Hypergeometric Transformations Research Articles

On Derkachov–Manashov R-matrices for the principal series of unitary representations

In 2001–2013 Derkachov and Manashov with coauthors obtained simple and natural expressions of R-matrices for the principal series of representations of the groups SL(2,C), SL(2,R), SL(n,C), SO(1, n). The Yang–Baxter identities for these intertwining operators are kinds of multivariate hypergeometric transformations. Derivations of the identities are based on calculations “of physical level of rigor” with divergent integrals. Our purpose is a formal mathematical justification of these results.

Supercongruences arising from a 7 F 6 hypergeometric transformation formula

Abstract Using a F 6 7 {{}_{7}F_{6}} hypergeometric transformation formula, we prove two supercongruences. In particular, one of these supercongruences confirms a recent conjecture of Guo, Liu and Schlosser, and gives an extension of a supercongruence of Long and Ramakrishna.

Recent Applications of Multiple Hypergeometric Transformations in Function Spaces and Associated Reduction Formulas

In this research paper we have obtained recent applications of multiple hypergeometric transformations in function spaces and associated reductions formulas of Kampé de Ferlet’s, Exton’s double series, and Srivastava triple series. Further, we have presented some interesting integrals involving the Lauricella’s , Lauricella-Saran’s , etc., Exton’s , Horn’s and Jain’s and generalized hypergeometric series arise as special cases of our main results. Also, some special cases were considered as an application of presented integral functions in function spaces.

Hypergeometric functions over finite fields

Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study hypergeometric functions over finite fields in a manner that is parallel to that of the classical hypergeometric functions. Using a comparison between the classical gamma function and its finite field analogue the Gauss sum, we give a systematic way to obtain certain types of hypergeometric transformation and evaluation formulas over finite fields and interpret them geometrically using a Galois representation perspective. As an application, we obtain a few finite field analogues of algebraic hypergeometric identities, quadratic and higher transformation formulas, and evaluation formulas. We further apply these finite field formulas to compute the number of rational points of certain hypergeometric varieties.

Symmetry of terminating basic hypergeometric series representations of the Askey–Wilson polynomials

In this paper, we explore the symmetric nature of the terminating basic hypergeometric series representations of the Askey–Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy. In particular we identify and classify the set of 4 and 7 equivalence classes of terminating balanced ϕ34 and terminating very-well-poised W78 basic hypergeometric series which are connected with the Askey–Wilson polynomials. We study the inversion properties of these equivalence classes and also identify the connection of both sets of equivalence classes with the symmetric group S6, the symmetry group of the terminating balanced ϕ34. We then use terminating balanced ϕ34 and terminating very-well poised W78 transformations to give a broader interpretation of Watson's q-analog of Whipple's theorem and its converse.

Certain Generalizations of Quadratic Transformations of Hypergeometric and Generalized Hypergeometric Functions

There have been numerous investigations on the hypergeometric series 2F1 and the generalized hypergeometric series pFq such as differential equations, integral representations, analytic continuations, asymptotic expansions, reduction cases, extensions of one and several variables, continued fractions, Riemann’s equation, group of the hypergeometric equation, summation, and transformation formulae. Among the various approaches to these functions, the transformation formulae for the hypergeometric series 2F1 and the generalized hypergeometric series pFq are significant, both in terms of applications and theory. The purpose of this paper is to establish a number of transformation formulae for pFq, whose particular cases would include Gauss’s and Kummer’s quadratic transformation formulae for 2F1, as well as their two extensions for 3F2, by making advantageous use of a recently introduced sequence and some techniques commonly used in dealing with pFq theory. The pFq function, which is the most significant function investigated in this study, exhibits natural symmetry.

Some new representations of Hikami’s second-order mock theta function D5(q)

In this paper, a second-order mock theta function $\mathfrak{D}_5(q)$ given by Hikami [11] is studied. By using basic hypergeometric transformation formulae, we attain some new representations of Hikami's mock theta function $\mathfrak{D}_5(q)$. Meanwhile, dual nature of bilateral series associated to mock theta function $\mathfrak{D}_5(q)$ is also discussed.

On the Direct Limit from Pseudo Jacobi Polynomials to Hermite Polynomials

In this short communication, we present a new limit relation that reduces pseudo-Jacobi polynomials directly to Hermite polynomials. The proof of this limit relation is based upon 2F1-type hypergeometric transformation formulas, which are applicable to even and odd polynomials separately. This limit opens the way to studying new exactly solvable harmonic oscillator models in quantum mechanics in terms of pseudo-Jacobi polynomials.

Some hypergeometric transformations and reduction formulas for the Gauss function and their applications involving the Clausen function

Some hypergeometric transformations and reduction formulas for the Gauss function and their applications involving the Clausen function

Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

Several new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Zudilin. More concretely, the results in this paper include q-analogues of supercongruences (referring to p-adic identities remaining valid for some higher power of p) established by Long, by Long and Ramakrishna, and several other q-supercongruences. The six basic hypergeometric transformation formulas which are made use of are Watson’s transformation, a quadratic transformation of Rahman, a cubic transformation of Gasper and Rahman, a quartic transformation of Gasper and Rahman, a double series transformation of Ismail, Rahman and Suslov, and a new transformation formula for a nonterminating very-well-poised {}_{12}phi _{11} series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new {}_{12}phi _{11} transformation formula is further utilized to obtain a generalization of Rogers’ linearization formula for the continuous q-ultraspherical polynomials.

General Fine transformations I

We describe an extension of Fine’s method of deriving basic hypergeometric series transformations and derive new transformations from the method. Combinatorial proofs of two of the examples are provided.

A new approach to hypergeometric transformation formulas

We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi's canonical form of the hypergeometric differential equation. Analogy for $q$-hypergeometric functions is also studied.

Some basic hypergeometric transformations and Rogers–Ramanujan type identities

ABSTRACT In this paper, we establish a general q-series identity based on Z.-G. Liu's work, from which a handful of basic hypergeometric transformations are derived. Further, their applications to Rogers–Ramanujan type identities will be discussed.

New Infinite Series for Calculating the Circumference of an Ellipse

In this paper a new formula for the circumference is derived which is faster than existing formulas and is based off of a cubic hypergeometric transformation.

Terminating Basic Hypergeometric Representations and Transformations for the Askey-Wilson Polynomials.

In this survey paper, we exhaustively explore the terminating basic hypergeometric representations of the Askey–Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy.

On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier–Legendre expansions

Motivated by our previous work on hypergeometric functions and the parbelos constant, we perform a deeper investigation on the interplay among generalized complete elliptic integrals, Fourier–Legendre (FL) series expansions, and Fqp series. We produce new hypergeometric transformations and closed-form evaluations for new series involving harmonic numbers, through the use of the integration method outlined as follows: Letting K denote the complete elliptic integral of the first kind, for a suitable function g we evaluate integrals such as∫01K(x)g(x)dx in two different ways: (1) by expanding K as a Maclaurin series, perhaps after a transformation or a change of variable, and then integrating term-by-term; and (2) by expanding g as a shifted FL series, and then integrating term-by-term. Equating the expressions produced by these two approaches often gives us new closed-form evaluations, as in the formulas involving Catalan's constant G∑n=0∞(2nn)2Hn+14−Hn−1416n=Γ4(14)8π2−4Gπ,∑m,n≥0(2mm)2(2nn)216m+n(m+n+1)(2m+3)=7ζ(3)−4Gπ2.

On Transformations of <i>A</i>-Hypergeometric Functions

We propose a systematic study of transformations of $A$-hypergeometric functions. Our approach is to apply changes of variables corresponding to automorphisms of toric rings, to Euler-type integral representations of $A$-hypergeometric functions. We show that all linear $A$-hypergeometric transformations arise from symmetries of the corresponding polytope. As an application of the techniques developed here, we show that the Appell function $F_4$ does not admit a certain kind of Euler-type integral representation.

Some supercongruences involving

ABSTRACTBy applying a hypergeometric transformation, Long proved that for any prime . In this paper we first reprove the above congruence by using Zeilberger's algorithm. We then give some generalizations of this supercongruence, such as where is a prime of the form 3k+2. This partially confirms a recent conjecture of Guo [Some generalizations of a supercongruence of van Hamme. Integral Transforms Spec. Funct. 28 (2017), pp. 888–899].

On a class of new hypergeometric transformations

On a class of new hypergeometric transformations

Supercongruences on some binomial sums involving Lucas sequences

In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic hypergeometric transformation, we prove that∑k=0p−1(p−1k)(2kk)2Pk8k≡0(modp2) for any prime p≡7(mod8), where Pk is the k-th Pell number. Further, we also propose three new congruences of the same type.