An extension of Warnaar’s matrix inversion

R., X.

doi:10.1090/s0002-9939-05-07912-8

Cited by 14 publications

(6 citation statements)

References 19 publications

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“…Because M nm (a, a) = δ nm and D nm (bc/q; b, c) = δ nm , we find after setting t = k in (6.5) that M (a, k)M (k, a) = 1. The inversion of the matrix M is reached thus by the permutation of parameters a and k (in the p = 0 case this fact was established in [68]; a more detailed discussion of such matrix inversions is given in [53,56,69]). Therefore, α(a, k) = β(k, a) and β(a, k) = α(k, a) define new Bailey pairs to which one can apply the transformation (6.4).…”

Section: Chains Of Symmetry Transformations For Functionsmentioning

confidence: 99%

Essays on the theory of elliptic hypergeometric functions

Spiridonov¹

2008

Russ. Math. Surv.

118

177

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Abstract. We give a brief review of the main results of the theory of elliptic hypergeometric functions -a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transformations for the corresponding elliptic hypergeometric functions of the higher order.

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Section: Chains Of Symmetry Transformations For Functionsmentioning

confidence: 99%

Essays on the theory of elliptic hypergeometric functions

Spiridonov¹

2008

Russ. Math. Surv.

118

177

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“…As the earlier work of [22] displays, the ( f, g)-inversion formula contains many known inverse relations useful to the study of q-series as special cases. The reader is referred to [7,13] for further details on inverse relations and the classical lagrange inversion formula, and to [22,23] for applications of the ( f, g)-inversion formulas.…”

Section: Preliminariesmentioning

confidence: 99%

Further study on elliptic interpolation formulas for the elliptic Askey-Wilson polynomials and allied identities

Wang,

2020

Preprint

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In this paper, we introduce the so-called elliptic Askey-Wilson polynomials which are homogeneous polynomials in two special theta functions. With regard to the significance of polynomials of such kind, we establish some general elliptic interpolation formulas by the methods of matrix inversions and of polynomial representations. Furthermore, we find that the basis of elliptic interpolation space due to Schlosser can be uniquely characterized via the elliptic Askey-Wilson polynomials. As applications of these elliptic interpolation formulas, we establish some new elliptic function identities, including an extension of Weierstrass' theta identity, a generalized elliptic Karlsson-Minton type identity, and an elliptic analogue of Gasper's summation formula for very-well-poised 6+2m φ 5+2m series.

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“…[27,58]), forms the main theme of the present paper. One of valid tools for this purpose, in our viewpoint, is the so-called ( f, g)-inversion formula initially appeared in [43], together with an important method often referred to as "Ismail's argument" [54] in the literature. So named because it is…”

Section: Df(x) DX X=0mentioning

confidence: 99%

Expansion Formulas of Basic Hypergeometric Series via the (1-xy,y-x)--Inversion and Its Applications

Wang,

2013

Preprint

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With the use of the ( f, g)-inversion formula under specializations that f = 1−xy, g = y−x, we establish an expansion of (modified) basic hypergeometric r φ s series in variablex t as a linear combination of r+2 φ s+1 series in t and its various specifications. These expansions can be regarded as common generalizations of Carlitz's, Liu's, and Chu's expansion in the setting of q-series. As direct applications, some new transformation formulas of q-series including new approach to the Askey-Wilson polynomials, the Rogers-Fine identity, Andrews' four-parametric reciprocity theorem and Ramanujan's 1 ψ 1 summation formula, as well as a transformation for certain well-poised Bailey pairs, are presented.

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An extension of Warnaar’s matrix inversion

Cited by 14 publications

References 19 publications

Essays on the theory of elliptic hypergeometric functions

Essays on the theory of elliptic hypergeometric functions

Further study on elliptic interpolation formulas for the elliptic Askey-Wilson polynomials and allied identities

Expansion Formulas of Basic Hypergeometric Series via the (1-xy,y-x)--Inversion and Its Applications

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