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.
Highlights
Introduction and PreliminariesThe Pochhammer symbol (ξ )η (ξ, η ∈ C) is defined, in terms of Gamma function Γ, by Γ(ξ + η )ξ + η ∈ C \ Z0−, η ∈ C \ {0}; ξ ∈ C \ Z0−, η = 0 (ξ )η = Γ(ξ ) ( (1) ( η = 0), =ξ ( ξ + 1) · · · ( ξ + n − 1)(η = n ∈ N), it accepted that (0)0 = 1
We select to give simple particular identities of some formulas presented here. By differentiating both sides of two chosen formulas presented here with respect to a specific parameter, among numerous ones, further, we demonstrate two identities associated with finite sums of 4 F3 (−1)
(k + α)s n ∈ N, s ∈ C, α ∈ C \ Z−, (s) where Hn (α) := Hn (α) and Hn (0) := Hn are the harmonic numbers of order s (see, e.g., [73], Equation (1.2))
Summary
The Pochhammer symbol (ξ )η (ξ, η ∈ C) is defined, in terms of Gamma function Γ (see, e.g., [1], p. 2 and p. 5), by. (iii) The method in (ii) is to obtain summation formulas for certain generalized hypergeometric functions of higher order from those of lower order. In connection with the method (iii), for a generalized hypergeometric function p Fq (z) with positive integral differences between certain numerator and denominator parameters, Karlsson [39] provided a formula expressing the p Fq (z) as a finite sum of lower-order functions as follows (see [42,46,47]): b1 + `1 , . 1+κ+λ− p p ∈ N0 , Rakha and Rathie ([53], Theorems 5 and 6) provided two generalizations of (14) which, with the aid of Legendre’s duplication formula for the Gamma function Owing to the range of its applications in fractional calculus, some scholars have nicknamed the Mittag–Leffler function the “Queen Function of the Fractional Calculus" in the past (see, e.g., [65])
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