On discrete inequalities of Wirtinger's type

AbstractDiscrete inequalities of Wirtinger's type are considered. Constants in the obtained inequalities are the best ones. In the special case the inequalities (1) and (2) are obtained. They are proved by K. Fan, O. Taussky, and J. Todd: Discrete analogs of inequalities of Wirtinger, Montash. Math. 59 (1955), 73–79

Weighted quadrature formulas for semi-infinite range integrals

Weighted quadrature formulas on the half line $(a,+\infty)$, $a>0$, for non-exponentially decreasing integrands are developed. Such $n$-point quadrature rules are exact for all functions of the form $x\mapsto x^{-2}P(x^{-1})$, where $P$ is an arbitrary algebraic polynomial of degree at most $2n-1$. In particular, quadrature formulas with respect to the weight function $x\mapsto w(x)=x^\beta\log^m x$ ($0\le \beta<1$, $m\in \mathbb{N}_0$) are considered and several numerical examples are included

An efficient computation of parameters in the Rys quadrature formula

We present an efficient procedure for constructing the so-called Gauss-Rys quadrature formulas and the corresponding polynomials orthogonal on (−1, 1) with respect to the even weight function w(t; x) = exp(−xt2), where x a positive parameter. Such GaussRys quadrature formulas were investigated earlier e.g. by M. Dupuis, J. Rys, H.F. King [J. Chem. Phys. 65 (1976), 111 − 116; J. Comput. Chem. 4 (1983), 154 − 157], D.W. Schwenke [Comput. Phys. Comm. 185 (2014), 762 − 763], and B.D. Shizgal [Comput. Theor. Chem. 1074 (2015), 178 − 184], and were used to evaluate electron repulsion integrals in quantum chemistry computer codes. The approach in this paper is based to a transformation of quadratures on (−1, 1) with N nodes to ones on (0, 1) with only [(N + 1)/2] nodes and their construction. The method of modified moments is used for getting recurrence coefficients. Numerical experiments are included.Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles. Sciences mathématiques. - 43, 151 (2018

Summation of series and Gaussian quadratures

Abstract. In 1985, Gautschi and the author constructed Gaussian quadrature formulae on (0, +∞) involving Einstein and Fermi functions as weights and applied then to the summation of slowly convergent series which can be represented in terms of the derivative of a Laplace transform, or in terms of the Laplace transform itself. A problem that may arise in this procedure is the determination of the respective inverse Laplace transform. For the class of slowly convergent series and R(s) is a rational function, Gautschi recently solved this problem. In the present paper, using complex integration and constructing Gauss-Christoffel quadratures on (0, +∞) with respect to the weight functions w 1 (t) = 1/ cosh 2 t and w 2 (t) = sinh t/ cosh 2 t, we reduce the and w 2 , respectively. We illustrate this method with a few numerical examples

A NEW PROOF OF A REDUCTION FORMULA FOR THE APPELL SERIES F3 DUE TO BAILEY

In this short note, we provide a new proof of an interesting and useful reduction formula for the Appell series $F_{3}$  due to Bailey [{\it On the sum of a terminating ${}_3F_2(1)$}, Quart. J. Math. Oxford Ser. (2) {\bf4} (1953), 237--240]

On Drazin inverse of operator matrices

AbstractIn this short paper, we offer (another) formula for the Drazin inverse of an operator matrix for which certain products of the entries vanish. We also give formula for the Drazin inverse of the sum of two operators under special conditions

A note on an error bound of Gauss-Turán quadrature with the Chebyshev weight

In two BIT papers error expansions in the Gauss and Gauss-Turan quadrature formulas with the Chebyshev weight function of the first kind, in the case when integrand is an analytic function in some region of the complex plane containing the interval of integration in its interior, have been obtained. On the basis of that, using a representation of the remainder term in the form of contour integral over confocal ellipses, the upper bound of the modulus of the remainder term, in the cases when certain parameter s (s є N0) takes the specific values s = 0,1,2, has been obtained. Its form for a general s (s є N0) has been supposed in one of the mentioned papers. Here, we prove that formula

EVALUATION OF A NEW CLASS OF EULERIAN'S TYPE INTEGRALS INVOLVING GENERALIZED HYPERGEOMETRIC FUNCTIONS

Very recently Masjed-Jamei and Koepf [Axioms 2018, 7 (2), 38] established interesting and useful generalizations of various classical summation theorems for the${}_{2}F_{1}$, ${}_{3}F_{2}$, ${}_{4}F_{3}$, ${}_{5}F_{4}$ and ${}_{6}F_{5}$ generalized hypergeometric series.The main aim of this paper is to establish eleven Eulerian's type integrals involving generalized hypergeometric functions by employing these theorems. Several special cases (known and unknown) have also been given

Quadrature with multiple nodes, power orthogonality, and moment-preserving spline approximation, part ii

The paper deals with new contributions to the theory of the Gauss quadrature formulas with multiple nodes that are published after 2001, including numerical construction, error analysis and applications. The first part was published in Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials (W. Gautschi, F. Marcellan, and L. Reichel, eds.) [J. Comput. Appl. Math. 127 (2001), no. 1-2, 267-286]