Some families of series identities and associated fractional differintegral formulas

Abstract

In many recent papers, a number of closely-related fractional differintegral formulas associated with power, composite and rational functions were investigated individually. The main object of the present sequel to these earlier works is to consider some general results and their applications and consequences leading to many of the aforementioned fractional differintegral formulas. We also investigate, in a systematic and unified manner, several families of series identities which emerged in connection with some of these fractional differintegral formulas.

Keywords:

• series identities
• operators of fractional calculus
• Gamma function
• power functions
• composite functions
• rational functions
• fractional differintegral formulas
• generalized hypergeometric functions
• Fox–Wright functions
• Gauss hypergeometric function
• nth derivative formula
• generalized Leibniz rule
• analytic functions
• index law
• linearity property
• principal value
• F and H functions
• hypergeometric reduction formulas
• Legendre’s duplication formula

2000 Mathematics Subject Classification :

• Primary: 26A33
• 33B15
• 33C05
• Secondary 33C20
• 33C60

Acknowledgements

The present investigation was supported, in part, by the National Science Council of the Republic of China under Grant NSC 97-2115-M-033-005 and, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.