Michael J. Schlosser and Nian Hong Zhou

On the infinite Borwein product raised to a positive real power

(20 pages)

Abstract.

In this paper, we study properties of the coefficients appearing in the q-series expansion of  n ≥1 [(1 − qn)  ∕ (1 − qpn)] δ, the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power δ. We use the Hardy-Ramanujan-Rademacher circle method to give an asymptotic formula for the coefficients. For  p = 3 we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent δ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product.


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