Rogers–Ramanujan type identities and Chebyshev polynomials of the third kind

Abstract

It is known that q-orthogonal polynomials play an important role in the field of q-series and special functions. While studying Dyson’s “favorite” identity of Rogers–Ramanujan type, Andrews pointed out that the classical orthogonal polynomials also have surprising applications in the world of q. By introducing Chebyshev polynomials of the third and the fourth kinds into Bailey pairs, Andrews derived a family of Rogers–Ramanujan type identities and also results related to mock theta functions and Hecke-type series. In this paper, by constructing a new Bailey pair involving Chebyshev polynomials of the third kind, we further extend Andrews’ way of studying Rogers-Ramanujan type identities. By inserting this Bailey pair into various weak forms of Bailey’s lemma, we obtain a companion identity for Dyson’s favorite identity and a number of Rogers–Ramanujan type identities. As a consequences, we also obtain results related to Appell–Lerch series and the generalized Hecke-type series. Furthermore, our key Bailey pair also fits in the bilateral versions of Bailey’s lemma due to Andrews and Warnaar, which leads to more identities for the generalized Hecke-type series and false theta functions.