Rogers-Ramanujan type identities and Chebyshev polynomials

Yao, Olivia

doi:10.1090/proc/17107

Rogers-Ramanujan type identities and Chebyshev polynomials
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by Olivia X. M. Yao;

Proc. Amer. Math. Soc. 153 (2025), 1215-1229

DOI: https://doi.org/10.1090/proc/17107

Published electronically: February 3, 2025

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Abstract:

Recently, Andrews [Ann. Comb. 23 (2019), pp. 443–464] discovered a family of Rogers-Ramanujan type identities by introducing Chebyshev polynomials of the third and the fourth kinds into Bailey pairs. Motivated by Andrews’ work, Sun [Ramanujan J. 60 (2023), pp. 761–794] obtained a companion identity for Dyson’s favorite identity and a number of Rogers-Ramanujan type identities based on a new Bailey pair involving Chebyshev polynomials of the third kind. In this paper, we establish many new identities involving Chebyshev polynomials of the second kind by constructing several new Bailey pairs and inserting them into various weak forms of Bailey’s lemma. Some special cases of those identities yield many new and known Rogers-Ramanujan type identities. In particular, we derive several identities which are analogous to Dyson’s favorite identity. It is interesting that the right hand sides of some identities are sums of modular forms with different weights.

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