An Identity Motivated by an Amazing Identity of Ramanujan

Abstract

Ramanujan stated an identity to the effect that if three sequences $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ are defined by $r_1(x)=:{\sum }_{n=0}^{\infty }{a}_n{x}^n$, $r_2(x)=:{\sum }_{n=0}^{\infty }{b}_n{x}^n$ and $r_3(x)=:{\sum }_{n=0}^{\infty }{c}_n{x}^n$ (here each $r_i(x)$ is a certain rational function in $x$), then \[ a_n^3+b_n^3-c_n^3=(-1)^n, \hspace{25pt} \forall \,n \geq 0. \] Motivated by this amazing identity, we state and prove a more general identity involving eleven sequences, the new identity being "more general" in the sense that equality holds not just for the power 3 (as in Ramanujan's identity), but for each power $j$, $1\le j\le 5$.

5 pages