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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) n λ . As applications, we deduce the corresponding q-algebraic and q-combinatorial identities which are the q-extensions of Chu' results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish several families of identities involving universal mock theta functions and express them in terms of Appell–Lerch sums. The subsequent consequences of these results are a variety of interesting identities on theta series and classical mock theta functions. [ABSTRACT FROM AUTHOR]

Several trigonometric sums are examined by means of partial fraction decompositions. Their generating functions will be evaluated in closed forms and the related asymptotic formulae in terms of Riemann zeta function will be established. [ABSTRACT FROM AUTHOR]

By employing the linearization method and partial fractions, we examine a large class of nonterminating exotic 3 F 2 -series extended by five integer parameters. A general summation formula is proved, and several contiguous series are evaluated in closed form. [ABSTRACT FROM AUTHOR]

Modal identification is a key step in modal analysis. It enables the researcher to extract modal parameters, such as natural frequency, amplitude, and damping from a given structure. There are a considerable number of techniques in the state of the art aiming to address this problem, where multi-mode approaches arise as an appealing choice due to their ability to deal with mode coupling. This tutorial paper focuses on the complex-curve fitting technique, originally conceived for an application distinct from modal analysis. It aims at guiding other researchers by providing a tutorial-like and in-depth analysis of this important method, associated with a nonlinear weighting procedure for improved precision. Additionally, this paper fills a gap on the original technique, which is limited to the ratio of two polynomials, by proposing an automatic parameter extraction technique. The original and improved methods are applied on both simulated and experimental data, highlighting the effectiveness of the proposed changes. The proposed procedure is also compared with the rational fraction polynomial method. [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]

We evaluate, in closed forms, twelve infinite series that contain a free variable x and binomial coefficients ( 3 n + ϵ n) in the denominators. Several summation formulae resembling Gosper's sum are derived as consequences. [ABSTRACT FROM AUTHOR]

A large class of infinite triple series are explicitly evaluated by computing definite double integrals. Several remarkable formulae are presented in terms of π and ln 2. [ABSTRACT FROM AUTHOR]

By making use of the Cauchy double alternant and the Laplace expansion formula, we establish two closed formulae for the determinants of factorial fractions that are then utilised to evaluate several determinants of binomial coefficients and Catalan numbers, including those obtained recently by Chammam ['Generalized harmonic numbers, Jacobi numbers and a Hankel determinant evaluation', Integral Transforms Spec. Funct.30(7) (2019), 581–593]. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Two open problems recently proposed by Xi and Luo (Adv. Differ. Equ. 2021:38, 2021) are resolved by evaluating explicitly three binomial sums involving harmonic numbers, that are realized mainly by utilizing the generating function method and symmetric functions. [ABSTRACT FROM AUTHOR]

In this paper, we give some extensions for Mortenson's identities in series with the Bell polynomial using the partial fraction decomposition. As applications, we obtain some combinatorial identities involving the harmonic numbers. [ABSTRACT FROM AUTHOR]

• Extend integration approach from polynomial to rational PH curves. • Linear conditions for rational integrability via partial fraction decomposition. • Optimal interpolation with shape control (no cusps) by solving a quadratic program. • Optimize curve energy or impose length constraints. Using a residuum approach, we provide a complete description of the space of the rational spatial curves of given tangent directions. The rational Pythagorean hodograph curves are obtained as a special case when the norm of the direction field is a perfect square. The basis for the curve space is given explicitly. Consequently a number of interpolation problems (G 1 , C 1 , C 2 , C 1 / G 2) in this space become linear, cusp avoidance can be encoded by linear inequalities, and optimization problems like minimal energy or optimal length are quadratic and can be solved efficiently via quadratic programming. We outline the interpolation/optimization strategy and demonstrate it on several examples. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of singularities caused by repeated decay constants in the Bateman equations for nuclide concentrations in the radioactive transformation chains. Using partial fraction decomposition with the Bateman equations, the general solution for the nuclide concentrations in the radioactive transformation linear chains with repeated decay constants has been derived. The derived equations are similar to the solutions obtained by other authors, with the difference being the improved calculation scheme. A comparative analysis was conducted to assess the computational efficiency in determining concentration changes within transmutation chains, particularly in the presence of repetitive reaction rates. The investigation involved evaluating several methodologies. • Partial fraction decomposition is used to derive general Bateman equations. • An improvement to the general Bateman equations is proposed. • The passage for the improved general Bateman equations was calculated. [ABSTRACT FROM AUTHOR]

• It presents a partial fraction decomposition of non commensurate systems. • Suitable inversion of each fraction is done in two ways: series and integer/fractional decomposition. A study of non-commensurate fractional linear system is done in a parallel way to the commensurate case. A partial fraction decomposition is accomplished using a recursive procedure. Each partial fraction is inverted in two different ways. The decomposition integer/fractional is done also. Some examples are presented. [ABSTRACT FROM AUTHOR]

Generalizing earlier results, sums over the products of the Gaussian q -binomial coefficients are computed. Some applications of the results for special choices of q are emphasized. The results are obtained by the elementary technique of partial fraction decomposition. [ABSTRACT FROM AUTHOR]

Recently, Melham computed some finite sums in which the denominator of the summand includes products of ‘sine’ or ‘cosine’. In this paper, generalizations of the sums, which he studied in 2016, are presented, by allowing arbitrary factors in the denominator of the summand. Our approach uses the elementary technique of partial fraction decomposition. Furthermore, some of the sums, which he studied in 2017, are treated in the same style. [ABSTRACT FROM AUTHOR]

All rational parametric curves with prescribed polynomial tangent direction form a vector space. Via tangent directions with rational norm, this includes the important case of rational Pythagorean hodograph curves. We study vector subspaces defined by fixing the denominator polynomial and describe the construction of canonical bases for them. We also show (as an analogy to the fraction decomposition of rational functions) that any element of the vector space can be obtained as a finite sum of curves with single roots at the denominator. Our results give insight into the structure of these spaces, clarify the role of their polynomial and truly rational (non-polynomial) curves, and suggest applications to interpolation problems. [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]

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

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]

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]

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]

In 2000, Andrews challenged mathematicians in the 21st century to elucidate the overlap between classes of q -series and modular forms. Motivated by Andrews' challenge, seeking q -hypergeometric series which are mock theta functions in the modern sense and finding their alternative representations have attracted the attention of many mathematicians in recent years. In this paper, we use partial fraction decomposition to give rise to several families of two-parameter mock theta functions and express them in the form of Appell-Lerch sums. In particular, we establish some identities involving several families of mock theta functions and Appell-Lerch sums which imply three identities proved by Cui and Gu. [ABSTRACT FROM AUTHOR]

We consider sums of the Gaussian -binomial coefficients with a parametric rational weight function. We use the partial fraction decomposition technique to prove the claimed results. We also give some interesting applications of our results to certain generalized Fibonomial sums weighted with finite products of reciprocal Fibonacci or Lucas numbers. [ABSTRACT FROM AUTHOR]

We provide new formulas for the coefficients in the partial fraction decomposition of the restricted partition generating function. These techniques allow us to partially resolve a recent conjecture of Sills and Zeilberger. We also describe upcoming work, giving a resolution to Rademacher's conjecture on the asymptotics of these coefficients. [ABSTRACT FROM AUTHOR]

In his book Topics in Analytic Number Theory (1973), Hans Rademacher considered the generating function of integer partitions into at most N parts and conjectured certain limits for the coefficients of its partial fraction decomposition. We carry out an asymptotic analysis that disproves this conjecture, thus confirming recent observations of Sills and Zeilberger (Journal of Difference Equations and Applications 19 (2013)), who gave strong numerical evidence against the conjecture. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a simple cover-up approach for computing partial fractions with linear or irreducible quadratic factors in the denominators is presented. It mainly involves polynomial divisions and substitutions only, without having to factorize the irreducible quadratic factors into linear factors with complex coefficients, to use differentiation or to solve a system of linear equations. Examples and its application to derive some useful formulas of partial fraction decompositions are included. [Copyright &y& Elsevier]

Partial fraction decomposition is a useful technique often taught at senior secondary or undergraduate levels to handle integrations, inverse Laplace transforms or linear ordinary differential equations, etc. In recent years, an improved Heaviside's approach to partial fraction decomposition was introduced and developed by the author. An important feature of this approach is that there is no need to solve a system of linear equations or to use differentiations to find the unknown coefficients of the partial fractions. In order to study its potential application in mathematics education at the undergraduate level, a pilot study of tryout at the Hong Kong Institute of Education has been conducted. The data are collected via quizzes, questionnaires and face-to-face interviews. In this article, the results and implications will be discussed. In general, according to the responses and feedbacks from the instructors and students concerned, the improved Heaviside approach is suitable to be introduced at the undergraduate level, as an alternative to the method of undetermined coefficients described in common undergraduate mathematics textbooks. [ABSTRACT FROM PUBLISHER]

In this note, a new method for computing the partial fraction decomposition of rational functions with irreducible quadratic factors in the denominators is presented. This method involves polynomial divisions and substitutions only, without having to solve for the complex roots of the irreducible quadratic polynomial or to solve a system of linear equations. Some examples of its applications are included. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a simple and efficient method for determining the rational solution of Riccati differential equation with coefficients rational. In case the differential Galois group of the differential equation , is reducible, we look for the rational solutions of Riccati differential equation , by reducing the number of check to be made and by accelerating the search for the partial fraction decomposition of the solution reserved for the poles of θ which are false poles of r. This partial fraction decomposition of solution can be used to code r. The examples demonstrate the effectiveness of the method. [Copyright &y& Elsevier]

Recently, McCarthy presented two algebraic identities involving binomial coefficients and harmonic numbers, one of which generalizes an identity used to prove the Apéry number supercongruence. In 2008, Prodinger provided human proofs of identities initially obtained by Osburn and Schneider using the computer program Sigma. In this paper, we establish q -analogs of a fair number of the identities appearing in McCarthy (Integers 11 (2011): A37) and Prodinger (Integers 8 (2008): A10) by making use of q -partial fractions. [ABSTRACT FROM AUTHOR]

Abstract: We implement the partial fraction decomposition scheme proposed by Croy and Saalmann [A. Croy, U. Saalmann, Phys. Rev. B 80 (2009) 073102] to the construction of the hierarchical equations of motion for bosonic quantum dissipation systems [R.X. Xu, Y.J. Yan, Phys. Rev. E 75 (2007) 031107]. The expansion of Bose–Einstein function in this scheme shows similar properties as it applies for Fermi function. The performance of the resulting quantum dissipation theory is exemplified with spin-boson systems. The new theoretical construction performs much better (about an order of magnitude faster) than the best available conventional formalism based on Matsubara spectral decomposition scheme. [Copyright &y& Elsevier]

Abstract: We find convergent double series expansions for Legendre''s third incomplete elliptic integral valid in overlapping subdomains of the unit square. Truncated expansions provide asymptotic approximations in the neighborhood of the logarithmic singularity if one of the variables approaches this point faster than the other. Each approximation is accompanied by an error bound. [Copyright &y& Elsevier]

• Method for the acoustic characterization of sound-absorbing materials of general nature. • Inspired from effective medium models of metamaterials. • Identification of dissipative and resonant components of the acoustic response, including elastic resonances of the sample. • Only acoustic measurements on a single material sample are required. • Successfully tested against various conventional and non-conventional porous materials. A method is presented to characterize general sound-absorbing materials through a pole-based identification of the equivalent fluid. This is accomplished by 1) determining the extended equivalent fluid of the material sample through the transfer function method (TFM), 2) identification of the acoustic response of the material through the poles of the extended effective density and compressibility, and 3) build the effective density and compressibility from the poles associated to the local acoustic response. Real pole pairs describe a dissipative medium (or equivalently an over-damped resonating medium), which is the natural behavior of rigid-frame porous materials, while complex-conjugate pole pairs describe a locally-resonant medium typical of metamaterials. Complex-conjugate poles associated to elastic resonances of the sample are discarded. We test the method for several non-conventional porous materials. In general, a better fit to the measured surface impedance is obtained than with an acoustics-based identification to the Johnson-Champoux-Allard-Pride-Lafarge model (JCAPL), and the method appears also to be robust to errors of the TFM. [ABSTRACT FROM AUTHOR]