Asymptotics for the Taylor coefficients of certain infinite products

Abstract

Let (m 1 , . . . , m J ) and (r 1 , . . . , r J ) be two sequences of J positive integers satisfying 1 ≤ r j < m j for all j = 1, . . . , J. Let (δ 1 , . . . , δ J ) be a sequence of J nonzero integers. In this paper, we study the asymptotic behavior of the Taylor coefficients of the infinite product J j=1 k≥11 − q r j +m j (k−1) 1 − q −r j +m j k δ j .

Vanishing coefficients in several $q$-series expansions related to the Rogers--Ramanujan continued fraction

“…However, the sign patterns (6.7) and (6.8) can hardly be confirmed completely by the above dissections. On the other hand, making use of an asymptotic formula provided by the first author in [6], we are able to show the validity of (6.7) and (6.8) for sufficiently large n.…”