On reducing the Heun equation to the hypergeometric equation

The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heun-to-hypergeometric reductions are similar to classical hypergeometric identities, but the conditions for the existence of a reduction involve features of the Heun equation that the hypergeometric equation does not possess; namely, its cross-ratio and accessory parameters. The reductions include quadratic and cubic transformations, which may be performed only if the singular points of the Heun equation form a harmonic or an equianharmonic quadruple, respectively; and several higher-degree transformations. This result corrects and extends a theorem in a previous paper, which found only the quadratic transformations. (SIAM J. Math. Anal. 10 (3) (1979) 655).     

Automatic generation of hypergeometric identities by the beta integral method

In this article, hypergeometric identities (or transformations) for _p+1F_p-series and for Kampé de Fériet series of unit arguments are derived systematically from known transformations of hypergeometric series and products of hypergeometric series, respectively, using the beta integral method in an automated manner, based on the Mathematica package HYP. As a result, we obtain some known and some identities which seem to not have been recorded before in literature.     

Multiple elliptic hypergeometric series. An approach from the Cauchy determinant

A multiple generalization of elliptic hypergeometric series is studied through the Cauchy determinant for the Weierstrass sigma function. In particular, a duality transformation for multiple hypergeometric series is proposed. As an application, two types of Bailey transformations for very well-poised multiple elliptic hypergeometric series are derived.     

Weighted norm inequalities for analytic functions

We present a weighted norm inequality involving convolutions of arbitrary analytic functions and certain confluent hypergeometric functions. This result implies a family of weighted norm inequalities both for entire functions of exponential type and for (generalized) hypergeometric series. The approach is based on author's general inequality for continuous functions and some hypergeometric transformations.     

Quartic theta hypergeometric series

Four classes of quartic theta hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several transformations are proved that express the quartic series in terms of well-poised, quadratic and cubic ones. Thirty new summation formulae for terminating quartic theta hypergeometric series are derived consequently.     

Abel's lemma on summation by parts and basic hypergeometric series

Basic hypergeometric series identities are revisited systematically by means of Abel's lemma on summation by parts. Several new formulae and transformations are also established. The author is convinced that Abel's lemma on summation by parts is a natural choice in dealing with basic hypergeometric series.     

Transformation and reduction formulae for double q‐Clausen hypergeometric series

By means of the Sears transformations, we establish eight general transformation theorems on bivariate basic hypergeometric series. Several transformation, reduction and summation formulae on the double q‐Clausen hypergeometric series are derived as consequences. Copyright © 2007 John Wiley & Sons, Ltd.     

A new hypergeometric transformation of the Rathie–Rakha type

A general transformation involving generalized hypergeometric functions has been recently found by Rathie and Rakha using simple arguments and exploiting Gauss’s summation theorem. In this sequel to the work of Rathie and Rakha, a new hypergeometric transformation formula is derived by their method and by appealing to Gauss’s second summation theorem. In addition, it is shown that the method fails to give similar hypergeometric transformations in the cases of the classical summation theorems of Kummer, Bailey, Watson and Dixon.     

双变量基本超几何级数的若干变换与求和公式

The purpose of this paper is to establish several transformation formulae for bivariate basic hypergeometric series by means of series rearrangement technique. From these transformations, some interesting summation formulae are obtained.     

Continued fraction proofs of <Emphasis Type="Italic">m</Emphasis>-versions of some identities of Rogers–Ramanujan–Slater type

We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two q-continued fractions previously investigated by the authors.     

Specialization of Appell's functions to univariate hypergeometric functions

Univariate specializations of Appell's hypergeometric functions F₁, F₂, F₃, F₄ satisfy ordinary Fuchsian equations of order at most 4. In special cases, these differential equations are of order 2 and could be simple (pullback) transformations of Euler's differential equation for the Gauss hypergeometric function. The paper classifies these cases, and presents corresponding relations between univariate specializations of Appell's functions and univariate hypergeometric functions. The computational aspect and interesting identities are discussed.     

Certain transformations and special values of hypergeometric functions over finite fields

The Ramanujan Journal - In this article we find finite field analogues of certain transformations satisfied by the classical hypergeometric series. Using properties of Gauss and Jacobi sums we...     

An integral recursive inequality and applications

An integral recursive inequality for two functions is obtained. It is used to describe the equality cases in the related inequalities. The applications involve some bi-Hermitian forms, integral transformations, and confluent hypergeometric functions.      

On certain transformations of poly-basic bilateral hypergeometric series

In this paper, we have established certain transformations of basic hypergeometric series with more than one base. Some of these lead to the relationship between product of two q-series. These results, in turn, lead to very interesting transformations of bi-basic and poly-basic q-series. A few of the results which are representative of the many results obtained are presented in this article.     

A certain class of q-series transformations

A large number of summation and transformation formulas for a certain class of double hypergeometric series are observed here to follow fairly readily from a single known result which, in turn, is a very special case of one of six general expansion formulas given in the literature. Generalizations and unifications of these expansion formulas involving series with essentially arbitrary terms are presented. It is also shown how the various series transformations considered in this paper admit themselves of $\text{q}$-extensions which are capable of unifying numerous scattered results in the theory of basic double hypergeometric functions.     

New transformations for elliptic hypergeometric series on the root system A n

Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system A _n , with different dimensions n. We give, with a new, elementary proof, an elliptic extension of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series. 2000 Mathematics Subject Classification Primary—33D67; Secondary—11F50     

General multi-sum transformations and some implications

We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence \(\{g(k)\}\)), to be reduced to an infinite q-product times a single basic hypergeometric sum. Various applications are given, including summation formulae for some q orthogonal polynomials and various multi-sums that are expressible as infinite products.     

Some results involving series representations of hypergeometric functions

In this paper, we prove some generalisations of several theorems given in [K.A. Driver, S.J. Johnston, An integral representation of some hypergeometric functions, Electron. Trans. Numer. Anal. 25 (2006) 115-120] and examine some special cases which correspond to a transformation given by Chaundy in [T.W. Chaundy, An extension of hypergeometric functions, I., Quart. J. Maths. Oxford Ser. 14 (1943) 55-78] and other transformations involving the Riemann zeta function and the beta function.     

Extension of a quadratic transformation due to Exton

By applying various known summation theorems to a general formula based upon Bailey’s transform theorem due to Slater, Exton has obtained numerous new quadratic transformations involving hypergeometric functions of two and of higher order. Some of the results have typographical errors and have been corrected recently by Choi and Rathie. In addition, two new quadratic transformation formulæ were also obtained [Junesang Choi, A.K. Rathie, Quadratic transformations involving hypergeometric functions of two and higher order, EAMJ, East Asian Math. J. 22 (2006) 71-77]. The aim of this research paper is to obtain a generalization of one of the Exton’s quadratic transformation. The results are derived with the help of generalized Kummer’s theorem obtained earlier by Lavoie, Grondin and Rathie. As special cases, we mention six interesting results closely related to that of Exton’s result.