We obtain Mehler’s formula and the Rogers formula for the continuous big qHermite polynomials Hn(x; a|q). Instead of working with the polynomials Hn(x; a|q) directly, we consider the equivalent forms in terms of the bivariate Rogers-Szegö polynomials hn(x, y|q) recently introduced by Chen, Fu and Zhang. It turns out that Mehler’s formula for Hn(x; a|q) involves a 3φ2 sum, and the Rogers formula...

Recently we interpreted five q-series identities of Rogers combinatorially by using partitions with “n + t copies of n” of Agarwal and Andrews [1]. In this paper we use lattice paths of Agarwal and Bressoud [2] to provide new combinatorial interpretations of the same identities. This results in five new 3-way combinatorial identities.

Ramanujan’s lost notebook contains several identities arising from the Rogers–Fine identity and/or Rogers’ false theta functions. Combinatorial proofs for many of these identities are given.

N ature is full of curves. So it is only natural that manmade devices meant to interact with nature emulate its elements of design. That is the line of reasoning behind the innovations of John Rogers, a member of the National Academy of Engineering and a professor of materials science at the University of Illinois, Urbana-Champaign. Rogers’ innovations have inspired an array of approaches to re...

In this paper, we consider composition operators on Hardy-Sobolev spaces in connections with BMO-quasiconformal mappings. Using the duality of Hardy and BMO-spaces, prove that mappings generate bounded from Hardy–Sobolev to Sobolev spaces.