Reduction Formulas for Karlsson–Minton-Type Hypergeometric Functions

We prove a master theorem for hypergeometric functions of Karlsson–Minton type, stating that a very general multilateral U(n) Karlsson–Minton-type hypergeometric series may be reduced to a finite sum. This identity contains the Karlsson–Minton summation formula and many of its known generalizations as special cases, and it also implies several Bailey-type identities for U(n) hypergeometric series, including multivariable ₁₀W₉ transformations of Denis and Gustafson and of Kajihara. Even in the one-variable case our identity is new, and even in this case its proof depends on the theory of multivariable hypergeometric series.     

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.     

Classes of series identities and associated hypergeometric reduction formulas

The main object of the present paper is to investigate some classes of series identities and their applications and consequences leading naturally to several (known or new) hypergeometric reduction formulas. We also indicate how some of these series identities and reduction formulas would yield several series identities which emerged recently in the context of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order).     

Summation formulae for elliptic hypergeometric series

Several new identities for elliptic hypergeometric series are proved. Remarkably, some of these are elliptic analogues of identities for basic hypergeometric series that are balanced but not very-well-poised.

     

Certain Classes of Infinite Series

 The authors evaluate some interesting families of infinite series by analyzing known identities involving generalized hypergeometric series. Several special cases of the main results are shown to be related to earlier works on the subject.     

Book review

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

More hypergeometric identities related to Ramanujan-type series

We find new hypergeometric identities which, in a certain aspect, are stronger than others of the same style found by the author in a previous paper. The identities in Sect. 3 are related to some Ramanujan-type series for 1/π. We derive them by using WZ-pairs associated to some interesting formulas by Wenchang Chu. The identities we prove in Sect. 4 are of the same style but related to Ramanujan-like series for 1/π ².     

Hypergeometric identities for 10 extended Ramanujan-type series

We prove, by the WZ-method, some hypergeometric identities which relate ten extended Ramanujan type series to simpler hypergeometric series. The identities we are going to prove are valid for all the values of a parameter a when they are convergent. Sometimes, even if they do not converge, they are valid if we consider these identities as limits.      

Generalized hypergeometric functions: product identities and weighted norm inequalities

We describe a method of obtaining weighted norm inequalities for generalized hypergeometric functions. This method is based upon our recent convolution theorem and some classical hypergeometric identities. In particular, it is shown that some product identities involving the divergent hypergeometric series lead to the convergent hypergeometric inequalities. A number of the new weighted norm inequalities for the Gaussian hypergeometric function, confluent hypergeometric function, and other generalized hypergeometric functions are presented.     

Certain Classes of Infinite Series

 The authors evaluate some interesting families of infinite series by analyzing known identities involving generalized hypergeometric series. Several special cases of the main results are shown to be related to earlier works on the subject. Received 16 December 1996; in revised form 21 May 1997     

Legendre inversions and balanced hypergeometric series identities

By means of Legendre inverse series relations, we prove two terminating balanced hypergeometric series formulae. Their reversals and linear combinations yield several known and new hypergeometric series identities.     

Strange evaluations of Fox–Wright function

By means of inversion techniques and four known hypergeometric series identities, eight summation formulas for the Fox–Wright function are established. They can give numerous summation formulas for 2-balanced hypergeometric series when the parameters are specified.     

Further bibasic hypergeometric transformations and their applications

In this paper, we give several new transformation formulae and generalize one result obtained by Singh [U.B. Singh, Certain bibasic hypergeometric transformations and their applications, J. Math. Anal. Appl. 201 (1996) 44-56] with the help of Bailey's transform. Further, some new multiple series identities of the Rogers-Ramanujan type are established.     

Legendre polynomials and Ramanujan-type series for 1/π

We resolve a family of recently observed identities involving 1/π using the theory of modular forms and hypergeometric series. In particular, we resort to a formula of Brafman which relates a generating function of the Legendre polynomials to a product of two Gaussian hypergeometric functions. Using our methods, we also derive some new Ramanujan-type 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.     

Some families of combinatorial and other series identities and their applications

The main object of this presentation is to show how some simple combinatorial identities can lead to several general families of combinatorial and other series identities as well as summation formulas associated with the Fox-Wright function _pΨ_q and various related generalized hypergeometric functions. At least one of the hypergeometric summation formulas, which is derived here in this manner, has already found a remarkable application in producing several interesting generalizations of the Karlsson-Minton summation formula. We also consider a number of other combinatorial series identities and rational sums which were proven, in recent works, by using different methods and techniques. We show that much more general results can be derived by means of certain summation theorems for hypergeometric series. Relevant connections of the results presented here with those in the aforementioned investigations are also considered.     

Abel's method on summation by parts and theta hypergeometric series

Abel's lemma on summation by parts is reformulated to investigate systematically terminating theta hypergeometric series. Most of the known identities are reviewed and several new transformation and summation formulae are established. The authors are convinced by the exhibited examples that the iterating machinery based on the modified Abel lemma is powerful and a natural choice for dealing with terminating theta hypergeometric series.     

Exceptional sets of hypergeometric series

For a family of transcendental hypergeometric series, we determine explicitly the set of algebraic points at which the series takes algebraic values (the so-called exceptional set). This answers a question of Siegel in special cases. For this, we first prove identities, each one relating locally one hypergeometric series to modular functions. In some cases, the identity and the theory of complex multiplication allow the determination of an infinite subset of the exceptional set. These subsets are shown to be the whole sets in using a consequence of Wüstholz's Analytic Subgroup Theorem together with mapping properties of Schwarz triangle functions. Further consequences of the identities are explicit evaluations of hypergeometric series at algebraic points. Some of them provide examples for Kroneckers Jugendtraum.     

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).     

Duality for hypergeometric functions and invariant Gauss-Manin systems

We present some basic identities for hypergeometric functions associatedwith the integrals of Euler type. We give a geometrical proof for formulaesuch as the identity between the single and double integrals expressingAppell's hypergeometric series F1 (a, b, b' c; x, y).