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In this work, by introducing several parameters, a new kernel function including both the homogeneous and non-homogeneous cases is constructed, and a Hilbert-type inequality related to the newly constructed kernel function is established. By convention, the equivalent Hardy-type inequality is also considered. Furthermore, by introducing the partial fraction expansions of trigonometric functions, some special and interesting Hilbert-type inequalities with the constant factors represented by the higher derivatives of trigonometric functions, the Euler number and the Bernoulli number are presented at the end of the paper.

This paper proposes an efficient method for computing selected generalized eigenpairs of a sparse Hermitian definite matrix pencil $(A,B)$. Based on Zolotarev's best rational function approximations of the signum function and conformal mapping techniques, we construct the best rational function approximation of a rectangular function supported on an arbitrary interval via function compositions with partial fraction representations. This new best rational function approximation can be applied to construct spectrum filters of $(A,B)$ with a smaller number of poles than a direct construction without function compositions. Combining fast direct solvers and the shift-invariant generalized minimal residual method, a hybrid fast algorithm is proposed to apply spectral filters efficiently. Compared to the state-of-the-art algorithm FEAST, the proposed rational function approximation is more efficient when sparse matrix factorizations are required to solve multi-shift linear systems in the eigensolver, since the smaller number of matrix factorizations is needed in our method. The efficiency and stability of the proposed method are demonstrated by numerical examples from computational chemistry.

It is proved that the sums $$ \sum_{k=1}^{n} \frac{1}{(z-a_{k})^{2}}\mspace{2mu}, \qquad \operatorname{Im}a_{k} < 0, \quad n \in \mathbb{N}, $$ are dense in all Hardy spaces $$H_{p}$$ , $$1

Mnemofunctions of the form f(x/ε), where f is the proper rational function without singularities on the real line, are considered in this article. Such mnemofunctions are called automodeling rational mnemofunctions. They possess the following fine properties: asymptotic expansions in the space of distributions can be written in explicit form and the asymptotic expansion of the product of such mnemofunctions is uniquely determined by the expansions of multiplicands.Partial fraction decomposition of automodeling rational mnemofunctions generates the so-called sloped analytical representation of a distribution, i.e. the representation of a distribution by a jump of the boundary values of the functions analytical in upper and lower half-planes. Sloped analytical representation is similar to the classical Cauchy analytical representation, but its structure is more complicated. The multiplication rule of such representations is described in this article.

We prove a partial fraction decomposition of a quotient of two functions E α ( i t x ) and I α ( i t ) which are defined in terms of the Bessel functions J α and J α + 1 of the first kind. This expansion leads naturally to the introduction of an orthonormal system with respect to the measure | x | 2 α + 1 d x 2 α + 1 Γ ( α + 1 ) in [ − 1 , 1 ] , which we call the Fourier–Dunkl system of the second kind. Euler–Dunkl polynomials E n , α ( x ) of degree n are defined by considering E α ( t x ) ∕ I α ( t ) as a generating function. It is shown that the sum ∑ m = 1 ∞ 1 ∕ j m , α 2 k , where j m , α are the positive zeros of J α , is equal (up to an explicit factor) to E 2 k − 1 , α ( 1 ) . For α = 1 ∕ 2 this leads to classical results of Euler since the function E 1 ∕ 2 ( x ) is the exponential function and E n , 1 ∕ 2 ( x ) are (essentially) the Euler polynomials. In the second part of the paper a sampling theorem of Whittaker–Shannon–Kotel’nikov type is established which is strongly related to the above-mentioned partial decomposition and which holds for all functions in the Payley–Wiener space defined by the Dunkl transform in [ − 1 , 1 ] .

It is known that for the best uniform approximation of real constants c by simple partial fractions ρn of order n on a real segment it is necessary and sufficient to have an alternance of n + 1 points on this segment for the difference ρn − c, n ≥ n0(c). We propose an algorithm for constructing the alternance and simple partial fraction of the best approximation of constants c.

Sums of products of two Gaussian q-binomial coefficients, 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.

In approximation theory, logarithmic derivatives of complex polynomials are called simple partial fractions (SPFs) as suggested by Dolzhenko. Many solved and unsolved extremal problems, related to SPFs, are traced back to works of Boole, Macintyre, Fuchs, Marstrand, Gorin, Gonchar, and Dolzhenko. Now many authors systematically develop methods for approximation and interpolation by SPFs and their modifications. Simultaneously, related problems, being of independent interest, arise for SPFs: obtaining inequalities of different metrics, estimation of derivatives, separation of singularities, etc. In introduction to this survey, we systematize some of these problems. In themain part, we formulate principal results and outline methods to prove them whenever possible.

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.

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.

We describe a method, based on the theory of Macdonald-Koornwinder polynomials, for proving bounded Littlewood identities. Our approach provides an alternative to Macdonald's partial fraction technique and results in the first examples of bounded Littlewood identities for Macdonald polynomials. These identities, which take the form of decomposition formulas for Macdonald polynomials of type (R,S) in terms Macdonald polynomials of type A, are q,t-analogues of known branching formulas for characters of the symplectic, orthogonal and special orthogonal groups, important in the theory of plane partitions. As applications of our results we obtain combinatorial formulas for characters of affine Lie algebras, Rogers-Ramanujan identities for such algebras complementing recent results of Griffin et al., and transformation formulas for Kaneko-Macdonald-type hypergeometric series.

We first establish a Hilbert-type inequality and its equivalent Hardy form with best possible constant factors by constructing a homogeneous kernel function involving hyperbolic functions. Furthermore, we introduce Bernoulli numbers and the partial fraction expansions of trigonometric functions, and then we present several special and interesting Hilbert-type inequalities, in which the constant factors are represented by Bernoulli numbers and by some higher derivatives of trigonometric functions.

The standard technique taught in calculus courses for partial fraction expansions uses undetermined coefficients to generate a system of linear equations; we present a derivative-based technique th...

This paper presents a simple algorithm for computing the cumulative distribution function of the sojourn time of a random customer in an M/GL/1 queue with bulk-service of exactly size L. Both theoretical and numerical aspects related to this problem were not discussed by Chaudhry and Templeton in their monograph (1983). Our analysis is based on the roots of the so-called characteristic equation of the Laplace-Stieltjes transform (LST) of the sojourn time distribution. Using the method of partial fractions and residue theorem, we obtain a closed-form expression of sojourn time distribution, from which we can calculate the value of the distribution function for any given time t ∈ [0, + ∞). Finally, to ensure the reliability of the analytical procedure, employing the work done by Gross and Harris (1985), an effective way to validate the correctness of our results along with some numerical examples are also provided.

Let f be a function and ρ be a simple partial fraction of degree at most n. Under linear-fractional transformations, the difference f − ρ becomes the difference of another function and a certain simple partial fraction of degree at most n with a quadratic weight. We study applications of this important property. We prove a theorem on uniqueness of interpolating simple partial fraction, generalizing known results, and obtain estimates for the best uniform approximation of certain functions on the real semi-axis ℝ+. For continuous functions of rather common type we first obtain estimates of the best approximation by differences of simple partial fractions on ℝ+. For odd functions we obtain such estimates on the whole axis ℝ.

By means of the modified Abel lemma on summation by parts, we investigate infinite series involving quadratic and cubic harmonic numbers. Several infinite series identities are established for $\pi^2$ and $\zeta(3)$ as consequences.

Classroom Capsules are short (1–3 pages) notes that contain new mathematical insights on a topic from undergraduate mathematics, preferably something that can be directly introduced into a college ...

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&rsquo, 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&rsquo, s constant, Apery&rsquo, s constant and a fast converging identity for the famous &zeta, ( 2 ) constant.

A new version of the parallel Alternating Direction Implicit (ADI) method by Peaceman and Rachford for solving systems of linear algebraic equations with positive-definite coefficient matrices represented as sums of two commuting terms is suggested. The algorithms considered are suited for solving two-dimensional grid boundary-value problems with separable variables, as well as the Sylvester and Lyapunov matrix equations. The approach to rising parallel efficiency proposed in the paper is based on representing rational functions as sums of partial fractions. An additive version of the factorized ADI method for solving Sylvester’s equation is described. Estimates of the speedup resulting from increasing the number of computer units are presented. These estimates demonstrate a potential advantage of using the additive algorithms when implemented on a supercomputer with large number of processors or cores.

In this paper, we define a finite sum analogue of multiple polylogarithms inspired by the work of Kaneko and Zaiger and prove that they satisfy a certain analogue of the shuffle relation. Our result is obtained by using a certain partial fraction decomposition due to Komori-Matsumoto-Tsumura. As a corollary, we give an algebraic interpretation of our shuffle product., 13 pages

We obtain the sharp quadrature formulas for integrals of complex rational functions over circles, segments of the real axis, and the real axis itself. Among them there are formulas for calculating the L 2-norms of rational functions. Using the quadrature formulas for rational functions, in particular, for simple partial fractions and polynomials, we derive some sharp inequalities between various metrics (Nikol’skiĭ-type inequalities).

Continuing a topic from two recent articles, we present three additional methods for evaluating a particular definite integral, using partial fractions from complex roots, an infinite series repres...

In this paper, we obtain a new identity using the partial fraction decomposition. As applications, some interesting binomial identities are also derived.

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

By means of the partial fraction decomposition method, a general reciprocal theorem on trigonometric sums is established. Several trigonometric reciprocities and summation formulae are derived as consequences.

We show that the necessary condition for the convergence of the series of simple partial fractions $$ {\displaystyle \sum_{k=1}^{\infty }{\left(z-{z}_k\right)}^{-1}} $$ in $$ Lp\left(\mathrm{\mathbb{R}}\right) $$ , 1 < p < ∞, is the convergence of the series $$ {\displaystyle \sum_{k=1}^{\infty }{\left|{z}_k\right|}^{-1/q}1{\mathrm{n}}^{-1-\varepsilon}\left(\left|{z}_k\right|+1\right)} $$ , e > 0. In the case 1 < p < 2, we obtain a convergence criterion in terms of the imaginary parts of poles under the condition that all the poles z k = x k + iy k belong to the angle |z k | ≤ α|y k | with a fixed α > 0.

Learning to compute integrals via the various techniques of integration (e.g., integration by parts, partial fractions, etc.) is difficult for many students. Here, we look at how students in a college level Calculus II course develop the ability to categorize integrals and the difficulties they encounter using a card sort-resort activity. Analysis of the data required the use of several non-standard techniques which provided interesting insights into the ways students develop categories in mathematics. One finding of note is that students may need a significant amount of time “off topic” to allow sufficient time to fully organize their schema for integration.

In the problem of approximating real functions by simple partial fractions of order on closed intervals , we obtain a criterion for the best uniform approximation which is similar to Chebyshev's alternance theorem and considerably generalizes previous results: under the same condition on the poles of the fraction of best approximation, we omit the restriction on the order of this fraction. In the case of approximation of odd functions on , we obtain a similar criterion under much weaker restrictions on the position of the poles : the disc is replaced by the domain bounded by a lemniscate contained in this disc. We give some applications of this result. The main theorems are extended to the case of weighted approximation. We give a lower bound for the distance from to the set of poles of all simple partial fractions of order which are normalized with weight on (a weighted analogue of Gorin's problem on the semi-axis).

We prove that for a continuous real-valued function f on the segment [−1, 1] a real-valued simple partial fraction R n with n distinct poles outside the unit disk is a fraction of degree at most n of best approximation and is unique if and only if for the difference f − R n on [−1, 1] there exists a Chebyshev alternance of n + 1 points. The result is applied to the problem on approximation of real constants.

We consider the problem of interpolation and best uniform approximation of constants c ≠ 0 by simple partial fractions ρ n of order n on an interval [a, b]. (All functions and numbers considered are real.) For the case in which n > 4|c|(b − a), we prove that the interpolation problem is uniquely solvable, obtain upper and lower bounds, sharp in order n, for the interpolation error on the set of all interpolation points, and show that the poles of the interpolating fraction lie outside the disk with diameter [a, b]. We obtain an analog of Chebyshev’s classical theorem on the minimum deviation of a monic polynomial of degree n from a constant. Namely, we show that, for n > 4|c|(b − a), the best approximation fraction ρ* n for the constant c on [a, b] is unique and can be characterized by the Chebyshev alternance of n+1 points for the difference ρ* n − c. For theminimum deviations, we obtain an estimate sharp in order n.

Using reduction to polynomial interpolation, we study the multiple interpolation problem by simple partial fractions. Algebraic conditions are obtained for the solvability and the unique solvability of the problem. We introduce the notion of generalized multiple interpolation by simple partial fractions of order ≤ n. The incomplete interpolation problems (i.e., the interpolation problems with the total multiplicity of nodes strictly less than n) are considered; the unimprovable value of the total multiplicity of nodes is found for which the incomplete problem is surely solvable. We obtain an order n differential equation whose solution set coincides with the set of all simple partial fractions of order ≤ n.

We obtain estimates for L p -norms of simple partial fractions in terms of their L r -norms on bounded and unbounded segments of the real axis for various p > 1 and r > 1 (S. M. Nikolskii type inequalities). We adduce examples and remarks concerning sharpness of the inequalities and area of their application.

We obtain new parametric quadrature formulas with variable nodes for integrals of complex rational functions over circles, segments of the real axis and the real axis itself. Basing on these formulas we derive $(q,p)$-inequalities of Jackson-Nikolskii type for various classes of rational functions, complex polynomials and their logarithmic derivatives (simple partial fractions). It is shown that our $(\infty,2)$- and $(\infty,4)$-inequalities are sharp in a number of main theorems. Our inequalities extend and refine several results obtained earlier by other authors., We changed the title and the structure of the paper. Several misprints were corrected

For arbitrary natural n ≥ 2 we construct an example of a real continuous function, for which there exists more than one simple partial fraction of order ≤ n of the best uniform approximation on a segment of the real axis. We prove that even the Chebyshev alternance consisting of n+1 points does not guarantee the uniqueness of the best approximation fraction. The obtained results are generalizations of known non-uniqueness examples constructed for n = 2, 3 in the case of simple partial fractions of an arbitrary order n.

Under the assumption that a certain algebraic identity holds for all $ n\in \mathbb{N} $ (it is verified for n ≤ 5), we prove that a real-valued simple partial fraction R n with n simple poles lying outside the unit disk is a simple partial fraction of degree at most n of the best uniform approximation of a continuous real-valued functions f on [−1, 1] provided that for the difference f − R n there is a Chebyshev alternance of n + 1 points on [−1, 1]. The result is applied to the problem of approximation of real constants. Bibliography: 8 titles.

The problem of the best uniform approximation of a real constant c by real-valued simple partial fractions Rn on a closed interval of the real axis is considered. For sufficiently small (in absolute value) c, |c| ≤ cn, it is proved that Rn is a fraction of best approximation if, for the difference Rn − c, there exists a Chebyshev alternance of n + 1 points on a closed interval. A criterion for best approximation in terms of alternance is stated.

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.

The paper is devoted to the expansion in partial fractions for a meromorphic function of one complex variable. It contains the results by the author on representing a meromorphic function as a sum of an entire function and the principal parts in its Laurent expansion at its poles.

When Newton's method, or Halley's method is used to approximate the $p${th} root of $1-z$, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010)., 10 pages

UDC 517.538 It is shown that an alternance consisting of less than 2n points does not guarantee best approximation by simple partial fractions of degree n. We obtain a sufficient condition for the uniqueness of a simple partial fraction of degree 3 of best approximation and propose a numerical method for verifying this condition. We also construct an example of nonuniqueness of a simple partial fraction of degree 3 of best approximation, as well as some other theoretical examples. Bibliography :4 titles.

For p ≥ 2 we obtain bounds for Lp-norms of the Fourier transform of real parts of simple partial fractions. For even p our estimate is sharp. We also prove a new inequality for Lp-norms of simple partial fractions which in some cases is stronger than the corresponding inequality obtained by V. Yu. Protasov.

We study a generalized interpolation of a rational function at n nodes by a simple partial fraction of degree n and reduce the consideration to the solvability question for a special difference equation. We construct explicit interpolation formulas in the case where the equation order is equal to 1. We show that for functions A(x − a) m , $ m \in \mathbb{N} $ , it is possible to reduce the consideration to a system of m + 1 independent first order equations and construct explicit interpolation formulas. Bibliography: 6 titles.

In this paper we provide a q-analog for a type of identity involving rational sums shown by Prodinger (Appl Anal Discrete Math 2(1):65–68, 2008). Our proof is algebraic and makes use of q-partial fractions and q-inverse pairs. A bijective proof involving a sign-changing involution is given for one of the main results.

We obtain a nontrivial estimate of the variance of the sum of bounded partial quotients appearing in the continued-fraction expansion of a rational number with fixed denominator. As a consequence, we obtain a law of large numbers for the sum of all partial quotients.

We consider a multiple interpolation by Pade simple partial fractions and propose a method for calculating the values of rational functions and polynomials on the basis of approximation by special rational functions (their numerator and denominator are represented as the differences between two simple partial fractions). We obtain an extrapolation formula for an analytic function h(z) in a neighborhood of the origin. For an extrapolation tool we use the expressions Σ k λ k h(λ k z), where λ k are calculated by a certain algorithm and are independent of the choice of h. Bibliography: 17 titles.

We obtain a sufficient condition on the number of points of a Chebyshev alternance for the uniqueness of a simple partial fraction of degree n of best uniform approximation of a real-valued function on a segment of the real axis. We discuss the case where this number is greater than n + 1 and some aspects concerning approximations of constants. Bibliography: 6 titles.

By partial fraction approach, we derive q-analog for several well known results on harmonic number sums.

A new estimate of the measure of irrationality of the number π is obtained. The previous result (M. Hata, 1993) is improved by means of another integral construction.