Generalizations of a terminating summation formula of. Summation and transformation formulas for elliptic. Hence, many examples from this very active field are given. Generalized hypergeometric series, saalschutzs theorem.
Applications of some hypergeometric summation theorems. Before deriving new summation and transformation formulas for elliptic hypergeometric series we need to prepare several, mostly elementary, results for the elliptic function e of 2. Some finite summation formulas involving multivariable hypergeometric polynomials article pdf available in integral transforms and special functions 144. Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, verywellpoised, elliptic hypergeometric series. A hypergeometric function is the sum of a hypergeometric series, which is. Modern algorithmic techniques for summation, most of which were introduced in the 1990s, are developed here and carefully implemented in the computer algebra system maple. The combination of these results gives orthogonal polynomials and hypergeometric and qhypergeometric special functions a solid algorithmic foundation. Sometimes the most general hypergeometric function pf q is called a generalized hypergeo. The series converges conditionally if z 1 with z 6 1 and. Currie a research report submitted to the faculty of science. Pdf the main objective of the present paper is to obtain some results. Generalized hypergeometric functions, clausen series, summation formulas, hypergeometric identities, integral parameter. Some summation theorems for generalized hypergeometric.
Identities for the gamma and hypergeometric functions. Equations involving hypergeometric functions are of great interest to mathematicians and scientists, and newly proven identities for these functions assist in finding solutions for many differential and integral equations. The materials covered are suitable for an introductory course on algorithmic summation and will appeal to students and researchers. The denominator of formula 1 represents the number of ways n objects can be selected from n objects. Pdf this paper continues the study of finite summation formulas of multivariable hypergeometric functions. Since the binomial theorem is at the foundation of most of the summation formulas for hypergeometric series, we then derive a qanalogue of it, called the qbinomial theorem, and use it to derive heines qanalogues of eulers transformation formulas, jacobis triple product identity, and summation formulas. Infinite summation formulas related to riemann zeta.
Summation and transformation formulas our approach to elliptic hypergeometric summation and transformation formulas is a standard one in the context of basic hypergeometric series, see, e. In mathematics, the gaussian or ordinary hypergeometric function 2 f 1 a,b. This represents the number of possible out comes in the experiment. Decision procedure for indefinite hypergeometric summation. To give just one example, here is a transformation formula for a splitpoised theta hypergeometric 12e 11 series 12e 11 a,qa1 2. The algorithms of fasenmyer, gosper, zeilberger, petkovsek and van hoeij for hypergeometric summation and recurrence equations, efficient multivariate summation as well as q analogues of the above algorithms are covered. This paper surveys recent applications of basic hypergeometric functions to partitions, number theory, finite vector spaces, combinatorial identities and physics. In this paper we explore many relations and explicitly derive closed form representations of integrals of logarithmic, hypergeometric functions and the lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers. An algorithmic approach to summation and special function identities, springer universitext series, 2014. Contemporary mathematics qextensions of erd elyis fractional integral representations for hypergeometric functions and some summation formulas for double qkamp edef eriet series george gasper dedicated to dick askey on the occasion of his 66th birthday. Construction of a summation formula allied with hyper geometric function and involving recurrence relation. Development of some summation formulae using hypergeometric function. This first paper, in which we simultaneously treat real, complex, and quaternionic analysis, is the result of our desire to present a complete theory of hypergeometric functions of.
There is an extensive literature on application of newtonandrews method for nding identities with harmonic numbers h n xn k1 1 k. Every secondorder linear ode with three regular singular points can be transformed into this. Several striking harmonic number identities discovered in 3 will be recovered and some new ones will be established. In this paper, we obtain two gaussian hypergeometric summation formulas and show how each of these summation formulas can be applied to derive several general doubleseries identities and various. Certain properties of the generalized gauss hypergeometric. Abstractthe main aim of this paper is to create a summation formula associated to recurrence relation and hypergeometric function.
A simple proof of a new summation formula for a terminating. Integrals of logarithmic and hypergeometric functions are intrinsically connected with euler sums. Pdf finite summation formulas of srivastavas general triple. Given a summand an, we seek the indefinite sum s n determined within an additive constant by formula.
Our principal objective in this article is to present a systematic investigation of several further. In section 3 we derive theta hypergeometric extensions of some of the summation and transformation formulas in 8, secs. Because hypergeometric series will play a central role in the present work, we reproduce its notation for those who are not familiar with it. Quadratic transformations of hypergeometric function and. An analogue of the gauss summation formula for hypergeometric. Some generalizations of pochhammers symbol and their. Kummer, dixon, watson, whipple, pfaffsaalschutz and dougall formulas and also obtain some new summation theorems in the sequel. Some of the identities presented here generalize corresponding formulas given in chapter 11 of the gasper and rahman book basic hyperge. Applications of basic hypergeometric functions siam. Eulers integral representation 3 can be used to prove gausss summation formula.
Integrals of logarithmic and hypergeometric functions in. Modern algorithmic techniques for summation, most of which were introduced in the. The main object of this paper is to derive a number of general double series identities and to apply each of these identities in order to deduce several hypergeometric reduction formulas for the srivastavadaoust double hypergeometric function. Top ten summation formulas name summation formula constraints 1. It is a solution of a secondorder linear ordinary differential equation ode. An alternative proof of the extended saalsch utz summation theorem. Applying gauss and watsons famous hypergeometric summation theorems, the authors establish two pattern infinite summation formulas involving generalized harmonic numbers related to riemann zeta function.