Lisa H. Sun

The bivariate Rogers-Szegö polynomials

We present an operator approach to deriving Mehlers formula and the Rogers formula for the bivariate Rogers–Szego polynomials hn(x, y|q). The proof of Mehlers formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials Hn(x; a|q) due to Askey, Rahman and Suslov. Mehlers formula for hn(x, y|q) involves a 32 sum and the Rogers formula involves a 21 sum. The proofs of these results are based on parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers–Szego polynomials hn(x|q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for hn(x, y|q). Finally, we give a change of base formula for Hn(x; a|q) which can be used to evaluate some integrals by using the Askey–Wilson integral.

An operator approach to the Al-Salam–Carlitz polynomials

We present an operator approach to Rogers-type formulas and Mehler’s formula for the Al-Salam–Carlitz polynomials Un(x,y,a;q). By using the q-exponential operator, we obtain a Rogers-type formula, which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler’s formula due to Al-Salam and Carlitz [“Some orthogonal q-polynomials,” Math. Nachr. 30, 47 (1965)]. By means of the Cauchy companion augmentation operator, we obtain an equivalent form of Mehler’s formula. We also give several identities on the generating functions for products of the Al-Salam–Carlitz polynomials, which are extensions of the formulas for the Rogers–Szego polynomials.

The method of combinatorial telescoping

We present a method for proving q-series identities by combinatorial telescoping, in the sense that one can transform a bijection or a classification of combinatorial objects into a telescoping relation. We shall illustrate this method by giving a combinatorial derivation of Watsons identity, which implies the Rogers-Ramanujan identities.

Stanley's Lemma and Multiple Theta Functions

We present an algorithmic approach to the verification of identities on multiple theta functions in the form of products of theta functions

Extended Zeilberger's algorithm for identities on Bernoulli and Euler polynomials

[(-1)^{\delta}a_1^{\alpha_1}a_2^{\alpha_2}\cdots a_r^{\alpha_r}q^{s}; q^{t}]_\infty

Ramanujan-type congruences for overpartitions modulo 5

, where

Quadratic forms and congruences for ℓ-regular partitions modulo 3, 5 and 7

\alpha_i

Congruences of multipartition functions modulo powers of primes

are integers,

Proving hypergeometric identities by numerical verifications

\delta=0

Stanley's Lemma and Multiple Theta Function Identities

or