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VANISHING COEFFICIENTS IN FOUR QUOTIENTS OF INFINITE PRODUCT EXPANSIONS

Published online by Cambridge University Press:  20 March 2019

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

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

Type
Research Article
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

Alladi, K. and Gordon, B., ‘Vanishing coefficients in the expansion of products of Rogers–Ramanujan type’, in: Proc. Rademacher Centenary Conf., Contemporary Mathematics, 166 (eds. Andrews, G. E. and Bressoud, D.) (American Mathematical Society, Providence, RI, 1994), 129139.Google Scholar
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Tang, D., ‘Vanishing coefficients in some q-series expansions’, Int. J. Number Theory, to appear, doi:10.1142/S1793042119500398.Google Scholar

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VANISHING COEFFICIENTS IN FOUR QUOTIENTS OF INFINITE PRODUCT EXPANSIONS
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