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A novel proof of two partial fraction decompositions
Advances in Difference Equations volume 2021, Article number: 274 (2021)
Abstract
In this paper, by constructing contour integral and using Cauchy’s residue theorem, we provide a novel proof of Chu’s two partial fraction decompositions.
1 Introduction
The generalized harmonic numbers are defined by
when \(r=1\), they reduce to the classical harmonic numbers \(H_{n}=H_{n}^{(1)}\).
For \(z \in \mathbb{C}\), the shifted factorial is defined by
The complete Bell polynomials \(\mathbf{B}_{n}(x_{1},x_{2}, \dots , x_{n})\) are defined by [10, p. 134]
with explicit expression
where \(\pi (n)\) denotes all partitions of n into nonnegative parts, that is, all nonnegative integer solutions of the equation
The way of the partial fraction decomposition plays an important role in studying the combinatorial identities and related questions (see, e.g., [1–9, 11, 12, 14, 16–22, 24] and the references therein).
In 2005, Chu [4] established the partial fraction decompositions of two rational functions \(\frac{1}{(x)^{\lambda }_{n+1}}\) and \(\frac{x^{\theta }}{(x)^{\lambda }_{n+1}}\) based on the induction principle and famous the Faà di Bruno formula and obtained several striking harmonic number identities from two partial fraction decompositions. He constructed the generalized Hermite–Padé approximates to the logarithm and therefore resolved completely the open problem of Driver et al. [13].
We now rewrite two main results of Chu.
Theorem A
([4, Theorem 2])
Let λ, θ, and n be three natural numbers such that \(0 \leqslant \theta < \lambda (n+1)\). Then we have the algebraic identity
where the Ω-coefficients are defined as
Theorem B
([4, Theorem 5])
Let λ, θ, and n be three natural numbers such that \(0 \leqslant \theta < \lambda (n+1)\). Then we have the algebraic identity
where Ω-coefficients are defined as
For definitions of \(\tilde{m}!,|\tilde{m}|, \| \tilde{m}\|\), and \(\mathcal{H}_{i}(x)\), see [4, p. 43, and p. 44, (1.4a)].
Comparing \(\Omega _{\ell }(\lambda ,-k)\), and \(\Omega _{\ell }(\lambda ,\theta ,-k)\) with expression (2) of complete Bell polynomials, it is not difficult to reformulate Theorem A and Theorem B as follows (let \(\theta \longmapsto M\)).
Theorem 1
Suppose that λ and n are positive integers and \(x \in \mathbb{C} \setminus \{0,-1,\dots ,-n\}\). Let \(N=\lambda (n+1)\). Then we have the partial fraction decomposition
where
Theorem 2
Let λ, M, and n be three natural numbers such that \(\lambda \leqslant M < \lambda (n+1)\). Then we have the partial fraction decomposition
where
In the present paper, we give a novel proof of Theorem 1 and Theorem 2 by constructing an appropriate contour integral.
We also use the following lemma in Sects. 2 and 3.
Lemma 3
([15])
Let \(P(z)\) and \(Q(z)\) be polynomials (in the complex variable z) of degrees m and n, respectively, given by
Suppose that \(P(z)\) and \(Q(z)\) have no common zeros. If \(\mathcal{C}\) is a simple closed path containing the poles of \(P(z)/Q(z)\) in its interior, then
2 The proof of Theorem 1
In this section, we give a novel proof of Theorem 1 using a contour integral and Cauchy’s residue theorem. We need two lemmas.
Lemma 4
Suppose that λ and n are positive integers and \(x \in \mathbb{C} \setminus \{0,-1,\dots ,-n\}\). Let \(N=\lambda (n+1)\). Then we have the algebraic identity
where
Proof
We first consider two polynomials \(P(z)\) and \(Q(z)\) of degrees 0 and \(N+1\), respectively, given by
We next construct the following contour integrals for the rational functions \(1/Q(z)\):
\(\oint _{ \Gamma } \frac{1}{Q(z)}\,\mathrm{d}z\), where Γ is a simple closed contour, which only surrounds the single pole x of \(1/Q(z)\);
\(\oint _{ {\Gamma } _{1}} \frac{1}{Q(z)}\,\mathrm{d}z\), where \({\Gamma }_{1}\) is a simple closed contour, which surrounds the poles \(0,-1, \dots , -n\) of \(1/Q(z)\).
Applying Cauchy’s residue theorem, we compute the contour integral \(\oint _{ {\Gamma }}\frac{1}{Q(z)}\,\mathrm{d}z\):
We compute the contour integral \(\oint _{{ {\Gamma }_{1}}}\frac{1}{Q(z)}\,\mathrm{d}z\). By utilizing Cauchy’s residue theorem, the power series expansion of logarithmic function
and the definition of complete Bell polynomials, we obtain
Let \(\mathcal{C}={\Gamma }+{\Gamma }_{1}\). Applying the second result of Lemma 3, we have that \(\oint _{\Gamma +{\Gamma }_{1}}\frac{1}{Q(z)}\,\mathrm{d}z=0\) or, equivalently, that \(\oint _{\Gamma }\frac{1}{Q(z)}\,\mathrm{d}z=-\oint _{{\Gamma }_{1}} \frac{1}{Q(z)}\,\mathrm{d}z\). Therefore we directly obtain the algebraic identity (7). □
Lemma 5
The complete Bell polynomials have the following recursive relations:
Proof
Let
Write \(y_{i}=x_{i}+\frac{(i-1)!}{(x+k)^{i}}\) in (8). By using the definition of complete Bell polynomials, power series expansion of the logarithmic function
and the geometric series
we get
The proof is complete. □
Proof of Theorem 1
Replacing (9) of Lemma 5 by (7) of Lemma 4, we immediately obtain Theorem 1. □
3 The proof of Theorem 2
In this section, we give a different proof of Theorem 2 in the same line. We first state the following result.
Theorem 6
Let M be a nonnegative integer, let λ and n be positive integers, and let \(x \in \mathbb{C} \setminus \{0,-1,\dots ,-n\}\). Let \(N=\lambda {n}\) for \(M<\lambda {n}\). Then we have the partial fraction decomposition
where
Proof
The proof is similar to that of Theorem 1, and thus we only present the important steps and omit many details.
We construct two polynomials \(P(z)\) and \(Q(z)\) of degrees M and \(N+1\), respectively, given by
We consider two 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 x of \(P(z)/Q(z)\);
\(\oint _{ {\Gamma } _{1}} \frac{P(z)}{Q(z)}\,\mathrm{d}z\), let \({\Gamma }_{1}\) be a simple closed contour which surrounds the poles \(-1,-2, \dots , -n\) of \(P(z)/Q(z)\).
We obtain contour integrals \(\oint _{ {\Gamma }}\frac{P(z)}{Q(z)}\,\mathrm{d}z\) and \(\oint _{ {\Gamma } _{1}} \frac{P(z)}{Q(z)}\,\mathrm{d}z\):
Applying the second result of Lemma 3, we have the following algebraic identity:
where
We can also obtain that
Substituting (13) into (12), we complete the proof of Theorem 6. □
Proof of Theorem 2
We obviously have
Letting \(M \longmapsto M-\lambda \) in Theorem 6, we immediately obtain Theorem 2. □
4 Conclusion
The basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-)hypergeometric polynomials, are known to have widespread applications, particularly, in several areas of number theory and combinatorial analysis such as the theory of partitions.
Recently, Srivastava [23] published a survey-cum-expository paper on the q-calculus and fractional q-calculus in geometric function theory of complex analysis. Remarkably, a considerably large group of authors have made use of the so-called \((p, q)\)-analysis by introducing a seemingly redundant parameter p in the already known results dealing with the classical q-analysis. On page 340, Professor Srivastava pointed 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 this concluding section, we also suggest the corresponding basic (or q-) extensions of the results of this paper to the interested reader.
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Acknowledgements
The authors greatly appreciate two referees for their valuable comments and suggestions.
Funding
The present investigation was supported by Natural Science Foundation General Project of Chongqing, China under Grant cstc2019jcyj-msxmX0143.
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Zhu, JM., Luo, QM. A novel proof of two partial fraction decompositions. Adv Differ Equ 2021, 274 (2021). https://doi.org/10.1186/s13662-021-03433-6
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DOI: https://doi.org/10.1186/s13662-021-03433-6