ON SOME SUM–TO–PRODUCT IDENTITIES.pdf

ON SOME SUM–TO–PRODUCT IDENTITIES SHAUN COOPER AND MICHAEL HIRSCHHORN We give new proofs of some sum–to–product identities due to Blecksmith, Brillhart and Gerst, as well as some other such identities found recently by us. 1. INTRODUCTION AND STATEMENT OF RESULTS In the first two of a sequence of papers, Blecksmith, Brillhart and Gerst [2,3] give five pairs of simple and beautiful sum–to–product identities, Theorems 2–6 below. We give another such in Theorem 1. We do several new things — we prove Theorem 1 (§2), we show that Theorem 2 follows from Theorem 1 (§3), we present new proofs of Theorems 3 and 4 (§4) and we show that Theorems 5 and 6 follow from Theorems 3 and 4 (§5). In order to state our results, we must explain some by–now fairly standard notation. (a; q) 2 ∞ = (1 − a)(1 − aq)(1 − aq ) · · · , (a , a , · · · a ; q) = (a ; q) (a ; q) · · · (a ; q) , 1 2 n ∞ 1 ∞ 2 ∞ n ∞ a , a , · · · , a (a , a , · · · , a ; q) 1 2 m 1 2 m ∞ ; q = , b , b , · · · , b (b , b , · · · , b ; q) 1 2 n 1 2 n ∞ ∞ (q)∞ = (q; q)∞ . We make use of Jacobi’s triple product identity [1, (2.2.10)] ∞ (aq, a−1 n n (n2 +n)/2 , q; q)∞ = (−1) a q −∞ and the quintuple prod