Asymptotic Formulas for Two Continued Fractions in Ramanujan's Lost Notebook

Abstract

Abstract. On page 45 of his lost notebook, Ramanujan recorded two asymptotic formulas for two continued fractions involving the Riemann zeta function and Dirichlet L-functions. In this paper, we prove a more general theorem and derive Ramanujan's claims as a corollary of our theorem

On Ramanujan’s continued fraction for (𝑞²;𝑞³)_{∞}/(𝑞;𝑞³)_{∞}

“…This asymptotic formula and a similar asymptotic formula for another continued fraction found on the same page were established by Berndt and Sohn [9], who deduced the results from a general theorem that they established. The second major purpose of this paper is to prove another asymptotic formula related to that for (1.1) found on the same page.…”

“…Note that, when λ = 0, the continued fraction in (1.1) is equal to 1/(1 − u 0 ), with q = e −x . In this case, the asymptotic formula (4.32) is identical to the aforementioned asymptotic formula also found on page 45 of [12] and proved by Berndt and Sohn in [9]. However, if λ > 0, the method of proof used in [9] does not generalize, and so in this particular situation we cannot verify Ramanujan's claim.…”

“…In this case, the asymptotic formula (4.32) is identical to the aforementioned asymptotic formula also found on page 45 of [12] and proved by Berndt and Sohn in [9]. However, if λ > 0, the method of proof used in [9] does not generalize, and so in this particular situation we cannot verify Ramanujan's claim. Note that if we set q = e −x and a = e −λx in Corollary 3.3, the continued fraction there is equivalent to the continued fraction e −λx u λ , where u λ is defined by (4.3).…”

“…In [9], Berndt and Sohn established an asymptotic expansion, as q → 1 − , for the continued fraction (1.1); this asymptotic series is found on page 45 in Ramanujan's lost notebook [12]. Elsewhere on page 45, Ramanujan gives an asymptotic expansion for a continued fraction which generalizes that of (1.1), but as we remarked in the Introduction, Ramanujan evidently derived his result from a recurrence relation, (4.2) below, satisfied by the continued fraction.…”

Ramanujan’s Lost Notebook

“…The three continued fractions are given by (7.1.1), (7.1.2), and (7.1.4) below. Our proofs are taken from papers by Berndt and J. Sohn [83] and Berndt and A.J. Yee [84].…”