Summation formulas of euler-maclaurin and abel-plana: Old and new results and applications

Abstract

Summation formulas of the Euler-Maclaurin and Abel-Plana and their connections with several kinds of quadrature rules are studied. Besides the history of these formulas, several of their modifications and generalizations are considered. Connections between the Euler-Maclaurin formula and basic quadrature rules of Newton-Cotes type, as well as the composite Gauss-Legendre rule and its Lobatto modification are presented. Besides the basic Plana summation formula a few integral modifications (the midpoint summation formula, the Binet formula, Lindelöf formula) are introduced and analysed. Starting from the moments of their weight functions and applying the recent MATHEMATICA package Orthogona lPolynomials, recursive coefficients in the three-term recurrence relation for the corresponding orthogonal polynomials are constructed, as well as the parameters (nodes and Christoffel numbers) of the corresponding Gaussian quadrature formula.