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

Robert Munafo mrob27 at gmail.com
Sun May 8 10:00:22 CEST 2016


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


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