General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.

On a Generalization of Kummer’s Second-Type F 1 1 and F 2 2

The aim of this paper is to establish a general form of Kummer’s second-type summation theory. By defining a new form for the divided of the Pochhammer symbol d + i n / d n , we can develop a general form of Kummer’s second-type summation theorem as e − x / 2 F 2 2 a , i + d ; x b + l , d ; in the form of a sum of e − x / 2 F 1 1 a ; x b + i ; for i , l = 0,1,2 , … , Then, some properties of the generalized Kummer’s second-type summation theorem can yield a number of known and novel results.

A new proof of classical Dixon's summation theorem for the series ${}_{3}F_{2}(1)$

The aim of this short note is to provide a new proof of classical Dixon's summation theorem for the series ${}_{3}F_{2}(1)$.

A new class of double integrals involving Generalized Hypergeometric Function 3F2

The aim of this research paper is to evaluate fifty double integrals invoving generalized hypergeometric function (25 each) in the form of\begin{align*}\int_{0}^{1}\int_{0}^{1} & x^{c-1}y^{c+\al-1} (1-x)^{\al- 1}(1-y)^{\be-1}\, (1-xy)^{c+\ell-\al-\be+1}\;\\ &\times \;{}_3F_2 \left[\begin{array}{c}a,\,\,\,\,\,b,\,\,\,\,\,2c+\ell+ 1 \\ \frac{1}{2}(a+b+i+1),\,\,2c+j \end{array}; xy\right]\,dxdy\end{align*}and\begin{align*}\int_{0}^{1}\int_{0}^{1} & x^{c+\ell}y^{c+\ell+\al} (1-x)^{\al-1}(1-y)^{\be-1}\, (1-xy)^{c- \al-\be}\;\\ &\times \;{}_3F_2 \left[\begin{array}{c}a,\,\,\,\,\,b,\,\,\,\,\,2c+\ell+ 1 \\ \frac{1}{2}(a+b+i+1),\,\,2c+j \end{array}; 1-xy\right]\,dxdy\end{align*}in the most general form for any $\ell \in \mathbb{Z}$ and $i, j = 0, \pm 1, \pm2$.The results are derived with the help of generalization of Edwards's well known double integral due to Kim, {\it et al.} and generalized classical Watson's summation theorem obtained earlier by Lavoie, {\it et al}.More than one hundred ineteresting special cases have also been obtained.

Generalizations of a formula due to Kummer with applications

The aim of this research paper is to obtain explicit expressions of 2F1 [a,b 1/2(a+b ? ?+1); 1+x/2] in the most general case for any ? = 0, 1, 2, ... For ? = 0, we have the well known, interesting and useful formula due to Kummer which was proved independently by Ramanujan. The results presented here are obtained with the help of known generalizations of Gauss?s second summation theorem for the series 2F1(1/2), which were given earlier by Rakha and Rathie [Integral Transforms Spec. Func. 22 (11) (2011), 823-840]. The results are further utilized to obtain new hypergeometric identities by using beta integral method developed by Krattenthaler & Rao [J. Comput. Appl. Math. 160 (2003), 159-173]. Several interesting results due to Ramanujan, Choi, et. al. and Krattenthaler & Rao follow special cases of our main findings.

ON SOME NEW Pδ-TRANSFORMS OF KUMMER'S CONFLUENT HYPERGEOMETRIC FUNCTIONS

The aim of our paper is to present Pδ -transforms of the Kummer’s confluent hypergeometric functions by employing the generalized Gauss’s second summation the-orem, Bailey’s summation theorem and Kummer’s summation theorem obtained earlier by Lavoie, Grondin and Rathie [9]. Relevant connections of certain special cases of the main results presented here are also pointed out.

Applications of the Generalized Kummer’s Summation Theorem to Transformation Formulas and Generating Functions

In this paper, we establish two general transformation formulas for Exton’s quadruple hypergeometric functions K5 and K12 by application of the generalized Kummer’s summation theorem. Further, a number of generating functions for Jacobi polynomials are also derived as an applications of our main results.