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FURTHER RESULTS ON VANISHING COEFFICIENTS IN INFINITE PRODUCT EXPANSIONS

Published online by Cambridge University Press:  11 November 2014

JAMES MC LAUGHLIN*
Affiliation:
Mathematics Department, 25 University Avenue, West Chester University, West Chester, PA 19383, USA email jmclaughl@wcupa.edu
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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.

Type
Research Article
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), 129139.CrossRefGoogle Scholar
Andrews, G. E. and Bressoud, D. M., ‘Vanishing coefficients in infinite product expansions’, J. Aust. Math. Soc. Ser. A 27(2) (1979), 199202.CrossRefGoogle Scholar
Richmond, B. and Szekeres, G., ‘The Taylor coefficients of certain infinite products’, Acta Sci. Math. (Szeged) 40(3–4) (1978), 347369.Google Scholar
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