  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.
