Rogers-Ramanujan type identities and Chebyshev Polynomials of the third kind
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Publication:6386747
DOI10.1007/S11139-022-00627-8arXiv2112.13482MaRDI QIDQ6386747
Publication date: 26 December 2021
Abstract: It is known that -orthogonal polynomials play an important role in the field of -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 . 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.
Combinatorial identities, bijective combinatorics (05A19) Basic hypergeometric functions in one variable, ({}_rphi_s) (33D15)
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