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rogers–ramanujan continued fraction

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THE ROGERS–RAMANUJAN CONTINUED FRACTION AND RELATED ETA-QUOTIENT REPRESENTATIONS
Part of
- Additive number theory; partitions
- Elementary number theory
SHANE CHERN, DAZHAO TANG
Journal:

Bulletin of the Australian Mathematical Society / Volume 103 / Issue 2 / April 2021

Published online by Cambridge University Press:

17 September 2020, pp. 248-259

Print publication:

April 2021
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We construct eta-quotient representations of two families of q-series involving the Rogers–Ramanujan continued fraction by establishing related recurrence relations. We also display how these eta-quotient representations can be utilised to dissect certain q-series identities.

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}

$\begin{eqnarray}h=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{12j-1})(1-q^{12j-11})}{(1-q^{12j-5})(1-q^{12j-7})}\end{eqnarray}$

and show that it has many properties that are analogues of corresponding results for Ramanujan’s function

k=k(q) $k=k(q)$ 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}

$\begin{eqnarray}k=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{10j-1})(1-q^{10j-2})(1-q^{10j-8})(1-q^{10j-9})}{(1-q^{10j-3})(1-q^{10j-4})(1-q^{10j-6})(1-q^{10j-7})}.\end{eqnarray}$

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THE ROGERS–RAMANUJAN CONTINUED FRACTION AND RELATED ETA-QUOTIENT REPRESENTATIONS

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

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THE ROGERS–RAMANUJAN CONTINUED FRACTION AND RELATED ETA-QUOTIENT REPRESENTATIONS

THE LEVEL 12 ANALOGUE OF RAMANUJAN’S FUNCTION kk

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