Related papers: Some summation theorems for Clausen's hypergeometr…
By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.
We offer some summation formulas that appear to have great utility in probability theory. The proofs require some recent results from analysis that have thus far been applied to basic hypergeometric functions.
The main aim of this paper is to derive some new summation theorems for terminating and truncated Clausen's hypergeometric series with unit argument, when one numerator parameter and one denominator parameter are negative integers. Further,…
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 give a combinatorial interpretation for the hypergeometric functions associated with tuples of rational numbers.
In a recent paper (Appl. Math. Comput. 215, 1622--1645, 2009), the authors proposed a method of summation of some slowly convergent series. The purpose of this note is to give more theoretical analysis for this transformation, including the…
We systematically exploit a new generalized hypergeometric identity to obtain new hypergeometric summation formulas. As a consistency test, alternative proofs for some special cases are also provided. As a byproduct new summation formulas…
A generalization of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli--type polynomials. Related families of…
We establish several summation formulae for hypergeometric and basic hypergeometric series involving noncommutative parameters and argument. These results were inspired by a recent paper of J. A. Tirao [Proc. Nat. Acad. Sci. 100 (14)…
We elaborate on the expansion of hypergeometric functions about rational parameters, where we focus mainly on the integer and half-integer case. The strategy and the basic steps of a recently developed algorithm for the expansion about…
In this article we developed a special topic of our pure-mathematics papers concerning the hypergeometric theory. Based upon a Roberts's reduction approach of hyperelliptic integrals to elliptic ones and on the simultaneous multivariable…
We introduce several new identities combining basic hypergeometric sums and integrals. Such identities appear in the context of superconformal index computations for three-dimensional supersymmetric dual theories. We give both analytic…
By some hypergeometric summation theorems, the authors establish a series of new infinite summation formulas involving generalized harmonic numbers related to Riemann-Zeta function, with three different patterns.
The theoretical computing of special values assumed by the hypergeometric functions has a high interest not only on its own, but also in sight of the remarkable implications to both pure Mathematics and Mathematical Physics. Accordingly, in…
Given the growing quantity of proposals and works of basic hypergeometric functions in the scope of $q$-calculus, it is important to introduce a systematic classification of $q$-calculus. Our aim in this article is to investigate certain…
We prove the following theorems: 1) The Laurent expansions in epsilon of the Gauss hypergeometric functions 2F1(I_1+a*epsilon, I_2+b*epsilon; I_3+p/q + c epsilon; z), 2F1(I_1+p/q+a*epsilon, I_2+p/q+b*epsilon; I_3+ p/q+c*epsilon;z),…
The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers including those discovered recently by Paule and Schneider (2003).
In this paper, using similar symbolical method of Burchnall and Chaundy formulas of expansion for the generalized hypergeometric function were constructed. By means of the found formulas of expansion the formulas of an analytic continuation…
In terms of the derivative operator and three hypergeometric series identities, several interesting summation formulas involving generalized harmonic numbers are established.
We develop a theoretical study of non-terminating hypergeometric summations with one free parameter. Composing various methods in complex and asymptotic analysis, geometry and arithmetic of certain transcendental curves and rational…