Transformations Of Hypergeometric Series Research Articles

Supercongruences arising from transformations of hypergeometric series

We derive two supercongruences on partial sums of hypergeometric series from transformations of hypergeometric series, which are similar to corresponding results of Long and Ramakrishna (2016) [11].

Summation identities and transformations for hypergeometric series

We find summation identities and transformations for the McCarthy’s p-adic hypergeometric series by evaluating certain Gauss sums which appear while counting points on the family $$\begin{aligned} Z_{\lambda }: x_1^d+x_2^d=d\lambda x_1x_2^{d-1} \end{aligned}$$ over a finite field $$\mathbb {F}_p$$ . Salerno expresses the number of points over a finite field $$\mathbb {F}_p$$ on the family $$Z_{\lambda }$$ in terms of quotients of p-adic gamma functions under the condition that $$d|p-1$$ . In this paper, we first express the number of points over a finite field $$\mathbb {F}_p$$ on the family $$Z_{\lambda }$$ in terms of McCarthy’s p-adic hypergeometric series for any odd prime p not dividing $$d(d-1)$$ , and then deduce two summation identities for the p-adic hypergeometric series. We also find certain transformations and special values of the p-adic hypergeometric series. We finally find a summation identity for the Greene’s finite field hypergeometric series.

Certain transformations for hypergeometric series in the p-adic setting

In [The trace of Frobenius of elliptic curves and the p-adic gamma function, Pacific J. Math. 261(1) (2013) 219–236], McCarthy defined a function nGn[⋯] using the Teichmüller character of finite fields and quotients of the p-adic gamma function. This function extends hypergeometric functions over finite fields to the p-adic setting. In this paper, we give certain transformation formulas for the function nGn[⋯] which are not implied from the analogous hypergeometric functions over finite fields.

Automatic generation of hypergeometric identities by the beta integral method

In this article, hypergeometric identities (or transformations) for p+1 F p -series and for Kampé de Fériet series of unit arguments are derived systematically from known transformations of hypergeometric series and products of hypergeometric series, respectively, using the beta integral method in an automated manner, based on the Mathematica package HYP. As a result, we obtain some known and some identities which seem to not have been recorded before in literature.

An integral of hypergeometric type

In a recent paper (2) I gave some integrals which represented general transformations of hypergeometric series of both ordinary and basic types. In this paper I give the most general form of this kind of integral, from which all the previously known general basic transformations can be deduced.In the usual notation for basic series,where x may be a function of n, for example, and Π is written for ‘Idem (a; b)' means that the expression immediately preceding is to be repeated with b written in place of a and a written in place of b.