In 1981, Andrews gave a four-variable generalization of Ramanujan's ${_1ψ_1}$ summation formula. We establish a six-variable generalization of Andrews' identity according to the transformation formula for two ${_8ϕ_7}$ series and Bailey's transformation formula for three ${_8ϕ_7}$ series. Then it is used to find a six-variable generalization of Ramanujan's reciprocity theorem, which is different from Liu's formula. We derive the generalizations of Bailey's two $_3ψ_3$ summation formulas in terms of two limiting relations and Bailey's another transformation formula for three $_8ϕ_7$ series. Based on the two limiting relations, some different results involving bilateral basic hypergeometric series are also deduced from the Guo--Schlosser transformation formula and other two transformation formulas.
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. From the terminating basic hypergeometric representations of these polynomials, and due to symmetry in their free parameters, we are able to exhaustively explore the terminating basic hypergeometric transformation formulae which these polynomials satisfy.
We derive two generalizations of Gasper's transformation formula for basic hypergeometric series. Using these generalized formulas, we give explicit expressions for the coefficients of three-term relations for the basic hypergeometric series {}_{2}phi_{1}, which are generalizations of the author's previous results on three-term relations for {}_{2}phi_{1}.
We describe a uniform way of obtaining basic hypergeometric functions as limits of the elliptic beta integral. This description gives rise to the construction of a polytope with a different basic hypergeometric function attached to each face of this polytope. We can subsequently obtain various relations, such as transformations and three-term relations, of these functions by considering geometrical properties of this polytope. The most general functions we describe in this way are sums of two very-well-poised ${}_{10}ϕ_9$'s and their Nassrallah-Rahman type integral representation.
We show an asymptotic formula of the divergent bilateral basic hypergeometric series ${}_1ψ_0 (a;-;q,\cdot)$ with using the $q$-Borel-Laplace method. We also give the limit $q\to 1-0$ of our asymptotic formula.
Here are the latest errata, etc., to the Gasper and Rahman "Basic Hypergeometric Series" book. Any additional errata will be added to the end of the last list.
We establish conditions for the discrete versions of logarithmic concavity and convexity of the higher order regularized basic hypergeometric function with respect simultaneous shift of all its parameters. For a particular case of Heine's basic hypergeometric function we prove logarithmic concavity and convexity with respect to the bottom parameter. We further establish a linearization identity for the generalized Turánian formed by a particular case of Heine's basic hypergeometric function. Its $q=1$ case also appears to be new.
Any three basic hypergeometric series {}_{2}phi_{1} whose respective parameters (a, b, c) differ by integer powers of the base q satisfy a linear relation with coefficients which are rational functions of a, b, c, q and the variable x. These relations are called three term relations for the basic hypergeometric series {}_{2}phi_{1}. This paper gives explicit expressions for the coefficients of these three term relations.
This article gives a classification scheme of algebraic transformations of Gauss hypergeometric functions, or pull-back transformations between hypergeometric differential equations. The classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of the Gauss hypergeometric function. The other transformations are considered more thoroughly in a series of supplementing articles.
Darboux Transformation, well known in second order differential operator theory, is applied here to the difference equation satisfied by the discrete hypergeometric polynomials(Charlier, Meixner-Krawchuk, Hahn).
The paper classifies algebraic transformations of Gauss hypergeometric functions with the local exponent differences $(1/2,1/4,1/4)$, $(1/2,1/3,1/6)$ and $(1/3,1/3,1/3)$. These form a special class of algebraic transformations of Gauss hypergeometric functions, of arbitrary high degree. The Gauss hypergeometric functions can be identified as elliptic integrals on the genus 1 curves $y=x^3-x$ or $y=x^3-1$. Especially interesting are algebraic transformations of the hypergeometric functions into themselves; these transformations come from isogenies of the respective elliptic curves.
The hypergeometric and Heun functions are classical special functions. Transformation formulas between them are commonly induced by pull-back transformations of their differential equations, with respect to some coverings P1-to-P1. This gives expressions of Heun functions in terms of better understood hypergeometric functions. This article presents the list of hypergeometric-to-Heun pull-back transformations with a free continuous parameter, and illustrates most of them by a Heun-to-hypergeometric reduction formula. In total, 61 parametric transformations exist, of maximal degree 12.
New duality transformation formulas are proposed for multiple elliptic hypergeometric series of type $BC$ and of type $C$. Various transformation and summation formulas are derived as special cases to recover some previously known results.
We study the divergent basic hypergeometric series which is a $q$-analog of divergent hypergeometric series. This series formally satisfies the linear $q$-difference equation. In this paper, for that equation, we give an actual solution which admits basic hypergeometric series as a $q$-Gevrey asymptotic expansion. Such an actual solution is obtained by using $q$-Borel summability, which is a $q$-analog of Borel summability. Our result shows a $q$-analog of the Stokes phenomenon. Additionally, we show that letting $q\to1$ in our result gives the Borel sum of classical hypergeometric series. The…
Hypergeometric equations with a dihedral monodromy group can be solved in terms of elementary functions. This paper gives explicit general expressions for quadratic monodromy invariants for these hypergeometric equations, using a generalization of Clausen's formula and terminating double hypergeometric sums. Besides, pull-back transformations for the dihedral hypergeometric equations are presented, including Klein's pullback transformations for the equations with a finite (dihedral) monodromy group.
This paper presents explicit algebraic transformations of some Gauss hypergeometric functions. Specifically, the transformations considered apply to hypergeometric solutions of hypergeometric differential equations with the local exponent differences $1/K,1/L,1/M$ such that $K,L,M$ are positive integers and $1/K+1/L+1/M<1$. All algebraic transformations of these Gauss hypergeometric functions are considered. We show that apart from classical transformations of degree 2, 3, 4, 6 there are several other transformations of degree 6, 8, 9, 10, 12, 18, 24. Besides, we present an algorithm to comput…
In this paper, a generalization of Ramanujan's cubic transformation, in the form of an inequality, is proved for zero-balanced Gaussian hypergeometric function $F(a,b;a+b;x)$, $a,b>0$.
We derive generalized generating functions for basic hypergeometric orthogonal polynomials by applying connection relations with one free parameter to them. In particular, we generalize generating functions for the Askey-Wilson, continuous $q$-ultraspherical/Rogers, little $q$-Laguerre/Wall, and $q$-Laguerre polynomials. Depending on what type of orthogonality these polynomials satisfy, we derive corresponding definite integrals, infinite series, bilateral infinite series, and $q$-integrals.
In this article we continue the work from arXiv:0902.0621. In that article Eric Rains and the present author considered the limits of the elliptic beta integral as p->0 while the parameters t_r have a p-dependence of the form t_r=u_rp^{α_r} (for fixed u_r and certain real numbers α_r). In this article we again consider such limits, but now we let p->0 along a geometric sequence p=xq^{sk} (for some integer s, while k -> \infty), and only allow α_r\in 2/s Z. These choices allow us to take many more limits. In particular we now also obtain bilateral basic hypergeometric series as possible limits,…
General duality relations for hypergeometric and basic hypergeometric series
Holon Institute of Technology（霍隆理工学院）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Department of Applied Mathematics, School of Mathematics and Physics, Xi’an Jiaotong-Liverpool University（西交利物浦大学数理学院应用数学系）
In this note we state (with minor corrections) and give an alternative proof of a very general hypergeometric transformation formula due to Slater. As an application, we obtain a new hypergeometric transformation formula for a ${}_5F_4(-1)$ series with one pair of parameters differing by unity expressed as a linear combination of two ${}_3F_2(1)$ series.
We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions possess. These integral representations were studied by Bailey, Slater, Askey, Roy, Gasper and Rahman and were also used to facilitate the computation of certain outstanding problems in the theory of basic hypergeometric orthogonal polynomials in the $q$-Askey scheme. We also generalize and give consequences and transformation formulas for some fundamental integ…
Recently, we formulated the $q$-Garnier system and its variations as translations of an extended affine Weyl group of type $A^{(1)}_{2n+1}\times A^{(1)}_1\times A^{(1)}_1$. On the other hand, those systems admit particular solutions in terms of the basic hypergeometric series ${}_{n+1}ϕ_n$. In this article, we investigate an action of the extended affine Weyl group on ${}_{n+1}ϕ_n$.
In this brief note, we show how to apply Kummer's and other quadratic transformation formulas for Gauss' and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that contain one or two continuous parameters. We also give a generating function of the sequence $\frac{(a)_n (1-a)_n}{(n!)^2}H_n$ as a combination of Gauss hypergeometric function and elementary functions.
We present here the $q$-analogues of certain transformations or reduction formulae for Srivastava-Daoust type double hypergeometric series. These reduction formulae are derived by utilizing the extended Bailey's Transform developed and studied by Joshi and Vyas [Int. J. Math. Sci., (12), 2005, 1909-1927]. A number of well-known $q$-hypergeometric transformations are also obtained as special cases of our results.
We prove a duality relation for generalized basic hypergeometric functions. It forms a $q$-extension of a recent result of the second and the third named authors and generalizes both a $q$-hypergeometric identity due to the third named author (jointly with Feng and Yang) and a recent identity for the Heine's ${}_2ϕ_{1}$ function due to Suzuki. We further explore various consequences of our identity leading to several presumably new multi-term relations for both terminating and non-terminating generalized basic hypergeometric series. Moreover, we give confluent versions of our results and furni…
In this article we prove a new elliptic hypergeometric integral identity. It previously appeared (as a conjecture) in articles by Rains, and Spiridonov and Vartanov. Moreover it gives a different proof of an identity in another article by Rains. We also give some basic hypergeometric and classical limits of this identity. The classical limit gives identities (some known, some new) between generalizations of the Selberg integral.
In this article we give a new transformation between elliptic hypergeometric beta integrals, which gives rise to a Weyl group symmetry of type F_4. The transformation is a generalization of a series transformation discovered by Langer, Schlosser, and Warnaar. Moreover we consider various limits of this transformation to basic hypergeometric functions obtained by letting p tend to 0.
After reviewing some fundamental facts from the theory of theta hypergeometric series we derive, using indefinite summation, several summation, transformation, and expansion formulas for multibasic theta hypergeometric series. Some of the identities presented here generalize corresponding formulas given in Chapter 11 of the Gasper and Rahman book [Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics And Its Applications 96, Cambridge University Press, Cambridge, 2004].
A proof of an unusual summation formula for a basic hypergeometric series associated to the affine root system $\tilde A_n$ that was conjectured by Warnaar is given. It makes use of Milne's $A_n$ extension of Watson's transformation, Ramanujan's $_1ψ_1$-summation, and a determinant evaluation of the author. In addition, a transformation formula between basic hypergeometric series associated to the affine root systems $\tilde A_n$ respectively $\tilde A_m$, which generalizes at the same time the above summation formula and an identity due to Gessel and the author, is proposed as a conjecture.
We extend expansion formulas of Liu given in 2013 to the context of multiple series over root systems. Liu and others have shown the usefulness of these formulas in Special Functions and number-theoretic contexts. We extend Wang and Ma's generalizations of Liu's work which they obtained using $q$-Lagrange inversion. We use the $A_n$ and $C_n$ Bailey transformation and other summation theorems due to Gustafson, Milne, Milne and Lilly, and others, from the theory of $A_n$, $C_n$ and $D_n$ basic hypergeometric series.
The beta integral method proved itself as a simple nonetheless powerful method of generating hypergeometric identities at a fixed argument. In this paper we propose a generalization by substituting the beta density with a particular type of Meijer's G function. By application of our method to known transformation formulas we derive about forty hypergeometric identities, majority of which are believed to be new. We further apply some of these transformations to obtain several new summation formulas.
Pull-back transformations between Heun and Gauss hypergeometric equations give useful expressions of Heun functions in terms of better understood hypergeometric functions. This article classifies, up to Mobius automorphisms, the coverings P1-to-P1 that yield pull-back transformations from hypergeometric to Heun equations with at least one free parameter (excluding the cases when the involved hypergeometric equation has cyclic or dihedral monodromy). In all, 61 parametric hypergeometric-to-Heun transformations are found, of maximal degree 12. Among them, 28 pull-backs are compositions of smalle…
We prove some new semi-finite forms of bilateral basic hypergeometric series. One of them yields in a direct limit Bailey's celebrated ${}_6ψ_6$ summation formula, answering a question recently raised by Chen and Fu ({\em Semi-Finite Forms of Bilateral Basic Hypergeometric Series}, Proc. Amer. Math. Soc., to appear).
Eric Rains conjectured several quadratic transformations between multivariate elliptic hypergeometric functions in "Elliptic Littlewood Identities", with the integrand multiplied by interpolation functions. In this article two of these conjectures are proven in the case where the interpolation functions are constant, and we obtain a third conjecture as a corollary. Two other equations for elliptic Selberg integrals with 10 parameters, two of which multiplying to pq/t, are given, as they are needed in the proof. The proofs consist essentially of a calculation which strings together many ellipti…
We first show how one can obtain Al-Salam--Chihara polynomials, continuous dual $q$-Hahn polynomials, and Askey--Wilson polynomials from the little $q$-Laguerre and the little $q$-Jacobi polynomials by using special transformations. This procedure is then extended to obtain multiple Askey--Wilson, multiple continuous dual $q$-Hahn, and multiple Al-Salam--Chihara polynomials from the multiple little $q$-Laguerre and the multiple little $q$-Jacobi polynomials.
Explicit solutions for the three-term recurrence satisfied by associated continuous dual $q$-Hahn polynomials are obtained. A minimal solution is identified and an explicit expression for the related continued fraction is derived. The absolutely continuous component of the spectral measure is obtained. Eleven limit cases are discussed in some detail. These include associated big $q$-Laguerre , associated Wall, associated Al-Salam-Chihara, associated Al-Salam-Carlitz I, and associated continuous $q$-Hermite polynomials.
The purpose of this article is to bring structure to (basic) hypergeometric biorthogonal systems, in particular to the q-Askey scheme of basic hypergeometric orthogonal polynomials. We aim to achieve this by looking at the limits as p->0 of the elliptic hypergeometric biorthogonal functions from Spiridonov, with parameters which depend in varying ways on p. As a result we get 38 systems of biorthogonal functions with for each system at least one explicit measure for the bilinear form. Amongst these we indeed recover the q-Askey scheme. Each system consists of (basic hypergeometric) rational fu…
In this paper, we propose a general method to express explicitly the inversion and the connection coefficients between two basic hypergeometric polynomial sets. As application, we consider some $d$-orthogonal basic hypergeometric polynomials and we derive expansion formulae corresponding to all the families within the $q$-Askey scheme.
In this very short note we will derive an inequality for a class of entire functions including all the confluent basic hypergeometric series and an inequality for a class of meromorphic functions including theta functions.
We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive, via inverse relations, several new identities for bilateral basic hypergeometric series.
We study the group of transformations of 4F3 hypergeometric functions evaluated at unity with one unit shift in parameters. We reveal the general form of this family of transformations and its group property. Next, we use explicitly known transformations to generate a subgroup whose structure is then thoroughly studied. Using some known results for 3F2 transformation groups, we show that this subgroup is isomorphic to the direct product of the symmetric group of degree 5 and 5-dimensional integer lattice. We investigate the relation between two-term 4F3 transformations from our group and three…
We prove a summation formula for a bilateral series whose terms are products of two basic hypergeometric functions. In special cases, series of this type arise as matrix elements of quantum group representations.
We construct a set $H$ of orthogonal polynomial sequences that contains all the families in the Askey scheme and the $q$-Askey scheme. The polynomial sequences in $H$ are solutions of a generalized first-order difference equation which is determined by three linearly recurrent sequences of numbers. Two of these sequences are solutions of a difference equation of order three and the other sequence satisfies a related difference equation of order five. We obtain explicit expressions for the coefficients of the orthogonal polynomials and for the generalized moments with respect to a basis of Newt…
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.
In the lecture notes we start off with an introduction to the $q$-hypergeometric series, or basic hypergeometric series, and we derive some elementary summation and transformation results. Then the $q$-hypergeometric difference equation is studied, and in particular we study solutions given in terms of power series at $0$ and at $\infty$. Factorisations of the corresponding operator are considered in terms of a lowering operator, which is the $q$-derivative, and the related raising operator. Next we consider the $q$-hypergeometric operator in a special case, and we show that there is a natural…
In this paper, we will obtain new algebraic transformations of the $_2F_1$-hypergeometric functions. The main novelty in our approach is the interpretation of identities among $_2F_1$-hypergeometric functions as identities among automorphic forms on different Shimura curves.
We obtain extensions of classical hypergeometric identities of Bailey and Whipple that transform nearly-poised and very-well-poised series to Saalschützian series, Saalschützian series to Saalschützian series, and very-well-poised and nearly-poised series to very-well-poised series. We employ a method in which summations and transformations of lower-order series are used to obtain transformations of higher-order series. By taking limits, we also obtain extensions of two classical quadratic transformations of Whipple and Bailey. Furthermore, we show how a number of other well-known results rega…
We give summation formulae for the bilateral basic hypergeometric series ${}_1ψ_1( a; b; q, z )$ through Ramanujan's summation formula, which are generalizations of nontrivial identities found in the physics of three-dimensional Abelian mirror symmetry on $\mathbf{R}P^2 \times S^1$. We also show the $q \to 1 - 0$ limit of our summation formulae.
The classical summation and transformation theorems for very well-poised hypergeometric functions, namely, $_{5}F_4(1)$ summation, Dougall's $_{7}F_6(1)$ summation, Whipple's $_{7}F_6(1)$ to $_{4}F_3(1)$ transformation and Bailey's $_{9}F_8(1)$ to $_{9}F_8(1)$ transformation are extended. These extensions are derived by applying the well-known Bailey's transform method along with the classical very well-poised summation and transformation theorems for very well-poised hypergeometric functions and the Rakha and Rathie's extension of the Saalschütz's theorem. To show importance and applications …
Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system An, with different dimensions n. We give, with a new, elementary, proof, an elliptic analogue 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.
