[seqfan] Re: The Ramanujan alpha, beta, and gamma series

Robert Munafo mrob27 at gmail.com
Tue May 17 19:01:37 CEST 2016


I agree - the confusion is all mine, from not knowing how to "reverse" the
GF and not knowing how to put a Laurent series formulation in a %F field.

The formula %F lines should have GF's that actually work, like you suggest.
I'd like to keep the extra x in the numerator so that the series starts
with the 9 term (since the 1 term, i.e. coefficient of x^0, is already in
A051028). Ramanujan's 9 is the coefficient of x^-1 so ours should the
coefficient of x^1.

Ramanujan's notebook (see mrob.com/pub/math/images/ram-ln-p82.jpg clearly
defines the alpha/beta/gamma sequences as Laurent series on the GF's that I
gave, i.e. the same GF's as the a/b/c series respectively. That is noted in
the comment %C, and is also represented in the existing programs that use
Series[] with "Infinity" as the convergence limit.

So I'll edit those (A272853, A272854, A272855).

On Tue, May 17, 2016 at 9:41 AM, Ron Knott <ron at ronknott.com> wrote:
> The “a” series A051028 has GF
>
> (1 + 53 x + 9 x^2)/(1 - 82 x - 82 x^2 + x^3) = 1 + 135 x + 11161 x^2 +
926271 x^3 + 76869289 x^4+…
>
> and replacing x by 1/x gives the “alpha” series
>
>  (x (9 + 53 x + x^2))/(1 - 82 x - 82 x^2 + x^3).
>
> The x factor is irrelevant as a GF.  Note that we have just reversed the
coefficients in the numerator and this is the GF of
>
>   (9 + 53 x + x^2)/(1 - 82 x - 82 x^2 + x^3) = 9 + 791 x + 65601 x^2 +
5444135 x^3 + 451797561 x^4+… i.e. A272853
>
> So perhaps the GFs should be changed to this form to be consistent with
the rest of OEIS and the second form used in A272853?
>
> Similarly for the betas: [...]
>
> Perhaps these changes might clear up some confusion between these 6
series and make things more consistent with the rest of OEIS?

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