On the expansion of Ramanujan's continued fraction
Ramanujan's continued fraction \(R(q)= 1+ \frac{q}{1+ \frac{q^2}{1+ \frac{q^3}{1+\dots}}}\) and its reciprocal \(R(q)^{-1}= \frac{1}{1+ \frac{q}{1+ \frac{q^2}{1+ \frac{q^3}{1+\dots}}}}\) can be expanded as series \[ \begin{aligned} R(q)&= 1+q-q^3+ q^5+\dots= \sum_{n=0}^\infty c(n)q^n \\ \text{and} R(q)^{-1}&= 1-q+q^2-q^4+\dots= \sum_{n=0}^\infty d(n) q^n.\end{aligned} \] M. Hirschhorn and G. Szekeres noticed that the sign of \(c(n)\) and \(d(n)\) for large \(n\) is periodic with period 5. \textit{B. Richmond} and \textit{G. Szekeres} [Acta Sci. Math. 40, 347-369 (1978; Zbl 0397.10046)] found an explicit expression in asymptotic form for \(c(n)\) and \(d(n)\). \textit{G. Andrews} [Adv. Math. 41, 186-208 (1981; Zbl 0477.33009)] proved a theorem with a combinatorial interpretation of \(c(n)\) and \(d(n)\), from which the periodicity follows easily. The author of the present paper proves the periodicity of \(c(n)\) and \(d(n)\) by finding another formulae for \(R(q)\) and \(R(q)^{-1}\), and he gives combinatorial interpretations of \(c(n)\) and \(d(n)\).
scientific article; zbMATH DE number 1251336
scientific article