Identities of nonterminating series by Zeilberger's algorithm

This paper argues that automated proofs of identities for non-terminating hypergeometric series are feasible by a combination of Zeilberger's algorithm and asymptotic estimates. For two analogues of Saalsch\"utz' summation formula in the non-terminating case this is illustrated.Comment: 12 page

Special functions and q-commuting variables

This paper is mostly a survey, with a few new results. The first part deals with functional equations for q-exponentials, q-binomials and q-logarithms in q-commuting variables and more generally under q-Heisenberg relations. The second part discusses translation invariance of Jackson integrals, q-Fourier transforms and the braided line.Comment: plain TeX, 35 pages; Univ. of Amsterdam, Dept. of Math.; to appear in "Special Functions, q-Series and Related Topics", The Fields Institute Communications Series, minor revision, mainly linguisti

Fractional Integral and Generalized Stieltjes Transforms for Hypergeometric Functions as Transmutation Operators

For each of the eight $n$-th derivative parameter changing formulas for Gauss hypergeometric functions a corresponding fractional integration formula is given. For both types of formulas the differential or integral operator is intertwining between two actions of the hypergeometric differential operator (for two sets of parameters): a so-called transmutation property. This leads to eight fractional integration formulas and four generalized Stieltjes transform formulas for each of the six different explicit solutions of the hypergeometric differential equation, by letting the transforms act on the solutions. By specialization two Euler type integral representations for each of the six solutions are obtained

Some details of proofs of theorems related to the quantum dynamical Yang-Baxter equation

This paper gives some further details of proofs of some theorems related to the quantum dynamical Yang-Baxter equation. This mainly expands proofs given in "Lectures on the dynamical Yang-Baxter equation" by P. Etingof and O. Schiffmann, math.QA/9908064. This concerns the intertwining operator, the fusion matrix, the exchange matrix and the difference operators. The last part expands proofs given in "Traces of intertwiners for quantum groups and difference equations, I" by P. Etingof and A. Varchenko, math.QA/9907181. This concerns the dual Macdonald-Ruijsenaars equations. This paper does not claim originality, priority or completeness. It is meant as a service to whoever may take profit of it.Comment: 14 page

Quadratic transformations for orthogonal polynomials in one and two variables

We discuss quadratic transformations for orthogonal polynomials in one and two variables. In the one-variable case we list many (or all) quadratic transformations between families in the Askey scheme or $q$-Askey scheme. In the two-variable case we focus, after some generalities, on the polynomials associated with root system $BC_2$, i.e., $BC_2$-type Jacobi polynomials if $q=1$ and Koornwinder polynomials in two variables in the $q$-case.Comment: v2: minor corrections, 24 page