Some identities involving the partial sum of q-binomial coefficients

We give some identities involving sums of powers of the partial sum of q-binomial coefficients, which are q-analogues of Hirschhorn’s identities [Discrete Math. 159 (1996), 273–278] and Zhang’s identity [Discrete Math. 196 (1999), 291–298].


Introduction
In [2], Calkin proved the following curious identity: Hirschhorn [5] established the following two identities on sums of powers of binomial partial sums: and the electronic journal of combinatorics 21(3) (2014), #P3.17 In [7], Zhang proved the following alternating form of (2): if n is even and n = 0, −2 2n−1 − (−1) (n−1)/2 n−1 (n−1)/2 , if n is odd. ( Several generalizations are given in [6,8,9].Later, Guo et al. [4] gave the following q-identities: Here and in what follows, n k q is the q-binomial coefficient defined by ) is the q-shifted factorial for n 0. The purpose of this paper is to study q-analogues of (1)-( 2) and establish a new q-version of (3).Our main results may be stated as follows.
In Sections 2 and 3, we will give proofs of Theorems 1.1 and 1.2 respectively by using the q-binomial theorem and generating functions.

Proof of Theorem 1.1
To give our proof of Theorem 1.1, we need to establish a result, which is a q-analogue of Chang and Shan's identity (see [3]).Lemma 3.For any positive integer n, we have Proof.According to the q-binomial theorem (see [1]), we have for all complex numbers z and q with |z| < 1 and |q| < 1, there holds It follows that the electronic journal of combinatorics 21(3) (2014), #P3.17 and Therefore, for any non-negetive integer k with k n−1, the coefficient of and the coefficient of Using the fact equating the coefficients of z n−1 and after some simplifications, we obtain Lemma 2.1.
where in the last step, we have used (8).We next show (5).By (8), we have and taking m = −1 in (4), we obtain the electronic journal of combinatorics 21(3) (2014), #P3.17 Hence, by Lemma 2.1, we get 3 Proof of Theorem 1.2 In order to prove the Theorem 1.2, we need the following result, which gives a q-analogue of alternating sums of Chang and Shan's identity.
Lemma 4. For any non-negative integer n, we have Proof.By (8), we find that Therefore, for any non-negetive integer k with k n − 1, the coefficient of the electronic journal of combinatorics 21(3) (2014), #P3.17 the coefficient of and the coefficient of z n−1 in (z 2 ; q 2 ) n where [2|n] is defined by Using the fact equating the coefficients of z n−1 and after some simplifications, we obtain Lemma 3.1.