Some Summation Formulas for the Hypergeometric
نویسندگان
چکیده
The aim of this paper is to obtain explicit expressions of the generalized hypergeometric function r+2Fr+1 [ a, b, 1 2 (a+ b+ j + 1), (fr +mr) (fr) ; 1 2 ] for j = 0,±1, . . . ,±5, where r pairs of numeratorial and denominatorial parameters differ by positive integers mr. The results are derived with the help of an expansion in terms of a finite sum of 2F1( 1 2 ) functions and a generalization of Gauss’ second summation theorem due to Lavoie et al. [J. Comput. Appl. Math. 72, 293–300 (1996)]. Some special and limiting cases are also given.
منابع مشابه
A New Generalization of Extended Beta and Hypergeometric Functions
Abstract: A new generalization of extended beta function and its various properties, integral representations and distribution are given in this paper. In addition, we establish the generalization of extended hypergeometric and confluent hypergeometric functions using the newly extended beta function. Some properties of these extended and confluent hypergeometric functions such as integral repr...
متن کاملShort Proofs of Summation and Transformation Formulas for Basic Hypergeometric Series
We show that several terminating summation and transformation formulas for basic hypergeometric series can be proved in a straigntforward way. Along the same line, new finite forms of Jacobi’s triple product identity and Watson’s quintuple product identity are also proved.
متن کاملOn Warnaar’s Elliptic Matrix Inversion and Karlsson–minton-type Elliptic Hypergeometric Series
Using Krattenthaler’s operator method, we give a new proof of Warnaar’s recent elliptic extension of Krattenthaler’s matrix inversion. Further, using a theta function identity closely related to Warnaar’s inversion, we derive summation and transformation formulas for elliptic hypergeometric series of Karlsson–Minton-type. A special case yields a particular summation that was used by Warnaar to ...
متن کاملNonterminating Basic Hypergeometric Series and the q-Zeilberger Algorithm
We present a systematic method for proving nonterminating basic hypergeometric identities. Assume that k is the summation index. By setting a parameter x to xqn, we may find a recurrence relation of the summation by using the q-Zeilberger algorithm. This method applies to almost all nonterminating basic hypergeometric summation formulas in the book of Gasper and Rahman. Furthermore, by comparin...
متن کاملTaylor Series for the Askey-wilson Operator and Classical Summation Formulas
Abstract. An analog of Taylor’s formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. As an application, a generalization of the binomial theorem is obtained. Besides, this method becomes quite useful to obtain summation formulas of basic hypergeometric series. New proofs of several well-known summation formulas ...
متن کامل