Operators of Fractional Calculus and Associated Integral Transforms of the $$({\mathfrak {r}}, {\mathfrak {s}})$$
- PDF / 395,789 Bytes
- 18 Pages / 439.37 x 666.142 pts Page_size
- 81 Downloads / 200 Views
Operators of Fractional Calculus and Associated Integral Transforms of the (r, s)-Extended Bessel–Struve Kernel Function Ritu Agarwal1 · Rakesh K. Parmar2 · S. D. Purohit3 Accepted: 3 November 2020 / Published online: 17 November 2020 © Springer Nature India Private Limited 2020
Abstract Present paper contains Marichev–Saigo–Maeda fractional integration and differentiation formulas and certain Integral transforms of the involving (r, s)-extended Bessel–Struve kernel function. Several particular cases of the leading findings are derived for the Saigo’s, Riemann–Liouville and Erdélyi–Kober fractional (arbitrary order) integration and fractional differentiation formulas. Keywords (r, s )-Extended Bessel–Struve kernel function · Fractional calculus operators Mathematics Subject Classification Primary 26A33 · 33B20 · 33C20; Secondary 26A09 · 33B15 · 33C05
Introduction Various extensions of Euler’s Beta function has been developed by several researchers. Particularly, (r, s)-extended Euler’s Beta integral is defined as [1]: 1 r s B(x, y; r, s) = t x−1 (1 − t) y−1 e− t − 1−t dt , (1) 0
B
S. D. Purohit [email protected] Ritu Agarwal [email protected] Rakesh K. Parmar [email protected]
1
Department of Mathematics, Malaviya National Institute of Technology, Jaipur, Rajasthan 302017, India
2
Department of HEAS, University College of Engineering and Technology, Bikaner, Rajasthan 334004, India
3
Department of HEAS, Rajasthan Technical University, Kota, Rajasthan, India
123
175
Page 2 of 18
Int. J. Appl. Comput. Math (2020) 6:175
(min{Re(r), Re(s)} ≥ 0); min{Re(x), Re(y)} > 0. If we set s = r in (1), we immediately obtain r-extended Euler’s Beta integral introduce by Chaudhry et al. [2]. Several applications and properties of r-extended Euler’s Beta integral and related r-extended functions in diverse areas of mathematical, physical, engineering and statistical sciences can be seen in Book of Chaudhry et al. [3] and recent papers [4,5]. Further by making use of (1), various higher transcendental hypergeometric type (r, s)-special functions including r-variant have been developed in [1,2,6–9]. More recently, Parmar and Pogány [10] introduced and studied the (r, s)-Bessel–Struve kernel function Sν, r,s (x) by showing integral representations, Mellin transformation, functional bound and various other characteristics in the form Sν,r,s (x) = √
xn n 1 1 . B + , ν + ; r, s 2 2 2 n! π (ν + 21 ) n≥0 (ν + 1)
(2)
with Re(ν) > − 21 and min{r, s} ≥ 0 when r = s = 0. Several other investigation of its properties, among others integral representations, bounding inequalities, Mellin transforms, complete monotonicity, Turán type inequality, associated non-homogeneous differential-difference, hypergeometric representations and as application various other (r, s)-variant, and in turn, when r = s the r-variant have been studied widely together with the set of related higher transcendental hypergeometric type special functions (see, [1,2,6–8] and the references therein). Clearly, when r = s = 0,
Data Loading...
