[seqfan] Re: The Ramanujan alpha, beta, and gamma series
Ron Knott
ron at ronknott.com
Sun May 8 23:00:13 CEST 2016
Have a look at the sequences A050787 - A050794 which cover Ramanujanâ€™s and more.
Ron Knott
> On 8 May 2016, at 09:00, Robert Munafo <mrob27 at gmail.com> wrote:
>
> In Ramanujan's "lost notebook", what I'm calling "page 82", but which is p.
> 341 in the 1988 book, is the page seen here:
>
> http://mrob.com/pub/math/images/ram-ln-p82.jpg
>
> There are three generating functions which can be expanded around x=0
> (Taylor series) to give three power series, with the sequences of
> coefficients that Ramanujan calls a_n, b_n, c_n.
>
> But they can also be expanded around x=Infinity (Laurent series) to give
> three power series with negative exponents of x, and with a different three
> sequences of coefficients, which he calls alpha_n, beta_n, gamma_n.
>
> All of them generate identities between three cubes, plus or minus one. The
> a, b, and c coefficients are A051028, A051029, A051030. The first
> nontrivial example represented by them is 135^3 + 138^3 = 172^3 - 1.
>
> The alpha, beta, and gamma coefficients begin with the famous 9^3 + 10^3 =
> 12^3 + 1. Amazingly, I could not find these coefficients in the OEIS or
> anywhere online, except for the first three tuples (9, 10, 12), (791, 812,
> 1010), and (65601, 67402, 83802) which are the examples that actually
> appear in Ramanujan's notebook.
>
> I added the three sequences as A272853, A272854, A272855, now pending
> review.
>
> I find it so hard to believe these haven't been written about before, that
> I'm asking the list to see if anyone knows of any references, links, etc.
>
> Also I'd appreciate it if someone could tell me if this is valid
> "Mathemetica" code:
>
> Series[(1+53*a+9*a^2)/(1-82*a-82*a^2+a^3), {a, Infinity, 10}]
>
> It works in Wolfram|Alpha but that's not the same as "Mathematica". Also,
> the recent advent of the "Wolfram Language" brand confuses me. If that
> really just another name for what we call "Mathematica program"? If so, i
> can move my %o fields to %t.
>
> --
> Robert Munafo -- mrob.com
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
More information about the SeqFan
mailing list