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By means of a general partial fraction decomposition expression, we will derive several striking algebraic and combinatorial identities including some results discovered recently by Abel (Aequ Math 94:163–167, 2020). We also recover some famous identities including Chu's identity and Melzak's identity. [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]

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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]

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]

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]

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.

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]

Abstract 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.

In the third part of his famous 1926 paper ‘Quantisierung als Eigenwertproblem’, Schrödinger came across a certain parametrized family of tridiagonal matrices whose eigenvalues he conjectured. A 1991 paper wrongly suggested that his conjecture is a direct consequence of an 1854 result put forth by Sylvester. Here we recount some of the arguments that led Schrödinger to consider this particular matrix and what might have led to the wrong suggestion. We then give a self-contained elementary (though computational) proof which would have been accessible to Schrödinger. It needs only partial fraction decomposition. We conclude this paper by giving an outline of the connection established in recent decades between orthogonal polynomial systems of the Hahn class and certain tridiagonal matrices with fractional entries. It also allows to prove Schrödinger’s conjecture.

Abstract 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.

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.

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]

• 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]

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]

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]

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]

Using partial fraction decomposition, limit methods and algebraic techniques, finding the absolutely minimal realization matrices for first-degree (multi-linear) nD SISO systems is transformed into solving a system of equations consisting of the all principal minors of an unknown matrix. Without using the symbolic approach by Gröbner basis, the necessary and sufficient conditions and construction of the absolutely minimal realizations for the degenerate and non-degenerate of transfer functions are derived. Five examples for first-degree 3D, 4D, and 5D SISO systems are presented to illustrate the basic ideas as well as the effectiveness of the proposed procedure, and computer programs written in Mathematica are also given. [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 brief, we consider the minimax design of infinite impulse response (IIR) filter. First, we apply the partial fraction method to decompose the transfer function into a sum of low order fractions, as a result the minimax IIR filter design problem can be formulated with the stability condition as constraints. Second, among different possible decompositions of the transfer function, we prove that the second order decomposition can always achieve good approximation to the optimal solution. Based on the second order decomposition formulation, several numerical experiments have been conducted, and better IIR filters can be designed using the proposed method compared with existing methods in the literature. [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]