9 research outputs found
Limits of elliptic hypergeometric biorthogonal functions
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
functions or polynomials.Comment: 27 pages. This is a self-contained article which can also be seen as
part 1 of a 3 part series on limits of (multivariate) elliptic hypergeometric
biorthogonal functions and their measure
Limits of elliptic hypergeometric biorthogonal functions
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 (2003), 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 functions or polynomials
Elliptic Racah polynomials
Upon solving a finite discrete reduction of the difference Heun equation, we
arrive at an elliptic generalization of the Racah polynomials. We exhibit the
three-term recurrence relation and the orthogonality relations for these
elliptic Racah polynomials. The well-known -Racah polynomials of Askey and
Wilson are recovered as a trigonometric limit.Comment: 18 page
Rahman’s Biorthogonal Rational Functions and Superconformal Indices
We study biorthogonal functions related to basic hypergeometric integrals with coupled continuous and discrete components. Such integrals appear as superconformal indices for three-dimensional quantum field theories and also in the context of solvable lattice models. We obtain explicit biorthogonal systems given by products of two of Rahman’s biorthogonal rational 10 W 9 -functions or their degenerate cases. We also give new bilateral extensions of the Jackson and q-Saalsch\ufctz summation formulas and new continuous and discrete biorthogonality measures for Rahman’s functions
Limits of multivariate elliptic beta integrals and related bilinear forms
In this article we consider the elliptic Selberg integral, which is a BC_n symmetric multivariate extension of the elliptic beta integral. We categorize the limits that are obtained as p → 0, for given behavior of the parameters as p → 0. This article is therefore the multivariate version of our earlier paper "Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions". The integrand of the elliptic Selberg integral is the measure for the BC_n symmetric biorthogonal functions introduced by the second author, so we also consider the limits of the associated bilinear form. We also provide the limits for the discrete version of this bilinear form, which is related to a multivariate extension of the Frenkel-Turaev summation
Limits of multivariate elliptic beta integrals and related bilinear forms
In this article we consider the elliptic Selberg integral, which is a BC_n symmetric multivariate extension of the elliptic beta integral. We categorize the limits that are obtained as p → 0, for given behavior of the parameters as p → 0. This article is therefore the multivariate version of our earlier paper "Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions". The integrand of the elliptic Selberg integral is the measure for the BC_n symmetric biorthogonal functions introduced by the second author, so we also consider the limits of the associated bilinear form. We also provide the limits for the discrete version of this bilinear form, which is related to a multivariate extension of the Frenkel-Turaev summation