Rogers-Ramanujan type identities and Chebyshev Polynomials of the third kind

It is known that \(q\)-orthogonal polynomials play an important role in the field of \(q\)-series and special functions. During 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 inserting 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 in the studying of Rogers--Ramanujan type identities. By fitting this Bailey pair into different weak forms of Bailey's lemma, we obtain a companion identity to Dyson's favorite one and also many other Rogers--Ramanujan type identities. Furthermore, as immediate consequences, we also obtain some results related to Appell--Lerch series and the generalized Hecke--type series.