On the equivalence of two fundamental theta identities

Two fundamental theta identities, a three-term identity due to Weierstrass and a five-term identity due to Jacobi, both with products of four theta functions as terms, are shown to be equivalent. One half of the equivalence was already proved by R.J. Chapman in 1996. The history and usage of the two identities, and some generalizations are also discussed.Comment: v3: 15 pages, minor errors corrected, references added, appendix on four-term theta identities added, accepted by Analysis and Application

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

Okounkov's BC-type interpolation Macdonald polynomials and their q=1 limit

This paper surveys eight classes of polynomials associated with $A$-type and $BC$-type root systems: Jack, Jacobi, Macdonald and Koornwinder polynomials and interpolation (or shifted) Jack and Macdonald polynomials and their $BC$-type extensions. Among these the $BC$-type interpolation Jack polynomials were probably unobserved until now. Much emphasis is put on combinatorial formulas and binomial formulas for (most of) these polynomials. Possibly new results derived from these formulas are a limit from Koornwinder to Macdonald polynomials, an explicit formula for Koornwinder polynomials in two variables, and a combinatorial expression for the coefficients of the expansion of $BC$-type Jacobi polynomials in terms of Jack polynomials which is different from Macdonald's combinatorial expression. For these last coefficients in the two-variable case the explicit expression in Koornwinder & Sprinkhuizen (1978) is now obtained in a quite different way.Comment: v5: 27 pages, formulas (10.7) and (10.14) correcte