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 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).
We show that several classical bilateral summation and transformation formulas have semi-finite forms. We obtain these semi-finite forms from unilateral summation and transformation formulas. Our method can be applied to derive Ramanujan's $_1ψ_1$ summation, Bailey's $_2ψ_2$ transformations, and Bailey's $_6ψ_6$ summation.
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.
We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For purpose of motivation, we review our previous simple proof ("A simple proof of Bailey's very-well-poised 6-psi-6 summation", Proc. Amer. Math. Soc., to appear) of Bailey's very-well-poised 6-psi-6 summation. Using a similar but different method, we now give elementary derivations of some transformations for bilateral basic hypergeometric series. In particular, these include M. Jackson's very-well-poised 8-psi-8 transformation, a very-well-poised …
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 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.
We provide new formulae for the degenerations of the bilateral basic hypergeometric function ${}_1ψ_1 ( a; b; q, z )$ with using the $q$-Borel-Laplace transformation. These are thought of as the first step to construct connection formulae of $q$-difference equation for ${}_1ψ_1 ( a; b; q, z )$. Moreover, we show that our formulae have the $q \to 1 - 0$ limit.
We discuss the spectral decomposition of the hypergeometric differential operators on the line $\mathrm{Re}\, z=1/2$. Such operators arise in the problem of decomposition of tensor products of unitary representations of the universal covering of the group $SL(2\,{\mathbb R}$. Our main purpose is a search of natural bases in generalized eigenspaces and variants of the inversion formula.
We study multivariable (bilateral) basic hypergeometric series associated with (type $A$) Macdonald polynomials. We derive several transformation and summation properties for such series including analogues of Heine's ${}_2ϕ_1$ transformation, the $q$-Pfaff-Kummer and Euler transformations, the $q$-Saalschütz summation formula and Sear's transformation for terminating, balanced ${}_4ϕ_3$ series. For bilateral series, we rederive Kaneko's analogue of the ${}_1ψ_1$ summation formula and give multivariable extensions of Bailey's ${}_2ψ_2$ transformations.
In [Electron. J. Combin. 10 (2003), #R10], the author presented a new basic hypergeometric matrix inverse with applications to bilateral basic hypergeometric series. This matrix inversion result was 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. The present paper features three different multivariable generalizations of the above result. These are extracted from Gustafson's A_r and C_r extensions and of the author's recent A_r extension of Bailey's 6-psi-6 summation formula. By combi…
Multidimensional matrix inversions provide a powerful tool for studying multiple hypergeometric series. In order to extend this technique to elliptic hypergeometric series, we present three new multidimensional matrix inversions. As applications, we obtain a new $A_r$ elliptic Jackson summation, as well as several quadratic, cubic and quartic summation formulas.
Using Krattenthaler's operator method, we give a new proof of Warnaar's recent elliptic extension of Krattenthaler's matrix inversion. Further, using a theta function identity closely related to Warnaar's inversion, we derive summation and transformation formulas for elliptic hypergeometric series of Karlsson-Minton-type. A special case yields a particular summation that was used by Warnaar to derive quadratic, cubic and quartic transformations for elliptic hypergeometric series. Starting from another theta function identity, we derive yet different summation and transformation formulas for el…
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 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.
Chu has recently shown that the Abel lemma on summations by parts can serve as the underlying relation for Bailey's ${}_6ψ_6$ bilateral summation formula. In other words, the Abel lemma spells out the telescoping nature of the ${}_6ψ_6$ sum. We present a systematic approach to compute Abel pairs for bilateral and unilateral basic hypergeometric summation formulas by using the $q$-Gosper algorithm. It is demonstrated that Abel pairs can be derived from Gosper pairs. This approach applies to many classical summation formulas.
A multidimensional generalization of Bailey's very-well-poised bilateral basic hypergeometric ${}_6ψ_6$ summation formula and its Dougall type ${}_5H_5$ hypergeometric degeneration for $q\to 1$ is studied. The multiple Bailey sum amounts to an extension corresponding to the case of a nonreduced root system of certain summation identities associated to the reduced root systems that were recently conjectured by Aomoto and Ito and proved by Macdonald. By truncation, we obtain multidimensional analogues of the very-well-poised unilateral (basic) hypergeometric Rogers ${}_6ϕ_5$ and Dougall ${}_5F_4…
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,…
In this paper, we study the solutions of the system of bilateral type matrix differential equations and presented these solutions in terms of Lauricella hypergeometric matrix functions of several variables and Srivastava's triple hypergeometric matrix functions of three variables. We also discuss the region of convergence and integral representation for these matrix functions.
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. Bas…
We deduce new q-series identities by applying inverse relations to certain identities for basic hypergeometric series. The identities obtained themselves do not belong to the hierarchy of basic hypergeometric series. We extend two of our identities, by analytic continuation, to bilateral summation formulae which contain Ramanujan's 1-psi-1 summation and a very-well-poised 4-psi-6 summation as special cases.
Given complex parameters $x$, $ν$, $α$, $β$ and $γ\notin -\mathbb{N}$, consider the infinite lower triangular matrix $\mathbf{A}(x,ν;α, β,γ)$ with elements $$ A_{n,k}(x,ν;α,β,γ) = \displaystyle (-1)^k\binom{n+α}{k+α} \cdot F(k-n,-(β+n)ν;-(γ+n);x) $$ for $1 \leqslant k \leqslant n$, depending on the Hypergeometric polynomials $F(-n,\cdot;\cdot;x)$, $n \in \mathbb{N}^*$. After stating a general criterion for the inversion of infinite matrices in terms of associated generating functions, we prove that the inverse matrix $\mathbf{B}(x,ν;α, β,γ) = \mathbf{A}(x,ν;α, β,γ)^{-1}$ is given by \begin{ali…
For any complex parameters $x$ and $ν$, we provide a new class of linear inversion formulas $T = A(x,ν) \cdot S \Leftrightarrow S = B(x,ν) \cdot T$ between sequences $S = (S_n)_{n \in \mathbb{N}^*}$ and $T = (T_n)_{n \in \mathbb{N}^*}$, where the infinite lower-triangular matrix $A(x,ν)$ and its inverse $B(x,ν)$ involve Hypergeometric polynomials $F(\cdot)$, namely $$ \left\{ \begin{array}{ll} A_{n,k}(x,ν) = \displaystyle (-1)^k\binom{n}{k}F(k-n,-nν;-n;x), \\ B_{n,k}(x,ν) = \displaystyle (-1)^k\binom{n}{k}F(k-n,kν;k;x) \end{array} \right. $$ for $1 \leqslant k \leqslant n$. Functional relation…
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.
A series expansion for Heckman-Opdam hypergeometric functions $φ_λ$ is obtained for all $λ\in \mathfrak a^*_{\mathbb C}.$ As a consequence, estimates for $φ_λ$ away from the walls of a Weyl chamber are established. We also characterize the bounded hypergeometric functions and thus prove an analogue of the celebrated theorem of Helgason and Johnson on the bounded spherical functions on a Riemannian symmetric space of the noncompact type. The $L^p$-theory for the hypergeometric Fourier transform is developed for $0<p<2$. In particular, an inversion formula is proved when $1\leq p <2$.
Given parameters $x \notin \mathbb{R}^- \cup \{1\}$ and $ν$, $\mathrm{Re}(ν) < 0$, and the space $\mathscr{H}_0$ of entire functions in $\mathbb{C}$ vanishing at $0$, we consider the family of operators $\mathfrak{L} = c_0 \cdot δ\circ \mathfrak{M}$ with constant $c_0 = ν(1-ν)x/(1-x)$, $δ= z \, \mathrm{d}/\mathrm{d}z$ and integral operator $\mathfrak{M}$ defined by $$ \mathfrak{M}f(z) = \int_0^1 e^{- \frac{z}{x}t^{-ν}(1-(1-x)t)} \, f \left ( \frac{z}{x} \, t^{-ν}(1-t) \right ) \, \frac{\mathrm{d}t}{t}, \qquad z \in \mathbb{C}, $$ for all $f \in \mathscr{H}_0$. Inverting $\mathfrak{L}$ or $\mat…
In this article, we derive some identities for multilateral basic hypergeometric series associated to the root system A_n. First, we apply Ismail's argument to an A_n q-binomial theorem of Milne and derive a new A_n generalization of Ramanujan's 1-psi-1 summation theorem. From this new A_n 1-psi-1 summation and from an A_n 1-psi-1 summation of Gustafson we deduce two lemmas for deriving simple A_n generalizations of bilateral basic hypergeometric series identities. These lemmas are closely related to the Macdonald identities for A_n. As samples for possible applications of these lemmas, we pro…
We give elementary derivations of some classical summation formulae for bilateral (basic) hypergeometric series. In particular, we apply Gauss' 2-F-1 summation and elementary series manipulations to give a simple proof of Dougall's 2-H-2 summation. Similarly, we apply Rogers' nonterminating 6-phi-5 summation and elementary series manipulations to give a simple proof of Bailey's very-well-poised 6-psi-6 summation. Our method of proof extends M. Jackson's first elementary proof of Ramanujan's 1-psi-1 summation.
We continue a program generalizing classical results from the analysis on symmetric cones to the Dunkl setting for root systems of type A. In particular, we prove a Dunkl-Laplace transform identity for Heckman-Opdam hypergeometric functions of type A and more generally, for the associated Cherednik kernel. This is achieved by analytic continuation from a Laplace transform identity for non-symmetric Jack polynomials which was stated, for the symmetric case, as a key conjecture in an unpublished manuscript of Macdonald (2013). Our proof for the Jack polynomials is based on Dunkl operator techniq…
We study the perturbative power-series expansions of the eigenvalues and eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d. The(small) expansion parameters are being the entries of the two diagonals of length d-1 sandwiching the principal diagonal, which gives the unperturbed spectrum. The solution is found explicitly in terms of multivariable (Horn-type) hypergeometric series of 3d-5 variables in the generic case, or 2d-3 variables for the eigenvalue growing from a corner matrix element. To derive the result, we first rewrite the spectral problem for a Jacobi matrix as an e…
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 ${}_4ϕ_3$ and terminating very-well poised ${}_8W_7$ 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 equ…
