In Praise of an Elementary Identity of Euler

TL;DR

This paper highlights Euler's elementary identity, reformulates it as Euler's Telescoping Lemma, and uses it to provide alternative, computer-free proofs of key hypergeometric summation theorems and identities, including some potentially new ones.

Contribution

The paper introduces Euler's Telescoping Lemma and demonstrates its broad applicability to hypergeometric series and related identities, offering new proofs without computer assistance.

Findings

Provided alternative proofs for key hypergeometric summation theorems

Derived identities for q-analogs of Fibonacci and Pell numbers

Identified some potentially new combinatorial identities

Abstract

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu--Vandermonde sum, the Pfaff--Saalch\" utz sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8\phi_7$ sum. Our proofs are conceptually the same as those obtained by the WZ method, but done without using a computer. We survey identities for Generalized Hypergeometric Series given by Macdonald, and prove several identities for $q$-analogs of Fibonacci numbers and polynomials and Pell numbers that have appeared in combinatorial contexts. Some of these identities appear to be new.