Journal of Integer Sequences, Vol. 19 (2016), Article 16.6.2

Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity

Alexander Kreinin
Risk Analytics
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Toronto, ON M5T 2C6


We consider integer sequences connected to the famous Laplace continued fraction for the function $R(t)=\int_t^\infty\varphi(x) \mathrm{d}x/\varphi(t)$, where $\varphi(t) = e^{-t^2/2}/\sqrt{2\pi}$ is the standard normal density. We compute the generating functions for these sequences and study their relation to the Hermite and Bessel polynomials. Using the master equation for the generating functions, we find a new proof of the Ramanujan identity.

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(Concerned with sequences A000012 A000165 A000217 A000457 A001147 A001879 A035101 A050534 A129890 A180048 A263384.)

Received November 11 2015; revised versions received April 5 2016; June 24 2016; June 27 2016. Published in Journal of Integer Sequences, June 27 2016.

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