Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications

Abstract

In this paper, by introducing two sequences of new numbers and their derivatives, which are closely related to the Stirling numbers of the first kind, and choosing to employ six known generalized Kummer’s summation formulas for 2F1(−1) and 2F1(1/2), we establish six classes of generalized summation formulas for p+2Fp+1 with arguments −1 and 1/2 for any positive integer p. Next, by differentiating both sides of six chosen formulas presented here with respect to a specific parameter, among numerous ones, we demonstrate six identities in connection with finite sums of 4F3(−1) and 4F3(1/2). Further, we choose to give simple particular identities of some formulas presented here. We conclude this paper by highlighting a potential use of the newly presented numbers and posing some problems.

Keywords

• Gamma function
• Psi function
• Pochhammer symbol
• hypergeometric function ₂<i>F</i>₁
• generalized hypergeometric functions _<i>t</i><i>F</i><i>_u</i>
• Gauss’s summation theorem for ₂<i>F</i>₁(1)