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.

Generalized Summation Formulas for the Kampé de Fériet Function

“…A similar process of the proof of Theorem 1 can establish the results here. Setting (i) x = 1/2 and γ = −α + i (i ∈ N 0 ) and (ii) x = 1/2 and γ = −α − i (i ∈ N 0 ) in (9) with the help of ( 10) and ( 11) produces, respectively, (44) and (45). The details are omitted.…”

Fractional Calculus Operators and the Mittag-Leffler Function

“…Numerous polynomials, numbers, their extensions, degenerations, and new polynomials and new numbers have been developed and studied, owing primarily to their potential applications and use in a diverse variety of research fields (see, e.g., [66][67][68][69][70][71] and the references therein). For example, Bernoulli polynomials and numbers are among most important and useful ones (see, e.g., [5], pp.…”

Certain Recurrence Relations of Two Parametric Mittag-Leffler Function and Their Application in Fractional Calculus

“…Since the results presented in Theorems 2-5 are quite general, they may reduce to yield a number of interesting identities. For example, setting µ = 1 in the results, with the aid of (10), we obtain several new relations among the generalized hypergeometric functions p F q , as stated in the following corollary (for current research of summation and reduction formulae for p F q , see [68,69]).…”