Related papers: A note on a hypergeometric transformation formula …
Euler's transformation formula for the Gauss hypergeometric function 2F1 is extended to hypergeometric functions of higher order. Unusually, the generalized transformation constrains the hypergeometric function parameters algebraically but…
New relations are established between families of three-variable Mahler measures. Those identities are then expressed as transformations for the $_5F_4$ hypergeometric function. We use these results to obtain two explicit $_5F_4$…
We derive two generalizations of Gasper's transformation formula for basic hypergeometric series. Using these generalized formulas, we give explicit expressions for the coefficients of three-term relations for the basic hypergeometric…
In this paper we continue investigation of the hypergeometric function ${}_4F_3(1)$ as the function of its seven parameters. We deduce several reduction formulas for this function under additional conditions that one of the top parameters…
Some multiple hypergeometric transformation formulas arising from the balanced du- ality transformation formula are discussed through the symmetry. Derivations of some transformation formulas with different dimensions are given by taking…
A master formula of transformation formulas for bilinear sums of basic hypergeometric series is proposed. It is obtained from the author's previous results on a transformation formula for Milne's multivariate generalization of basic…
The purpose of this note is to provide an alternative proof of two transformation formulas contiguous to that of Kummer's second transformation for the confluent hypergeometric function ${}_1F_1$ using a differential equation approach.
The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature.…
A summation formula is derived for the hypergeometric series of unit argument ${}_3F_2(1,1,c;d,n+2;1)$, where $n=0, 1, 2, \ldots$ and $\Re (d-c+n)>0$.
In 1984, the second author conjectured a quadratic transformation formula which relates two hypergeometric 2F1 functions over a finite field F_q. We prove this conjecture and give an application. The proof depends on a new linear…
New explicit as well as manifestly symmetric three-term summationformulas are derived for the Clausenian hypergeometric series $_3F_2(1)$ with negative integral parameter differences. Our results generalize and naturally extend several…
We prove a new Bailey-type transformation relating WP-Bailey pairs. We then use this transformation to derive a number of new 3- and 4-term transformation formulae between basic hypergeometric series.
The large variety of Fourier transforms in geometric algebras inspired the straight forward definition of ``A General Geometric Fourier Transform`` in Bujack et al., Proc. of ICCA9, covering most versions in the literature. We showed which…
Very recently a new series representation of Humbert's double hypergeometric series $\Phi_3$ in series of Gauss's $_2F_1$ function was given by one of us. The aim of this short research note is to provide an alternative proof of the result.…
A summation formula is derived for the sum of the first m+1 terms of the 3F2(a,b,c;(a+b+1)/2,2c;1) series when c = -m is a negative integer. This summation formula is used to derive a formula for the sum of a terminating double…
We prove a transformation formula relating two determinants involving elliptic shifted factorials. Similar determinants have been applied to multiple elliptic hypergeometric series.
With the use of the $(f,g)$-matrix inversion under specializations that $f=1-xy,g=y-x$, we establish an $(1-xy,y-x)$-expansion formula. When specialized to basic hypergeometric series, this $(1-xy,y-x)$-expansion formula leads us to some…
We extend Gegenbauer Polynomials technique to evaluate a class of complicated Feynman diagrams. New results in the form of $_3F_2$-hypergeometrical series of unit argument, are presented. As a by-product, we present a new transformation…
Expressions for the summation of a new series involving the Laguerre polynomials are obtained in terms of generalized hypergeometric functions. These results provide alternative, and in some cases simpler, expressions to those recently…
We study multivariable (bilateral) basic hypergeometric series associated with (type $A$) Macdonald polynomials. We derive several transformation and summation properties for such series including analogues of Heine's ${}_2\phi_1$…