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FURTHER RESULTS ON VANISHING COEFFICIENTS IN INFINITE PRODUCT EXPANSIONS

Part of: Basic hypergeometric functions Additive number theory; partitions

Published online by Cambridge University Press:  11 November 2014

JAMES MC LAUGHLIN
JAMES MC LAUGHLIN*
Affiliation:
Mathematics Department, 25 University Avenue, West Chester University, West Chester, PA 19383, USA email [email protected]
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Abstract

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We extend results of Andrews and Bressoud [‘Vanishing coefficients in infinite product expansions’, J. Aust. Math. Soc. Ser. A27(2) (1979), 199–202] on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if

$$\begin{eqnarray}\frac{(q^{r-tk},q^{mk-(r-tk)};q^{mk})_{\infty }}{(\pm q^{r},\pm q^{mk-r};q^{mk})_{\infty }}=:\mathop{\sum }_{n=0}^{\infty }c_{n}q^{n}\end{eqnarray}$$
for certain integers $k$, $m$, $s$ and $t$, where $r=sm+t$, then $c_{kn-rs}$ is always zero. Our theorems also partly give a simpler reformulation of results of Alladi and Gordon [‘Vanishing coefficients in the expansion of products of Rogers–Ramanujan type’, in: The Rademacher Legacy to Mathematics (University Park, PA, 1992), Contemporary Mathematics, 166 (American Mathematical Society, Providence, RI, 1994), 129–139], but also give results for cases not covered by the theorems of Alladi and Gordon. We also give some interpretations of the analytic results in terms of integer partitions.

Keywords

q-seriesinfinite productsinfinite q-productsvanishing coefficients

MSC classification

Primary: 33D15: Basic hypergeometric functions in one variable, $_rphi_s$
Secondary: 11P82: Analytic theory of partitions 11P84: Partition identities; identities of Rogers-Ramanujan type
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 98 , Issue 1 , February 2015 , pp. 69 - 77
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Alladi, K. and Gordon, B., ‘Vanishing coefficients in the expansion of products of Rogers–Ramanujan type’, in: The Rademacher Legacy to Mathematics (University Park, PA, 1992), Contemporary Mathematics, 166 (American Mathematical Society, Providence, RI, 1994), 129–139.CrossRefGoogle Scholar
Andrews, G. E. and Bressoud, D. M., ‘Vanishing coefficients in infinite product expansions’, J. Aust. Math. Soc. Ser. A 27(2) (1979), 199–202.Google Scholar
Richmond, B. and Szekeres, G., ‘The Taylor coefficients of certain infinite products’, Acta Sci. Math. (Szeged) 40(3–4) (1978), 347–369.Google Scholar