A q-ANALOG OF A GENERAL RATIONAL SUM IDENTITY Toufik Mansour Department of Mathematics, University of Haifa, 31905 Haifa, Israel tmansour@univ.haifa.ac.il Mark Shattuck Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA shattuck@math.utk.edu Chunwei Song School of Mathematical Sciences, LMAM, Peking University, Beijing 100871, P. R. China csong@math.pku.edu.cn abstract In this paper we provide a q-analog for a type of identity involving rational sums shown by Prodinger (Appl. Anal. Discrete Math. 2 (1) (2008), 65-68). Our proof is algebraic and makes use of q-partial fractions and q-inverse pairs. A bijective proof involving a sign-changing involution is given for one of the main results. AMS Mathematical Subject Classifications: 05A19, 11B65. Keywords and Phrases: q-identity, partial fractions, q-inversion, combinatorial proof. 1. Introduction Partially motivated by the studies of [3] and [6], in this paper we consider some particular cases of the more general problem of developing a q-theory for identities involving rational sums. Our main tools are q-partial fractions and q-inverse pairs, which we find here to be convenient and powerful. In Section 2, q-analogs of results concerning the rational sum identities shown in [6] are given. As a corollary, we obtain an identity relating q-binomial coefficients and q-analogs of the harmonic numbers. In Section 3, we provide a bijective proof of the first main theorem, which entails defining a specific sign-changing involution and seems to be new in the case q = 1 as well. We remark that while in this paper we treat the identities appearing in [3] and [6], the methodology could be applied elsewhere. Here, we will use the standard notation [i] q := (1 − q i )/(1 − q)=1+ q + ··· + q i−1 , [i] q ! := [1] q [2] q ··· [i] q ,  i k  q := [i] q ! [k] q ![i − k] q ! , respectively, for the q-analog of a positive integer i, the q-factorial, and the q-binomial coefficient where 0 ≤ k ≤ i. Let [0] q = 0, [0] q ! = 1, and  i k  q = 0 if k>i ≥ 0 or k< 0. Below we will omit the base q and write [i], [i]!, etc., since the context is clear. Furthermore, empty sums take the value 0 and empty products the value 1, by convention. 1