This paper classifies algebraic transformations of Gauss hypergeometric functions and pull-back transformations between hypergeometric differential equations. This classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of the Gauss hypergeometric function.

In this paper our aim is to deduce some Turán type inequalities for q-hypergeometric and q-confluent hypergeometric functions. In order to obtain the main results we apply the methods developed in the case of classical Kummer and Gauss hypergeometric functions. c ⃝ 2013 Elsevier Inc. All rights reserved.

The considered problem is uniform convergence of sequences of hypergeometric series. We give necessary and sufficient conditions for uniformly dominated convergence of infinite sums of proper bivariate hypergeometric terms. These conditions can be checked algorithmically. Hence the results can be applied in Zeilberger type algorithms for nonterminating hypergeometric series.

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...

It is known that the ratio of Gaussian hypergeometric functions can be represented by means of g -fractions. In this work, the ratio of q -hypergeometric functions are represented by means of g -fractions that lead to certain results on q -starlikeness of the q -hypergeometric functions defined on the open unit disk. Corresponding results for the q -convex case are also obtained.

We present the τ -functions for the hypergeometric solutions to the qPainlevé system of type E (1) 8 in a determinant formula whose entries are given by Rahman’s q-hypergeometric integrals. By using the symmetry of the q-hypergeometric integral, we can construct fifty-six solutions and describe the action of W (E (1) 7 ) on the solutions.

The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms F (n, k) is extended to certain nonhypergeometric terms. An expression F (n, k) is called hypergeometric term if both F (n + 1, k)/F (n, k) and F (n, k+1)/F (n, k) are rational functions. Typical examples are ratios of products of exponentials, factorials, Γ function terms, b...

We consider the following problem: Given a nested sum expression, find a sum representation such that the nested depth is minimal. We obtain symbolic summation algorithms that solve this problem for sums defined, e.g., over hypergeometric, q-hypergeometric or mixed hypergeometric expressions.

In this paper, we obtain the extended Wright generalized Hypergeometric function using extended Beta function. We also obtain certain integral representations, Mellin transform and some derivative properties of extended Wright generalized Hypergeometric function. Further, we represent extended Wright generalized Hypergeometric function in the form of Laguerre polynomials and Whittaker function.