Abstract
In this paper, by using the residue method of complex analysis, we obtain a q-analogue for partial-fraction decomposition of the rational function \(\frac{x^{M}}{(x+1)^{\lambda}_{n}}\). As applications, we deduce the corresponding q-algebraic and q-combinatorial identities which are the q-extensions of Chu’ results.
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1 Introduction and main result
Throughout this paper, we always make use of the following notation: \(\mathbb{N}=\{1,2,3, \ldots \}\) denotes the set of natural numbers, \(\mathbb{N}_{0}=\{0,1,2,3, \ldots \}\) denotes the set of nonnegative integers, \(\mathbb{C}\) denotes the set of complex numbers.
For \(a \in \mathbb{C}\), the shifted factorial are defined by
The q-shifted factorial are defined by
The q-numbers and q-numbers factorial are defined by
respectively. Clearly,
The q-shifted factorial of q-numbers defined by
Clearly,
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 \(q \mapsto q^{a}\) and then divide \((1-q)^{n}\). Therefore \((a;q)_{n}\) becomes \(([a]_{q})_{n} = \frac {(q^{a};q)_{n}}{(1-q)^{n}}\) which is a q-analogue of the shifted factorial \((a)_{n}\).
The q-binomial coefficient is defined by
which satisfies the following relationships:
The above q-standard notation can be found in [1] and [11].
The generalized harmonic numbers are defined by
when \(r=1\), they reduce to the classical harmonic numbers as \(H_{n}=H_{n}^{(1)}\).
Two q-generalized harmonic numbers are respectively defined by
and
when \(r=1\), they reduce to the q-harmonic numbers \(H_{n;q}=H_{n;q}^{(1)}\) and \(\tilde{H}_{n;q}=\tilde{H}_{n;q}^{(1)}\), respectively.
The complete Bell polynomials \(\mathbf{B}_{n}(x_{1},x_{2}, \dots, x_{n})\) are defined by (see [9, p. 133–137] or [16, p. 173–174])
with an exact expression being
where \(\pi (n)\) denotes a partition of n, usually denoted \(1^{k_{1}}2^{k_{2}} \cdots n^{k_{n}}\), with \(k_{1}+2k_{2}+\cdots +nk_{n}=n\). The complete Bell polynomials \(\mathbf{B}_{n}(x_{1},x_{2}, \dots, x_{n})\) for \(n=0\) to 5 are respectively given by:
From (2) we easily obtain
For convenience, we define the above sum as equal to zero for \(n<0\), i.e., \(\mathbf{B}_{n}(x_{1},x_{2},\dots, x_{n})=0\) when \(n<0\).
Chu [3] established the partial fraction decompositions of two rational functions \(\frac{1}{(z)^{\lambda}_{n+1}}\) and \(\frac{z^{M}}{(z)^{\lambda}_{n+1}}\) 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, \(z \in \mathbb{C} \setminus \{0,-1,\dots,-n\}\). Then the following partial fraction decomposition holds:
where
Theorem 2
([3, Theorem 5])
Suppose that \(n, M\), and λ are three natural numbers with \(\lambda \leqslant M < \lambda (n+1)\), \(z \in \mathbb{C} \setminus \{0,-1,\dots,-n\}\). 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 \(\frac{x^{M}}{(x+1)^{\lambda}_{n}}\). 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 \(N=\lambda{n}\), and z is a complex number such that \(z \in \mathbb{C} \setminus \{-1,-2,\dots,-n\}\). Then the following partial fraction decomposition holds:
where
When \(M-N \geq 0\) in Theorem 3, i.e., the rational function \(\frac{x^{M}}{(x+1)^{\lambda}_{n}}\) is improper, putting \(a_{j}= \frac{{\mathbf{{B}}}_{j}(x_{1},\dots,x_{j})}{j!}\) and \(a_{k,j}=\frac{(-1)^{\lambda{k}}}{j!(n!)^{\lambda}}\binom{n}{k}^{ \lambda}(-k)^{\lambda +M}{\mathbf{{B}}}_{j}(y_{1}, y_{2}, \dots,y_{j})\), (6) becomes the following explicit form:
which is an explicit result of the polynomial \(x^{M}\) divided by the polynomial \((x+1)^{\lambda}_{n}\). 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 \(N=\lambda{n}\), 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 \(N=n \lambda \), then the following q-algebraic identity holds:
where
Remark 5
2 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 \(N=n \lambda \), then the following q-algebraic identity holds:
where
When \(M \geq N\), 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 \(N=n\lambda \). Then the following q-algebraic identity holds:
where
Proof
Letting \(z \mapsto q^{z}-1\) in Theorem 4, we obtain Theorem 11 immediately. □
Remark 12
Formula (19) is another q-analogue of Luo and Zhu result (6).
Theorem 13
If z is a complex variable, M is a nonnegative integer, n and λ are two positive integers such that \(N=\lambda (n+1)\), then the following q-algebraic identity holds:
where
Proof
Letting \(z \mapsto q^{z}-1\) and \(M \mapsto M-\lambda \), noting that \(\sum^{n}_{k=1}=\sum^{n}_{k=0}\) in Theorem 4, and applying the relation
we obtain Theorem 13 immediately. □
When \(M< N\), and noting that \(\sum^{n}_{k=1}=\sum^{n}_{k=0}\) 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 \(N=\lambda (n+1)\), 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).
Taking \(\lambda =1\) in (27), we have
which is a q-analogue of the well-known binomial identity (e.g., see [14]):
Taking \(M=0\), \(n \mapsto n+1, z \mapsto z-1\), and then letting \(z \mapsto zq^{-1}\) in Theorem 4, we obtain the following corollary.
Corollary 18
If z is a complex variable, n and λ are two positive integers such that \(N=\lambda (n+1)\), 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 \(\lambda =1\) and letting \(M \mapsto m\) 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:
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 q-harmonic numbers.
Corollary 21
If M is a nonnegative integers, n and λ are two positive integers such that \(N=n\lambda \), then we have the following q-combinatorial identities:
where
Taking \(z=1\) in (7), we obtain the following q-combinatorial identities involving q-harmonic numbers:
Taking \(z=-1\) in (7), we obtain the following q-combinatorial identities involving q-harmonic numbers:
Taking \(z=1\) in (22), we obtain the following q-combinatorial identities involving q-harmonic numbers:
Multiplying both sides of (10) by z and then letting \(z \to \infty \), we immediately establish the following combinatorial identities:
Corollary 22
If M is a nonnegative integer, n and λ are two positive integers such that \(N=n\lambda \), then we have the following q-combinatorial identities:
where
Noting that \(\sum_{k=1}^{n}=\sum_{k=0}^{n}\) in (41), we say that the above q-combinatorial identities are the corresponding q-analogues of Chu’s result [3, Corollary 7]:
where
Taking \(\lambda =1,2,3,4\) in (41), respectively, we obtain the following q-combinatorial identities involving q-harmonic numbers:
Taking \(z=0\) in (30), we obtain the following q-combinatorial identity:
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 \(z \to \infty \), we establish the following q-combinatorial identities:
Corollary 23
Suppose that λ and n are positive integers. We have the following q-combinatorial identity:
where \(y_{i}\) are given by (31).
Remark 24
The combinatorial identity (51) is a q-analogue of Chu’s result [3, Corollary 3]:
where
Taking \(\lambda =1\) in (51), we have
which is a q-analogue of the well-known identity
4 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 \(N=n \lambda \), 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 \(\alpha \neq q^{-1}-1,q^{-2}-1,\ldots,q^{-n}-1\).
We next construct three contour integrals for the rational functions \(P(z)/Q(z)\):
-
\(\oint _{\Gamma}\frac{P(z)}{Q(z)}\,\mathrm{d}z\), where Γ is a simple closed contour which only surrounds the single pole α of \(P(z)/Q(z)\);
-
\(\oint _{\Gamma '} \frac{P(z)}{Q(z)}\,\mathrm{d}z\), where \({\Gamma}'\) is a simple closed contour which only surrounds the poles \(q^{-1}-1,q^{-2}-1, \dots, q^{-n}-1\) of \(P(z)/Q(z)\);
-
\(\oint _{\Gamma ''} \frac{P(z)}{Q(z)}\,\mathrm{d}z\), where \({\Gamma}''\) 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 \(\oint _{\Gamma}\frac{P(z)}{Q(z)}\,\mathrm{d}z, \oint _{{\Gamma}'} \frac{P(z)}{Q(z)}\,\mathrm{d}z\), and \(\oint _{{\Gamma}''}\frac{P(z)}{Q(z)}\,\mathrm{d}z\), respectively.
We compute the contour integral \(\oint _{ {\Gamma}}\frac{P(z)}{Q(z)}\,\mathrm{d}z\) as follows:
We calculate the contour integral \(\oint _{{\Gamma}''} \frac{P(z)}{Q(z)}\,\mathrm{d}z\) as follows:
If \(M-N<0\), then \(t=0\) is not a pole, and so we have
If \(M-N=0\), then \(t=0\) is a single pole of order 1, so we have
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
We get
We now calculate the contour integral \(\oint _{ {\Gamma}'}\frac{P(z)}{Q(z)}\,\mathrm{d}z\). 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
Noting that
it follows that
Therefore, by substituting (57)–(59) into (56), and then changing \(\alpha \mapsto z\), we obtain Lemma 25. This proof is complete. □
Lemma 26
The following recursion formula of complete Bell polynomial holds true:
where
Proof
Write \(y_{s}=w_{s}+(-1)^{s} \frac{(s-1)!}{(z-q^{-k}+1)^{s}}\) in (54). From the definition of a complete Bell polynomial, we obtain
In the above process, we apply the binomial theorem:
□
Lemma 27
The following recursion formula of complete Bell polynomial holds true:
where
Proof
Write \(z_{s}=v_{s}+(s-1)!z^{s}\) in (55). Using the definition of a complete Bell polynomial, we obtain
In the above process, we apply the binomial theorem:
□
Proof
Proof of Theorem 4 Substituting (60) and (61) into (53), and then changing \(w_{j} \mapsto y_{j} \) and \(v_{j} \mapsto x_{j} \), we obtain Theorem 4. This proof is complete. □
5 Conclusions
As it is well known, the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-)hypergeometric polynomials, have widespread applications, particularly in several areas of number theory and combinatorial analysis such as the theory of partitions. In [19, p. 340], professor Srivastava points out an important demonstrated observation that any \((p,q)\)-variations of the proposed q-results 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 \(\frac{x^{M}}{(x+1)^{\lambda}_{n}}\).
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The authors are grateful to Professor H. M. Srivastava for his references.
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Luo, ZQ., Luo, QM. A q-analogue for partial-fraction decomposition of a rational function and its application. Adv Cont Discr Mod 2024, 18 (2024). https://doi.org/10.1186/s13662-024-03814-7
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DOI: https://doi.org/10.1186/s13662-024-03814-7