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VANISHING COEFFICIENTS IN FOUR QUOTIENTS OF INFINITE PRODUCT EXPANSIONS

Part of: Discontinuous groups and automorphic forms Series expansions

Published online by Cambridge University Press:  20 March 2019

DAZHAO TANG
DAZHAO TANG*
Affiliation:
College of Mathematics and Statistics, Chongqing University, Huxi Campus LD204, Chongqing 401331, PR China email dazhaotang@sina.com
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Abstract

Motivated by Ramanujan’s continued fraction and the work of Richmond and Szekeres [‘The Taylor coefficients of certain infinite products’, Acta Sci. Math. (Szeged)40(3–4) (1978), 347–369], we investigate vanishing coefficients along arithmetic progressions in four quotients of infinite product expansions and obtain similar results. For example, $a_{1}(5n+4)=0$, where $a_{1}(n)$ is defined by

$$\begin{eqnarray}\displaystyle {\displaystyle \frac{(q,q^{4};q^{5})_{\infty }^{3}}{(q^{2},q^{3};q^{5})_{\infty }^{2}}}=\mathop{\sum }_{n=0}^{\infty }a_{1}(n)q^{n}. & & \displaystyle \nonumber\end{eqnarray}$$

Keywords

vanishing coefficientsquotients of infinite productJacobi’s triple product identityquintuple product identity

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms
Secondary: 30B10: Power series (including lacunary series)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 2 , October 2019 , pp. 216 - 224
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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Footnotes

This work was supported by the National Natural Science Foundation of China (No. 11501061) and the Fundamental Research Funds for the Central Universities (No. 2018CDXYST0024).

References

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