Jñānābha‎, Vol. 50 (1) (2020), 232-242

CONTOUR INTEGRAL REPRESENTATIONS OF TWO VARIABLE GENERALIZED HYPERGEOMETRIC FUNCTION OF SRIVASTAVA AND DAOUST WITH THEIR APPLICATIONS TO INITIAL VALUE PROBLEMS OF ARBITRARY ORDER

By

R. C. Singh Chandel and Hemant Kumar*
Former Head, Department of Mathematics
D. V. Postgraduate College Orai- 285001, Uttar Pradesh, India
Email:rc_chandel@yahoo.com
*Department of Mathematics
D. A-V. Postgraduate College Kanpur- 208001, Uttar Pradesh, India
Email:palhemant2007@rediffmail.com
(Received : June 01, 2020 ; Revised: June 21, 2020)


Abstract

In this paper, we establish two contour integral representations involving Mittag - Leffler functions (i) for a two variable generalized hypergeometric function of Srivastava and Daoust and (ii) a sum of the Kummer’s confluent hypergeometric functions. Then, we make their appeal to obtain the contour integrals for many generating functions and bilateral generating relations. Further, in development and extensions of fractional calculus, we obtain various relations of contour integrals with fractional derivatives and integral operators to use them in solving of any order initial value problems.

2010 Mathematics Subject Classification: 33C15, 33C20, 33C60, 33E12, 26A33.
Keywords and phrases: Srivastava and Daoust two variable function, Kummer’s confluent hypergeometric function, contour integral representations, Mittag - Leffler functions, arbitrary order initial value problems.