Dyson’s “Favorite” Identity and Chebyshev Polynomials of the Third and Fourth Kind

The combinatorial and analytic properties of Dyson’s “favorite” identity are studied in detail. In particular, a q -series analog of the anti-telescoping method is used to provide a new proof of Dyson’s results with mock theta functions popping up in intermediate steps. This leads to the appearance of Chebyshev polynomials of the third and fourth kind in Bailey pairs related to Bailey’s Lemma. The natural relationship with L.J. Rogers’s pioneering work is also presented.