Stieltjes trans‑ form and Hankel transform. By applications of the theorems established in this paper four new integral formulae involving special functions are obtained. Due to the general nature of the theorems established in this paper, several other integrals involving special functions may be evaluated. Keywords Stieltjes transform · Hankel transform · Bessel functions · Struve’s functions · Lommel’s functions · Integral formulae Mathematics Subject Classification 44A05
1 Introduction ̈ Srivastava [8–10], Srivastava and Tuan [11], Srivastava and Yurekli [12] and Yakubovich and Martins [13] have studied and explored Laplace, Meijer, Stieltjes and Hankel transforms at large in the form of generalizations, convolution and inter‑ connecting theorems. Being motivated by the work of Srivastava [8, 9], we establish four theorems connecting Stieltjes transform and Hankel transform in this paper. Now, we define the Stieltjes transform and Hankel transform. 1.1 Definition The Stieltjes transform [2, 10, 11] of a function f (x) ∈ L(0, ∞) is defined in the fol‑ lowing manner. Communicated by H. M. Srivastava. * Virendra Kumar [email protected] 1
Defence Research and Development Organization, India, D‑436, Shastri Nagar, Ghaziabad, U.P. 201002, India
13
Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
G(f ; y) =
∫0
∞
(x + y)−1 f (x)dx,
(1)
where y is a complex variable. 1.2 Definition The Hankel transform [2, 8, 9] of order v of a function f (x) ∈ L(0, ∞) is defined in the following manner.
hv (f ; 𝜁) =
∫0
∞
(𝜁x)1∕2 Jv (𝜁x) f (x)dx,
𝜁 > 0,
(2)
where Jv (z) stands for the Bessel function of the first kind [1, p. 4, Eq. (2)].
2 Main theorems In this section we establish four theorems connecting Stieltjes transform and Hankel transform. 2.1 theorem f and hv (f ; 𝜁) belong to L(0, ∞) , −1∕2 < Re(v) < 3∕2 and |arg y| < 𝜋 , then
G{xv−1∕2 f (x); y} =
∫0
∞
K(y; 𝜁) hv (f ; 𝜁)d𝜁,
(3)
where
K(y; 𝜁) = 2−1 𝜋𝜁 1∕2 yv 𝐬𝐞𝐜(v𝜋)[H−v (𝜁y) − Y−v (𝜁y)].
(4)
H−v (z) and Y−v (z) stand for the Struve’s function [1, p. 38, Eq. (55)] and Bessel func‑ tion of the second kind [1, p. 4, Eq.(4)] respectively. Proof Since f ∈ L(0, ∞) , we have by the Hankel inversion theorem [7] that
f (x) =
∫0
∞
(𝜁x)1∕2 hv (f ; 𝜁)Jv (𝜁x)d𝜁 .
(5)
Hence
G{xv−1∕2 f (x); y} =
∫0
∞
𝜁 1∕2 hv (f ; 𝜁) G{xv Jv (𝜁x); y}d𝜁 .
(6)
The change of order of integration is justified because xv (x + y)−1 ∈ L(0, ∞) if |arg y| < 𝜋 , −1∕2 0 , −1∕2 < Re(v) < 3∕2 and |arg y| < 𝜋 we arrive at the desired result (3), where −1∕2 < Re(v) < 3∕2 and |arg y| < 𝜋 . ◻ 2.2 Theorem If f and hv (f ; 𝜁) belong to L(0, ∞) , Re(v) > −3∕2 and |arg y| < 𝜋 , then
G{x−v−1∕2 f (x); y} =
∫0
∞
(10)
K(y; 𝜁) hv (f ; 𝜁)d𝜁,
where
K(y; 𝜁) = 𝜁
1∕2
[
−1
−v
2 𝜋y
] 21−v y−v S (𝜁y) . {Hv (𝜁y) − Yv (𝜁y)} − 𝛤 (v) v−1
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