An Identity Motivated by an Amazing Identity of Ramanujan

TL;DR

This paper generalizes Ramanujan's remarkable identity involving three sequences and a cubic relation to a broader family of identities involving eleven sequences and powers from 1 to 5.

Contribution

It introduces a new, more general identity that extends Ramanujan's original cubic relation to higher powers and more sequences, with a formal proof.

Findings

Generalized Ramanujan identity for powers 1 to 5

Involves eleven sequences with specific relations

Extends classical identities to broader algebraic structures

Abstract

Ramanujan stated an identity to the effect that if three sequences $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ are defined by $r_1(x)=:\sum_{n=0}^{\infty}a_nx^n$, $r_2(x)=:\sum_{n=0}^{\infty}b_nx^n$ and $r_3(x)=:\sum_{n=0}^{\infty}c_nx^n$ (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 $j$, $1\leq j \leq 5$.