A q -analogue for partial-fraction decomposition of a rational function and its application

In this paper, by using the residue method of complex analysis, we obtain a q -analogue for partial-fraction decomposition of the rational function x M ( x +1) λ n . As applications, we deduce the corresponding q -algebraic and q -combinatorial identities which are the q -extensions of Chu’ results


Introduction and main result
Throughout this paper, we always make use of the following notation: N = {1, 2, 3, . ..} denotes the set of natural numbers, N 0 = {0, 1, 2, 3, . ..} denotes the set of nonnegative integers, C denotes the set of complex numbers.
The q-shifted factorial of q-numbers defined by [a] q n := Clearly, lim q→1 [a] q n = (a) n , i.e., q-numbers shifted factorial ([a] q ) n is an q-analogue of the shifted factorial (a) n .
The q-shifted factorial (a; q) n is not a q-analogue of the shifted factorial (a) n .Let q → q a and then divide (1q) n .Therefore (a; q) n becomes ([a] q ) n = (q a ;q) n (1-q) n which is a q-analogue of the shifted factorial (a) n .
The q-binomial coefficient is defined by n k q = (q; q) n (q; q) n-k (q; q) k , which satisfies the following relationships: The above q-standard notation can be found in [1] and [11].
Chu [3] established the partial fraction decompositions of two rational functions 1 based on the induction principle and famous Faà di Bruno formula and obtained several striking harmonic number identities from two partial fraction decompositions, therefore resolved completely the open problem of Driver et al. [10].It is not difficult, by using (2), to reformulate two main results of Chu as follows: Theorem 1 ([3, Theorem 2]) Suppose that λ and n are positive integers, z ∈ C \ {0, -1, . . ., -n}.Then the following partial fraction decomposition holds: where Theorem 2 ([3, Theorem 5]) Suppose that n, M, and λ are three natural numbers with λ ≤ M < λ(n + 1), z ∈ C \ {0, -1, . . ., -n}.Then the following partial fraction decomposition holds: where The partial fraction decomposition plays an important role in the study of the combinatorial identities and related questions (for example, see [2-8, 12, 15, 17-23] and the references therein).But if a rational fraction is improper, the method of the partial fraction decomposition is invalid.So how do we decompose an improper rational fraction into partial fractions?Zhu and Luo answered this question using the contour integral and Cauchy's residue theorem and gave an explicit decomposition for the general rational function x M (x+1) λ n .We rewrite the main result of Zhu and Luo as follows: Theorem 3 ([24, Theorem 1]) Suppose that M is any nonnegative integer, λ and n are any positive integers such that N = λn, and z is a complex number such that z ∈ C \ {-1, -2, . . ., -n}.Then the following partial fraction decomposition holds: where When M -N ≥ 0 in Theorem 3, i.e., the rational function x M (x+1) λ n is improper, putting . ., y j ), ( 6) becomes the following explicit form: which is an explicit result of the polynomial x M divided by the polynomial (x + 1) λ n .Therefore we say that Theorem 3 provides a new idea and method for the division of two general polynomials.
When M -N < 0, i.e., the degree of the numerator polynomial M is smaller than the degree of the denominator polynomial N = λn, we deduce Chu's results (4) and ( 5).Therefore we say that Theorem 3 is an interesting extension of Chu's results.
In the present paper, we will provide a q-analogue of Theorem 3 using the contour integral and Cauchy's residue theorem.As some applications, we obtain the corresponding q-algebraic and q-combinatorial identities.
We state our main result in the following theorem.
Theorem 4 If z is a complex variable, M is a nonnegative integer, n and λ are two positive integers such that N = nλ, then the following q-algebraic identity holds: where Remark 5 Formula ( 7) is a q-analogue of formula (6).

q-Algebraic identities
In this section, we will provide the q-analogues of some algebraic identities.When M < N , we obtain the following q-algebraic identity: Corollary 6 If z is a complex variable, M is a nonnegative integer, n and λ are two positive integers such that N = nλ, then the following q-algebraic identity holds: where When M ≥ N , we give the following new and interesting q-algebraic identities: for M = N , we have Corollary 7 Suppose that z is a complex variable, n and λ are two positive integers.Then the following q-algebraic identity holds: where For M = N + 1, we have Corollary 8 Suppose that z is a complex variable, n and λ are two positive integers.Then the following q-algebraic identity holds: where For M = N + 2, we have Corollary 9 Suppose that z is a complex variable, n and λ are two positive integers.Then the following q-algebraic identity holds: where Taking M = 0 in (7), we deduce Corollary 10 Suppose that z is a complex variable, n and λ are two positive integers.Then the following q-algebraic identity holds: where Theorem 11 If z is a complex variable, M is a nonnegative integer, n and λ are two positive integers such that N = nλ.Then the following q-algebraic identity holds: where Proof Letting z → q z -1 in Theorem 4, we obtain Theorem 11 immediately.
Theorem 13 If z is a complex variable, M is a nonnegative integer, n and λ are two positive integers such that N = λ(n + 1), then the following q-algebraic identity holds: where Proof Letting z → q z -1 and M → Mλ, noting that n k=1 = n k=0 in Theorem 4, and applying the relation (q z -1) M (q z+1 ; q) λ n ≡ (-1) M (1q z ) M+λ (q z ; q) λ n+1 , we obtain Theorem 13 immediately.
When M < N , and noting that n k=1 = n k=0 in Theorem 13, we have Corollary 14 If z is a complex variable, n and M are nonnegative integers, λ is a positive integer such that N = λ(n + 1), then the following q-algebraic identity holds: where Remark 15 The algebraic identity (25) is just a q-analogue of Chu's result (5).
Taking M = 0 in Corollary 14, we have Corollary 16 If z is a complex variable, n and λ are two positive integers, then the following q-algebraic identity holds: where Remark 17 The algebraic identity ( 27) is just a q-analogue of Chu's result (4).
Corollary 18 If z is a complex variable, n and λ are two positive integers such that N = λ(n + 1), then the following q-algebraic identity holds: where Remark 19 The algebraic identity (30) is just another q-analogue of Chu's result (4).
Taking λ = 1 and letting M → m in Theorem 4, we have Corollary 20 If z is a complex variable, m is a nonnegative integer, and n is a positive integer, then the following q-algebraic identity holds: where Taking m = 0 in (32), we obtain the following q-algebraic identity: (q; q) n ((z + 1)q; q) n = n k=1 3 Further q-combinatorial identities In this section, we will give the q-analogues of some combinatorial identities of Chu.
Taking z = 0 in (7), we obtain the following q-combinatorial identities involving qharmonic numbers.
Corollary 21 If M is a nonnegative integers, n and λ are two positive integers such that N = nλ, then we have the following q-combinatorial identities: where Taking z = 1 in ( 7), we obtain the following q-combinatorial identities involving qharmonic numbers: Taking z = -1 in (7), we obtain the following q-combinatorial identities involving qharmonic numbers: Taking z = 1 in (22), we obtain the following q-combinatorial identities involving qharmonic numbers: Multiplying both sides of (10) by z and then letting z → ∞, we immediately establish the following combinatorial identities: Corollary 22 If M is a nonnegative integer, n and λ are two positive integers such that N = nλ, then we have the following q-combinatorial identities: where Noting that n k=1 = n k=0 in (41), we say that the above q-combinatorial identities are the corresponding q-analogues of Chu's result [3,Corollary 7]: where Taking λ = 1, 2, 3, 4 in (41), respectively, we obtain the following q-combinatorial identities involving q-harmonic numbers: (-1) n (q;q) n q ( n+1 2 ) Taking z = 0 in (30), we obtain the following q-combinatorial identity: q -(k+1)j B j (y 1 , . . ., y j ) j! = (q; q) λ n , where y i is given by (31).
Taking z = -1 in (30), we have Taking z = q in (30), we have Taking z = -q in (30), we have Multiplying both sides of (30) by z and then letting z → ∞, we establish the following q-combinatorial identities: Corollary 23 Suppose that λ and n are positive integers.We have the following qcombinatorial identity: where y i are given by (31).

Proof of Theorem 4
Lemma 25 If z is a complex variable, M is a nonnegative integer, n and λ are two positive integers such that N = nλ, then the following q-algebraic identity holds: where Proof We first construct two polynomials P(z) and Q(z) of degree M and N + 1, respectively, which are given by such that α = q -1 -1, q -2 -1, . . ., q -n -1.
We next construct three contour integrals for the rational functions P(z)/Q(z): , where is a simple closed contour which only surrounds the single pole α of P(z)/Q(z); , where is a simple closed contour which only surrounds the poles q -1 -1, q -2 -1, . . ., q -n -1 of P(z)/Q(z); where is a simple closed contour which only surrounds the pole ∞ of P(z)/Q(z).In the extended complex plane, since the total sum of residues of a rational function at all finite poles and that at infinity is equal to zero [13, Theorem 2], we have or equivalently, Below we compute the contour integrals P(z) Q(z) dz, P(z) Q(z) dz, and Q(z) dz, respectively.
If M -N > 0, then t = 0 is a single pole of order M -N .By utilizing Cauchy's residue theorem, noting that the power series expansion of the logarithmic function is and using the definition of complete Bell polynomials, we obtain j=1 (1t(q -j -1)) λ = -2πi t M-N 1 (1tα) n j=1 (1t(q -j -1)) λ out an important demonstrated observation that any (p, q)-variations of the proposed qresults would be trivially inconsequential because the additional parameter p is obviously redundant.
In the last section (see [24,Conclusions]), Zhu and Luo suggested an open problem which would yield the corresponding basic (or q-) extensions of Theorem 3 (see [24, Theorem 1]).In the present paper, we here have answered this question applying the contour integral and Cauchy's residue theorem and given a q-explicit analogue by decomposing the general rational function x M (x+1) λ n .