On a general q-identity

In this paper, by means of the q-Rice formula we obtain a general q-identity which is a unified generalization of three kinds of identities. Some known results are special cases of ours. Meanwhile, some identities on q-generalized harmonic numbers are also derived.


Introduction
Three kinds of identities will be introduced in this paper.
In the paper [21], Van Hamme gave the following identity One of the generalizations of (1.1) was given by Dilcher [6]: (1.2) Prodinger [16] gave another generalization of (1.1): where 0 m n.Many works have been devoted to the study of the generalizations of these identities.See for example [8,9,17,23].Recently, Guo and Zhang [12] made use of the Lagrange interpolation formula to give a generalization of Prodinger's identity (1.3).They also gave a generalization of Dilcher's identity (1.2).See Theorems 1.1 and 1.2 in [12], respectively.Ismail and Stanton used the theory of basic hypergeometric functions to generalize Dilcher's identity.See Theorem 2.2 in [13].
In the paper [5], Díaz-Barrero et al. obtained two identities involving rational sums: Recently, Prodinger [18] made use of partial fraction decomposition and inverse pairs to present a more general formula: where s k,j = k α=1 (x + α) −j .Almost at the same time, Chu and Yan [2] presented a generalization with multiple λ-fold sum: (1.5) A direct proof of (1.5) can be found in Chu [1].More recently, Mansour et al. [15] established a q-analog for the rational sum identity (1.4): where s k,j (q) = k α=1 q jα [x + α] −j q .In particular, they gave a very nice bijective proof for the case λ = 1.
In the recent paper [19], Prodinger established an interesting identity involving harmonic numbers: where and n are the generalized harmonic numbers defined by Mansour [14] obtained a general rational sum to generalize this identity.He also obtained a q-analog of this result involving q-harmonic numbers.Motivated by these interesting work, by means of the q-Rice formula used in [16,17], we will establish a general q-identity which is a common generalization of those three kinds of identities introduced before.
Theorem 1.1.Let λ be any positive integer.For 0 m n and 0 ) This is a very general q-series sum identity involving five parameters λ, l, m, x and z.It contains several known identities by choosing different parameters, which will be shown in the third section.By means of our identity, we will also obtain some identities on q-generalized harmonic numbers.
Throughout this paper, we will use the standard notation.For any real number x and any integer m, define For any nonnegative integer n, define 2 Proof of Theorem 1.1 In the very interesting paper [16], Prodinger introduced the following formula where C encircles the poles q −1 , q −2 ,. . ., q −n and no other.It is a q-analog of Rice's formula [7,20]: where C encircles the poles 1, 2,. . ., n and no other.Indeed, by Cauchy's integral formula one is not hard to find that for any integer m ∈ {0, 1, . . ., n} there holds where C encircles the poles q −j , j ∈ {0, 1, . . ., n} − {m} and no other.Prodinger first applied the q-analog of Rice's formula to prove many identities such as the identities of Van Hamme, Uchimura, Dilcher, Andrews-Crippa-Simon, and Fu-Lascoux, see [16,17] and references therein.It was shown that this formula is a very powerful and useful tool.Now, in this section we will use this important formula and present a proof of Theorem 1.1.
Remark 2.1.Actually, careful checking the proof of Theorem 1.1, one can find that Theorem 1.1 still holds for λ = 0 if in this case we assume the sum of the right hand side of (1.8) is equal to 1.This implies that for 0 l n − 1 there holds 3 Consequences of Theorem 1.1 Theorem 1.1 can help us to find some new identities or retrieve some well known identities.Let λ = 1 and x = 1.(1.8) reduces to the following identity.
Corollary 3.1.For 0 m n and 0 l n, there holds Guo and Zhang [12] made use of Lagrange interpolation formula to obtain this identity which generalizes the identity (1.3) due to Prodinger.It is obvious that (3.1) reduces to (1.3) when l = 0 and z → 0.
Let x = 1, l = λ − 1 and z = q −n in (1.8).We have Corollary 3.2.Let λ be any nonnegative integer.For 0 m n, there holds , where This identity is a q-analog of Prodinger's identity (1.7).An alternative form of this q-identity was presented in [14].
Corollary 3.3.Let λ be any nonnegative integer.There holds , the electronic journal of combinatorics 21(2) (2014), #P2.28 we apply Faà di Bruno's formula [4] to obtain Comparing (3.3) with (3.4), there holds Therefore, (3.2) can be rewritten as where By the theory of basic hypergeometric functions Ismail and Stanton [13] found Eq. (3.5) which reduces to the Dilcher identity [6] when x = 1.In fact, it has been recently pointed out in [11] that the Ismail-Stanton result (3.5) is the i = 1 (with m = λ + 1) case of following formula due to Zeng [23]: q; q) i−1 (q; q) n (q; q) i (zq; q) n h m−1 q i 1 − zq i , . . ., q n 1 − zq n , where 1 i n and h k (x 1 , . . ., x n ) is the kth homogeneous symmetric polynomial in x 1 , x 2 , . . ., x n defined by This more general formula can not follow from Theorem 1.1 and it can be viewed as a different generalization of the Ismail-Stanton result (3.5).Since Eq. (3.2) can be rewritten as the electronic journal of combinatorics 21(2) (2014), #P2.28 Using the q-inverse pair formula [10] we obtain the inverse of (3.2) Replacing x by q x , we rediscover an identity due to Mansour et al. [15]: Corollary 3.4.Let λ be any nonnegative integer.There holds This identity is a q-analog for the rational sum identity (1.4) due to Prodinger.If we further replace n by n + 1 and x by x − 1 in (3.6), then a q-analog of Chu-Yan's identity (1.5) is derived: Corollary 3.5.Let λ be any nonnegative integer.There holds Let the generalized q-harmonic numbers Recently, the q-generalized harmonic number sums have been useful in studying Feynman diagram contributions an relations among special functions [3].Taking x = 0 in (3.6), we have the following identities on q-generalized harmonic numbers: Corollary 3.6.For λ 1, there holds the electronic journal of combinatorics 21(2) (2014), #P2.28 The first few cases are listed as follows.