The Euler Series Transformation and the Binomial Identities of Ljunggren, Munarini and Simons

TL;DR

This paper demonstrates how Euler's series transformation explains several binomial identities, including those of Simons, Ljunggren, and Munarini, and explores their connections to Legendre polynomials.

Contribution

It reveals that Simons' identity can be derived from Euler's transformation and extends the approach to derive identities of Munarini, linking classical series transformations to binomial identities.

Findings

Simons' identity follows from Euler's series transformation

Ljunggren's identity is related to Simons' identity

Generalized Euler transformation yields recent binomial identities

Abstract

It is shown that the curious identity of Simons follows immediately from Euler's series transformation formula and also from an identity due to Ljunggren. The relation of Simons' identity to Legendre's polynomials is also discussed. At the end we use the generalized Euler series transformation to obtain two recent binomial identities of Munarini.