ABSTRACT
In the area of fractional calculus of arbitrary order, integrals and derivatives are studied and used in various applications. In the theory of fractional differential and integral equations, the special functions of the higher transcendental function are very important. In this chapter, our aim is to present the systematic study of the generalized Saigo's hypergeometric fractional calculus operators to establish a number of key results for the families of extended Hurwitz-Lerch Zeta function defined by many authors, including (for example) Srivastava et al. [Integral Transform. Spec. Funct. 22(7), (2011), 487{506], Garg et al. [Algebras Groups Geom. 25 (2008), 311{319], Lin and Srivastava [Appl. Math. Comput. 154 (2004), 725{733] and Goyal and Laddha [Ganita Sandesh 11 (1997), 99{108]. The new (presumably) and useful (potentially) results that are obtained in this chapter have been expressed in terms of the I-function introduced by Rathie [28]. Since the extended Hurwitz-Lerch zeta function and generalized Saigo's hypergeometric fractional calculus operators are of very general nature, therefore on specializing the parameters, a large number of special cases involving Riemann-Liouville and Erdélyi-Kober fractional integral and differential operators and Hurwitz-Lerch zeta function have been expressed in terms of I-function and https://www.w3.org/1998/Math/MathML"> H ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003368069/e37564ec-62a9-4ca6-84ef-69ab9c2b9ce0/content/UnEq0206-09.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> -function, as our main findings.
