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In this article, we present a new proof of the partial fraction expansion of the cotangent function using the theory of hypergeometric functions. [ABSTRACT FROM AUTHOR]

By making use of the multisection series method, four classes of alternating infinite series are evaluated, in closed form, by the Riemann zeta function and the Dirichlet beta function. [ABSTRACT FROM AUTHOR]

In this work, we first construct a half-discrete kernel function, which is defined in the whole plane and involves both the homogeneous and the non-homogeneous cases. By employing the method of weight coefficient and some classical techniques of real analysis, a class of halfdiscrete Hilbert-type inequalities with the newly constructed kernel as well as the equivalent inequalities of Hardy's type are established. In addition, we prove that all the constant factors in the newly established inequalities are the best possible. Lastly, assigning special values to the parameters, and using the partial fraction expansions of cotangent function and cosecant function, some new half-discrete Hilbert-type inequalities with special kernels defined in the whole plane are presented at the end of the paper. [ABSTRACT FROM AUTHOR]

A rational function can always be integrated, that is, the integral of such a function is always an elementary function. The integration procedure is complex and consists of four steps: elimination of the common zero-points of the numerator and denominator, reduction to a true rational function, decomposition into partial fractions and integration of the obtained expressions using direct integration, substitution method or partial integration method. Integrating rational functions is important because integrals of rational functions of trigonometric functions as well as integrals of some irrational functions are reduced to integrals of rational functions by appropriate transformations.

By means of partial fraction decompositions, a shorter proof is presented for the important interpolation formula of trigonometric polynomials discovered by Riesz (1914). [ABSTRACT FROM AUTHOR]

In this paper, by using the residue method of complex analysis, we obtain an explicit partial fraction decomposition for the general rational function x M (x + 1) n λ (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. [ABSTRACT FROM AUTHOR]

This paper presents an explicit and straightforward method for finding the sojourn-time distribution of a random customer in an M / G a / 1 queue with a fixed-size batch service. The exhibited process is much more straightforward than the approach discussed by Yu and Tang (Methodology and Computing in Applied Probability 20(4):1503–1514, 2018). We obtain two closed-form expressions for probability density functions by using the inside and outside roots of the underlying characteristic equation. Applying partial fractions and residue theorem, we determine an explicit form of sojourn-time distribution and evaluate the distribution function for any specific time. In illustrative examples, we compare the results obtained by both methods and find that the results match excellently. [ABSTRACT FROM AUTHOR]

New oscillation criteria for a class of third-order neutral differential equations with distributed deviating arguments are established. Examples illustrating the results are provided and some suggestions for further research are indicated. [ABSTRACT FROM AUTHOR]

In this paper, we consider systems of impulsive conformable partial fractional differential equations with a damping term and establish sufficient conditions for the oscillation of these systems under the Dirichlet boundary condition. The main results are illustrated by an example. [ABSTRACT FROM AUTHOR]

In this paper, we present new kinds of reciprocal sums of finite products of general second order linear recurrences. In order to evaluate explicitly them by q-calculus, first we convert them into their q-notation and then use the methods of partial fraction decomposition and creative telescoping. [ABSTRACT FROM AUTHOR]

Sums of products of two Gaussian q-binomial coeficients, are investigated, one of which includes two additional parameters, with a parametric rational weight function. By means of partial fraction decomposition, first the main theorems are proved and then some corollaries of them are derived. Then these q-binomial identities will be transformed into Fibonomial sums as consequences. [ABSTRACT FROM AUTHOR]

In this paper, the partial fractions expansions for the generalized hypergeometric function will be presented. Our results generalized several well-known results in the literature. [ABSTRACT FROM AUTHOR]

Abstract The hook length formula for d -complete posets states that the P -partition generating function for them is given by a product in terms of hook lengths. We give a new proof of the hook length formula using q -integrals. The proof is done by a case-by-case analysis consisting of two steps. First, we express the P -partition generating function for each case as a q -integral and then we evaluate the q -integrals. Several q -integrals are evaluated using partial fraction expansion identities and the others are verified by computer. [ABSTRACT FROM AUTHOR]

Algebraic functions are defined as those given by the solutions of systems of equations that have a unique solution. In this work we find a characterization of the clone of algebraic functions in the three-element bounded lattice over the language with one constant symbol. [ABSTRACT FROM AUTHOR]

We study uniform approximation of real-valued functions f, f(∞) = 0, on ℝ+ and ℝ by real-valued simple partial fractions (the logarithmic derivatives of polynomials). We obtain a criterion for the best approximation on ℝ+ and ℝ in terms of the Chebyshev alternance. This criterion is similar to the known criterion on finite segments. For the problem of approximating odd functions on ℝ we construct an alternance criterion with a weakened condition on the poles of fractions. We present a criterion for the best approximation by simple partial fractions on ℝ+ and ℝ in terms of Kolmogorov. We prove analogs of the de la Vallee-Poussin alternation theorem. [ABSTRACT FROM AUTHOR]

An asymptotics of the error of interpolation of real constants at Chebyshev nodes is obtained. Some well-known estimates of the best approximation by simple partial fractions (logarithmic derivatives of algebraic polynomials) of real constants in the closed interval [−1, 1] and complex constants in the unit disk are refined. As a consequence, new estimates of the best approximation of real polynomials on closed intervals of the real axis and of complex polynomials on arbitrary compact sets are obtained. [ABSTRACT FROM AUTHOR]

We consider the longest run of either successes or failures in a sequence of Bernoulli trials. The exact distribution of this random variable is obtained using probability generating function techniques. We consider approximating the tail areas of this random variable using a partial fraction approximation and a saddlepoint approximation with various continuity corrections. We investigate and compare the performance of the approximations for various values of sequence length and various values of the probability of success. [ABSTRACT FROM AUTHOR]

Reflection positivity constitutes an integral prerequisite in the Osterwalder–Schrader reconstruction theorem which relates quantum field theories defined on Euclidean space to their Lorentzian signature counterparts. In this work, we rigorously prove the violation of reflection positivity in a large class of free scalar fields with a rational propagator. This covers, in particular, higher-derivative theories where the propagator admits a partial fraction decomposition as well as degenerate cases including, e.g., p4-type propagators. [ABSTRACT FROM AUTHOR]

The paper deals with the developing of the methodological backgrounds for the modeling and simulation of complex dynamical objects. Such backgrounds allow us to perform coordinate transformation and formulate the algorithm of its usage for transforming the serial mathematical model into parallel ones. This algorithm is based on partial fraction decomposition of the transfer function of a dynamic object. Usage of proposed algorithms is one of the ways to decrease calculation time and improve PC usage while a simulation is being performed. We prove our approach by considering the example of modeling and simulating of fourth order dynamical object with various eigenvalues. This example shows that developed parallel model is stable, well-convergent, and high-accuracy model. There is no defined any calculation errors between well-known serial model and proposed parallel one. Nevertheless, the proposed approach's usage allows us to reduce calculation time by more than 20% by using several CPU's cores while calculations are being performed. [ABSTRACT FROM AUTHOR]

In this paper, a new method is developed to obtain explicit and integral expressions for the kernel of the ( κ , a ) -generalized Fourier transform for κ = 0 . In the case of dihedral groups, this method is also applied to the Dunkl kernel as well as the Dunkl Bessel function. The method uses the introduction of an auxiliary variable in the series expansion of the kernel, which is subsequently Laplace transformed. The kernel in the Laplace domain takes on a much simpler form, by making use of the Poisson kernel. The inverse Laplace transform can then be computed using the generalized Mittag–Leffler function to obtain integral expressions. In case the parameters involved are integers, explicit formulas are obtained using partial fraction decomposition. New bounds for the kernel of the ( κ , a ) -generalized Fourier transform are obtained as well. [ABSTRACT FROM AUTHOR]

ABSTRACT: Viscoelasticity is a mechanical phenomenon where the material modulus varies with time and temperature. Modern experimental methods can determine material properties within certain time and temperature ranges, but modeling the viscoelastic behavior remains challenging, mainly because the data processing is complex and different materials have distinct properties. Using polyetherimide as an example and based on the change in the secondary bonds of polyetherimide in different viscoelastic stages, we proposed a new shift factor model in Arrhenius format with alterable activation energy. We also used two methods based on nonlinear least squares to obtain the Maxwell model of the polyetherimide, and we then used a novel method integrated with Laplace transforms and partial fraction decomposition to convert the Maxwell model into the Voigt model. The results of our model are reliable and self‐consistent, showing its potential for modeling the viscoelasticity of other materials. © 2017 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2018, 135, 46102. [ABSTRACT FROM AUTHOR]

In this article, we investigate some Ulam's type stability concepts for the Darboux problem of partial fractional differential inclusions with a nonconvex valued right hand side. Our results are based upon Covitz-Nadler fixed point theorem and fractional version of Gronwall's inequality. [ABSTRACT FROM AUTHOR]

By means of the partial fraction decomposition method, we present a new and elementary proof of the Riesz interpolation formula for trigonometric polynomials, including another interpolation formula as a by-product. [ABSTRACT FROM AUTHOR]

The multipole representation provides an analytical way of Doppler broadening cross sections for neutron interactions and relies on a partial fraction decomposition that represents R-matrix resonance parameters by poles and residues. The windowed multipole method selectively broadens impactful poles and approximates the rest to increase the efficiency of the process. However, this process requires knowledge of the resonance parameters and certain limitations occur when additional channels are opened in the defined resolved resonance region such that only 2/3 of the ENDF/B-VII.1 nuclides can be processed. Some important nuclides needed in reactor applications such as Oxygen-16 and Hydrogen-1 are directly represented in point-wise form in the evaluation nuclear data library hindering the practicality of this approach. This paper presents a new fitting method for nuclear point-wise cross sections data that yields a pole and residue form analogous to the multipole representation. The Vector Fitting technique which originates from the field of signal processing was applied to the fitting of point-wise cross sections. Poles and residues from Vector Fitting were generated for Oxygen-16, Hydrogen-1, Boron-10 and Boron-11 and processed in the windowed multipole format. These new libraries were tested by direct comparison of the microscopic cross sections before and after Doppler Broadening, and by integral comparisons using a typical PWR pin cell from the BEAVRS benchmark. Results indicate that the new libraries are equivalent to the point-wise representation with the added benefit of allowing on-the-fly Doppler broadening without the need for temperature interpolation. Runtimes are approximately ∼28% slower using the multipole representation on this example problem compared to a single temperature ACE file. [ABSTRACT FROM AUTHOR]

In this paper, we establish some relations involving q-Euler type sums, q-harmonic numbers and q-polylogarithms. Then, using the relations obtained with the help of q-analog of partial fraction decomposition formula, we develop new closed form representations of sums of q-harmonic numbers and reciprocal q-binomial coefficients. Moreover, we give explicit formulas for several classes of q-harmonic sums in terms of q-polylogarithms and q-harmonic numbers. The given representations are new. [ABSTRACT FROM AUTHOR]

A class of double sums with binomial coefficients are evaluated by combining finite differences with partial fraction decompositions. [ABSTRACT FROM AUTHOR]

An analytical approach for the synthesis of narrow-band lossy filters with arbitrary transmission zeros is proposed in this paper. The scattering parameters are deduced from filtering polynomials once the transmission zeros are assigned. The formulas defining the transformation from scattering matrix to admittance matrix can be used by reconstructing the non-paraconjugate transmission zeros as paraconjugate. A canonical $$N + 2$$ transversal or $$N \times N$$ coupling matrix is then identified using partial fraction expansion of the normalized admittance functions. An increased order of the final network (with respect to the order of the characteristic polynomials) is required for obtaining this result, provided there are non-paraconjugate transmission zeros. Because of its versatility, the proposed approach can also be employed for lossless networks that have a transfer function with symmetrical transmission zeros. Two typical examples are presented to validate the proposed technique. [ABSTRACT FROM AUTHOR]

In this paper, we present some Euler-like sums involving partial sums of the harmonic and odd harmonic series. First, we give a brief historical account of Euler’s work on the subject followed by notations used in the body of the paper. After discussing some alternating Euler sums, we investigate the connection of integrals of inverse trigonometric and hyperbolic type functions to generate many new Euler sum identities. We also give some new identities for Catalan’s constant, Apery’s constant and a fast converging identity for the famous ζ ( 2 ) constant.

Here we produce an interesting fractional means scalar inequality. [ABSTRACT FROM AUTHOR]

We present a method for finding partial fraction decompositions of rational functions with linear or quadratic factors in the denominators by means of repeated polynomial divisions. This method does not involve differentiation or solving linear equations for obtaining the unknown partial fraction coefficients, which is very suitable for either hand or machine calculations. Some illustrative examples are provided. [ABSTRACT FROM AUTHOR]

Sums of products of two Gaussian q-binomial coefficients with a parametric rational weight function are considered. The partial fraction decomposition technique is used to evaluate the sums in closed form. Interesting applications of these results to certain generalized Fibonomial and Lucanomial sums are provided. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring and L is a Lie ideal of R such that LZ(R). A map F:R⋺R is called a multiplicative generalized derivation if there exists a map d:R⋺R such that F(xy)=F(x)y+xd(y), for all x,y∈R. In the present paper, we shall prove that d is a commuting map on L if any one of the following holds: i) F(uv)=±uv, ii) F(uv)=±vu, iii) F(u)F(v)=±uv, iv) F(u)F(v)=±vu, v) F(u)F(v)±uv∈Z, vi) F(u)F(v)±vu∈Z, vii) [F(u),v]±[u,G(v)]=0, for all u,v∈L. [ABSTRACT FROM AUTHOR]

Recently Melham [1] gives closed formulæ for certain finite reciprocal sums. In this paper, we present a different approach to compute these sums in closed form. Our approach is straight-forward and simple. First we convert the sums in q-notation, then use partial fraction decomposition and telescoping to derive closed formulæ. [ABSTRACT FROM AUTHOR]

In this paper, a penalty function method for solving non-linear optimization problems was introduced to obtain a reliability indicator and to calibrate the partial factors, avoiding the complexity associated with the calibration process of the partial factors and maintaining the accuracy of the reliability indicator. The proposed method was employed to calibrate the partial factors of gravity dam sliding stability analysis. The results indicate that using the penalty function-based calculation method to determine the reliability indicator not only ensures correctness and accuracy but also makes the calibration process of partial factors simple and easy to perform. [ABSTRACT FROM AUTHOR]

A critical but often overlooked question in the study of ligands binding to proteins is whether the parameters obtained from analyzing binding data are practically identifiable (PI), i.e., whether the estimates obtained from fitting models to noisy data are accurate and unique. Here we report a general approach to assess and understand binding parameter identifiability, which provides a toolkit to assist experimentalists in the design of binding studies and in the analysis of binding data. The partial fraction (PF) expansion technique is used to decompose binding curves for proteins with n ligand-binding sites exactly and uniquely into n components, each of which has the form of a one-site binding curve. The association constants of the PF component curves, being the roots of an n-th order polynomial, may be real or complex. We demonstrate a fundamental connection between binding parameter identifiability and the nature of these one-site association constants: all binding parameters are identifiable if the constants are all real and distinct; otherwise, at least some of the parameters are not identifiable. The theory is used to construct identifiability maps from which the practical identifiability of binding parameters for any two-, three-, or four-site binding curve can be assessed. Instructions for extending the method to generate identifiability maps for proteins with more than four binding sites are also given. Further analysis of the identifiability maps leads to the simple rule that the maximum number of structurally identifiable binding parameters (shown in the previous paper to be equal to n) will also be PI only if the binding curve line shape contains n resolved components. [ABSTRACT FROM AUTHOR]

In this paper, by using the method of partial fraction decomposition and integral representations of series, we establish some expressions of series involving harmonic numbers and binomial coefficients in terms of zeta values and harmonic numbers. Furthermore, we can obtain some closed form representations of sums of products of quadratic (or cubic) harmonic numbers and reciprocal binomial coefficients, and some explicit evaluations are given as applications. The given representations are new. [ABSTRACT FROM AUTHOR]

By means of finite differences and partial fraction decomposition, we establish two binomial transformations, that extend, with four free parameters, the recent results due to Prodinger (2010) and Dahlberg-Ferdinands-Tefera (2010). [ABSTRACT FROM AUTHOR]

Identities involving finite sums of products of hypergeometric functions and their duals have been studied since 1930s. Recently Beukers and Jouhet have used an algebraic approach to derive a very general family of duality relations. In this paper we provide an alternative way of obtaining such results. Our method is very simple and it is based on the non-local derangement identity. [ABSTRACT FROM AUTHOR]

We show that the dilogarithm has at most one zero on each branch, that each zero is close to a root of unity, and that they may be found to any precision with Newton's method. This work is motivated by applications to the asymptotics of coefficients in partial fraction decompositions considered by Rademacher. We also survey what is known about zeros of polylogarithms in general. [ABSTRACT FROM AUTHOR]

Partial fraction decomposition method is applied to evaluate a general determinant of shifted factorial fractions, which contains several Gaussian binomial determinant.El método de descomposición en fracción parciales aplicado para evaluar un determinante general de fracciones factoriales trasladadas, la cual contiene varias identidades determinante binomial Gaussiano.

The generating function for , the number of partitions of into at most parts, may be written as a product of factors. We find the behavior of coefficients in the partial fraction decomposition of this product as by applying the saddle-point method, where the saddle-point we need is associated to a zero of the analytically continued dilogarithm. Our main result disproves a conjecture of Rademacher. [ABSTRACT FROM AUTHOR]