Hypergeometric function representation of the roots of a certain cubic equation

The aim in this note is to obtain new hypergeometric forms for the functions (√z − 1 − √z) b ± (√z − 1 − √z)−b, (√z − 1 + √z)b ± (√z − 1 + √z)−b, where b is an arbitrary parameter, in terms of Gauss hypergeometric functions. An application of these results (when b =1/3) is made to obtain the hypergeometric form of the roots of the cubic equation r3 − r + 2/3√2/3 = 0. This complements the entry in the compendium of Prudnikov et al. on page 472, entry (68) of the table, where only the middle root (either real or purely imaginary) is given in hypergeometric form.Publisher's Versio

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

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