Partial-fraction decomposition of a rational function and its application

In this paper, by using the residue method of complex analysis, we obtain an explicit partial fraction decomposition for the general rational function xM(x+1)nλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{x^{M}}{(x+1)^{\lambda}_{n}}$\end{document} (M is any nonnegative integer, λ and n are any positive integers). As applications, we deduce the corresponding algebraic identities and combinatorial identities which are the corresponding extensions of Chu’ results. We also give some explicit formulas of Apostol-type polynomials and harmonic Stirling numbers of the second kind.

When (x+1) λ n into partial fractions? In the present paper, by using the contour integral and Cauchy's residue theorem, we will answer the above question and give an explicit decomposition for the general rational function x M (x+1) λ n . As applications, we deduce the corresponding algebraic and combinatorial identities which are just some extensions of Chu's results. We give some explicit formulas of Apostol-type polynomials and harmonic Stirling numbers of the second kind.
Theorem 1 Suppose M is any nonnegative integer, λ and n are any positive integers such that N = λn, and x is a complex number such that x ∈ C \ {-1, -2, . . . , -n}. Then the following partial fraction decomposition holds: where • When M -N ≥ 0 in Theorem 1, i.e., the degree of the numerator polynomial, M, is not smaller than the degree of the denominator polynomial, N = λn, we say that Theorem 1 is a more general and new decomposition of the rational function x M (x+1) λ n . For example, below we give the first four special cases.
When M -N = 0, we have When M -N = 1, we have x -λn 2 (n + 1) 2 8 × n(n + 1)λ 2 + 2(2n + 1)λ + 4 then Theorem 1 becomes the following explicit form: which is an explicit result when the polynomial x M is divided by polynomial (x + 1) λ n , i.e., the improper rational fraction x M (x+1) λ n is decomposed into a polynomial of order M -N plus a proper rational fraction. Therefore we say that Theorem 1 implies a new and interesting method for division of two 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 obtain the following Chu's result: Theorem 5]) Suppose M is any nonnegative integer, λ and n are any positive integers such that M < N = λn, and x is a complex number such that x ∈ C \ {-1, -2, . . . , -n}. Then the following partial fraction decomposition holds: Remark 4 If put M = 0 and then let x − → x -1 and n − → n + 1 in Theorem 1, noting that the empty sum is zero, we observe that Theorem 1 reduces to Theorem 2 of Chu [5, p. 44, (1.5)]: Remark 5 It is easily seen that Theorem 1 includes Chu's results, but when M -N ≥ 0 Theorem 1 is a new result. Therefore, we say that Theorem 1 is an interesting extension of Chu's results. We also see that Theorem 1 is not obtained using the partial fraction decomposition.
Setting λ = 1 and letting M → m in Theorem 1, we deduce the following result:

Corollary 6
Suppose m is any nonnegative integer, n is any positive integer, and x is a complex number such that x ∈ C \ {-1, -2, . . . , -n}. Then the following partial fraction decomposition holds: Furthermore, taking m = 0 in Corollary 6, we obtain the following algebraic identity: or, equivalently, the following well-known combinatorial identity (e.g., see [13]): Next we obtain the following combinatorial identities from Theorem 1: • Setting x = 0 and letting M − → M + 1, we get We can also get the following special cases of Theorem 1:

Proof of Theorem 1
Lemma 7 Suppose M is any nonnegative integer, λ and n are any positive integers such that N = λn, and x is a complex number such that x ∈ C \ {-1, -2, . . . , -n}. Then the following algebraic identity holds: where Proof We first construct two polynomials P(z) and Q(z) of degrees M and N + 1, respectively, which are given by such that x = 0, -1, -2, . . . , -n. 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 x of 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 [12, p. 25, Theorem 2], we have or equivalently, Below we compute the contour integrals Q(z) dz, respectively. Applying Cauchy's residue theorem, we compute the contour integral P(z) Q(z) dz as follows: We now compute the contour integral 1 P(z) Q(z) dz. By utilizing Cauchy's residue theorem, noting that the power series expansion of the logarithmic function is (-1) n-1 z n n |z| < 1 and using the definition of complete Bell polynomials, we obtain Calculating the contour integral 2 P(z) 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 + 1, and so we have Therefore, by replacing (9), (10), and (11) into (8), we obtain Lemma 7. This proof is complete.

Lemma 8
The following recursion formula of complete Bell polynomial holds true: Proof Let Write w i = y i + (i-1)! (x+k) i in (7). By the definition of complete Bell polynomial, we obtain In the above process, we apply the geometric series, z n , and the expansion of the logarithmic function, This proof is complete.

Lemma 9
The following recursion formula of complete Bell polynomial holds true: Proof Let (6). By the definition of complete Bell polynomial, we obtain This proof is complete.
From Lemmas 7, 8, and 9, we obtain Theorem 1 immediately. This completes the proof of Theorem 1.

Applications to Apostol-type polynomials
In the present section, using the contour integrals and the main result (Theorem 1), we first obtain two lemmas, and then we give some explicit formulas for Apostol-type polynomials.
Proof First we construct two polynomials z n+m and (z -1) n of degrees n + m and n, respectively. Next, we construct the following contour integrals for the rational functions z n+m (z-1) n : z n+m (z-1) n dz, where is a simple closed contour which only surrounds the single pole z = 1 of order n of z n+m (z-1) n ; z n+m (z-1) n dz, where is a simple closed contour which only surrounds the pole z = ∞ of z n+m (z-1) n . By utilizing Cauchy's residue theorem, we obtain In the extended complex plane, we calculate the residue of the rational function z n+m (z-1) n at z = ∞.

Lemma 11
The following algebraic identity holds true: Proof Putting n = 1 and letting M → m and λ → n in Theorem 1, noting (3), we get that x i and y i become respectively. Making use of (14) and (15) from Lemma 10, we obtain (19) immediately.
• The Apostol-Bernoulli polynomials B (α) n (x; λ) of order α are defined by means of the generating function (cf. Luo and Srivastava [10]): |z| < 2π when λ = 1; |z| < | log λ| when λ = 1 with, of course, where B (α) n (λ) denote the so-called Apostol-Bernoulli numbers of order α. Let x → -λ exp(x) in (19). We have Multiplying both sides of (22) by x n and noting (20), we have Letting k → kn and k → kj in the first and second terms on the right, respectively, we obtain Comparing the coefficients of x k k! on both sides of (24), we obtain the following new formula of Apostol-Bernoulli polynomials B (α) n (x; λ): Taking λ = 1 in (25), we obtain the following formula of the generalized Bernoulli polynomials at the nonnegative integers: Setting k = n in (26), we have Further setting m = n in (27), we have Taking n = 1 in (26), we obtain the following formula of Bernoulli polynomials at the nonnegative integers: • The Apostol-Euler polynomials E (α) n (x; λ) of order α are defined by means of the generating function(cf. Luo [9]): with, of course, where E (α) n (λ) denote the so-called Apostol-Euler numbers of order α.
Multiplying both sides of (32) by 2 n and using (30), we have Comparing the coefficients of x k k! on both sides of (33), we obtain the following new formula of Apostol-Euler polynomials E (α) n (x; λ): Taking λ = 1 in (34), we obtain the following formula of the generalized Euler polynomials at the nonnegative integers: Taking k = n in (35), we have Further setting m = n in (36), we have Taking n = 1 in (35), we obtain the following formula of Euler polynomials at the nonnegative integers: • The Apostol-Genocchi polynomials of higher order are defined by means of the generating function (cf. Luo and Srivastava [11]): with, of course, where G n (λ), G (α) n (λ), and G n (x; λ) denote the so-called Apostol-Genocchi numbers, Apostol-Genocchi numbers of order α, and Apostol-Genocchi polynomials, respectively.
Multiplying both sides of (41) by 2 n x n and noting (39), we have Let k → kn and k → kj in the first and second terms on the right, respectively, we Comparing the coefficients of x k k! on both sides of (43), we obtain the following new formula of Apostol-Genocchi polynomials G (α) n (x; λ): Taking λ = 1 in (44), we obtain the following formula of the generalized Genocchi polynomials at the nonnegative integers: Taking k = n in (45), we have Further setting m = n in (46), we have Taking n = 1 in (45), we obtain the following formula of Genocchi polynomials at the nonnegative integers: • The generalized Apostol-type polynomials of (real or complex) order α are defined by means of the following generating function (cf. Luo and Srivastava [11]): n (x; λ; μ; ν) z n n! |z| < log(-λ) ; 1 α := 1 , with, of course, we have and Letting x → λ exp(x) in (19), we have Multiplying both sides of (53) by 2 nμ x nν and noting (49), we get ∞ k=0 F (n) k (m; λ; μ; ν) Letting k → knν and k → kjν in the first and second terms on the right, respectively, we have Comparing the coefficients of x k k! on both sides of (55), we obtain the following new formula of Apostol-type polynomials F (n) k (m; λ; μ; ν): Taking λ = 1 in (56), we obtain the following formula of Apostol-type polynomials at the nonnegative integers; analytic number theory and applied mathematics, as well as some operators of fractional calculus, related special functions, and integral transformations. In [19, p. 340], professor Srivastava pointed out an important demonstrated observation that any (p, q)-variation of the proposed q-results would be trivially inconsequential, because the additional parameter p is obviously redundant. Hence we suggest the corresponding basic (or q-) extensions of the results of this paper. In this concluding section, we mention that for a general improper rational function, with m and n being the degrees of the numerator and denominator polynomials, p(z) and q(z), of this improper rational function, respectively, decomposing such a function into a polynomial plus a proper rational fraction is usually very difficult. However, can we decompose a general improper rational function p(z)/q(z) into partial fractions?