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THE LEVEL 12 ANALOGUE OF RAMANUJAN’S FUNCTION $k$

Part of: Discontinuous groups and automorphic forms Combinatorics Other special functions

Published online by Cambridge University Press:  22 January 2016

SHAUN COOPER  and
DONGXI YE
SHAUN COOPER*
Affiliation:
Institute of Natural and Mathematical Sciences, Massey University-Albany, Private Bag 102904, North Shore Mail Centre, Auckland, New Zealand email s.cooper@massey.ac.nz
DONGXI YE
Affiliation:
Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706, USA email lawrencefrommath@gmail.com
*
✉s.cooper@massey.ac.nz
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Abstract

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We provide a comprehensive study of the function $h=h(q)$ defined by

$$\begin{eqnarray}h=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{12j-1})(1-q^{12j-11})}{(1-q^{12j-5})(1-q^{12j-7})}\end{eqnarray}$$
and show that it has many properties that are analogues of corresponding results for Ramanujan’s function $k=k(q)$ defined by
$$\begin{eqnarray}k=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{10j-1})(1-q^{10j-2})(1-q^{10j-8})(1-q^{10j-9})}{(1-q^{10j-3})(1-q^{10j-4})(1-q^{10j-6})(1-q^{10j-7})}.\end{eqnarray}$$

Keywords

Dedekind eta functionDomb numbersEisenstein serieselliptic integralhypergeometric functionJacobi triple product identitymodular equationquintuple product identityRamanujan’s cubic continued fractionRogers–Ramanujan continued fraction

MSC classification

Primary: 11F11: Holomorphic modular forms of integral weight
Secondary: 05A10: Factorials, binomial coefficients, combinatorial functions 33C75: Elliptic integrals as hypergeometric functions 33E05: Elliptic functions and integrals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 101 , Issue 1 , August 2016 , pp. 29 - 53
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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