In praise of an elementary identity of Euler (Q547803)

Let \(u_k, v_k \) and \(w_k \) be three sequences, such that \(u_k - v_k = w_k. \) Then we have: \[ \sum_{k=0}^{n} \frac{w_k}{w_0} \frac{u_0 u_1 \cdots u_{k-1}}{v_1 v_2 \cdots v_k} = \frac{u_0}{w_0} \left( \frac{u_1 u_2 \cdots u_n}{v_1 v_2 \cdots v_n} - \frac{v_0}{u_0} \right), \] provided none of the denominators in the expression above are zero. This theorem is called Euler's Telescoping Lemma. In this survey paper, the author used Euler's Telescoping Lemma to give alternate proofs of almost all the key summation theorems for terminating hypergeometric series and basic hypergeometric series, including the terminating binomial theorem, the Chu-Vandermonde sum, the Pfaff-Saalschütz sum, and their analogues. He also gave a proof of Jackson's \(q\)-analogue of Dougall's sum, the sum of a terminating, balanced, very-well-poised \(_8 \phi_7 \) sum. The WZ method can be used to prove most of the identities provided in this article but the author provided proofs with no help from a computer. From the connection between Euler's Telescoping Lemma and three term recurrence relations, the author also proved several identities for \(q\)-analogues of Fibonacci numbers and polynomials and Pell numbers that have appeared in combinatorial contexts. Some results seem to be new.