A remarkable identity found in Ramanujan's third notebook

Berndt, B. C. and Bhargava, S. (1992) A remarkable identity found in Ramanujan's third notebook. Glasgow Mathematical Journal, 34 (3). pp. 341-345.

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Abstract

For each positive even integer n, let Fn(a,b,c,d)=(a+b+c)n+(b+c+d)n−(c+d+a)n−(d+a+b)n+(a−d)n−(b−c)n. The remarkable identity referred to in the title is the following: If a,b,c and d are numbers satisfying ad=bc then 64F6(a,b,c,d)F10(a,b,c,d)=45F28(a,b,c,d).(1) Ramanujan, as was his custom, gave no proof. The authors first verified that (1) is true with the aid of the symbolic algebra system Mathematica. However, as they point out, such a proof is less than satisfactory, since it gives no information as to how (1) might have been discovered. In the paper they give an elementary proof of (1) which gives some insight as to why it exists.

Item Type:	Article
Subjects:	E Mathematical Science > Mathematics
Divisions:	Department of > Mathematics
Depositing User:	Users 23 not found.
Date Deposited:	15 May 2021 06:34
Last Modified:	05 Jul 2022 05:47
URI:	http://eprints.uni-mysore.ac.in/id/eprint/16425

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