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Background

The central factorial numbers are described on OEIS sequence A008955. Among the references, "Ramanujan's notebooks, part 1" (edited by Bruce Berndt) is listed. Upon checking this source, I have not been able to find his work on the central factorial numbers.

Overall, I am curious as to why this source was included in the OEIS sequence.

Questions

  1. Where and how in his first notebook did Ramanujan employ the central factorial numbers? And what results did he derive?
  2. What results did Berndt derive by means of the central factorial numbers? How do they relate to Ramanujan's work?

Note: A week ago, I've asked a similar question on MSE.

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    $\begingroup$ The OEIS page you link to refers to pages 263 and 306. This should help. $\endgroup$ Commented Nov 21 at 14:16
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    $\begingroup$ I am unable to find anything relevant to central factorials in these two pages... $\endgroup$ Commented Nov 21 at 16:40

1 Answer 1

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There is no evidence that Ramanujan studied central factorials.

The reference to Berndt in the OEIS entry is there because, as shown by Peter Bala, the power series of $(x+\sqrt{1+x^2})^{-n}$ which Ramanujan derived in his first quarterly report (pages 263 and 306 of Berndt) can be used to obtain the generating function of the central factorials.

Analytic continuation to $n\in\mathbb{C}$ of Ramanujan's series gives the power series of $e^{a\arcsin x}$, since $\arcsin x=-i\log(ix+\sqrt{1-x^2})$. The even terms of this power series give the expansion $$\cosh\left(\frac{2}{\sqrt t}\arcsin[\tfrac{1}{2}z\sqrt{t}]\right)=1 + z^2/2! + (1 + t)z^4/4! + (1 + 5t + 4t^2)z^6/6! + (1 + 14 t + 49 t^2 + 36 t^3)z^8/8!+(1 + 30 t + 273 t^2 + 820 t^3 + 576 t^4)z^{10}/10!+\cdots.$$ This produces the OEIS list of central factorials, 1, 1, 1, 1, 5, 4, 1, 14, 49, 36, 1, 30, 273, 820, 576, ...

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