Efficient recurrence relations for univariate and multivariate Taylor series coefficients

Richard D. Neidinger

doi:10.3934/proc.2013.2013.587

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2013, Volume 2013: 587-596. Doi: 10.3934/proc.2013.2013.587 open access

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Efficient recurrence relations for univariate and multivariate Taylor series coefficients

Richard D. Neidinger^1,

1.
Davidson College, Box 7002, Davidson, NC 28035-7002

Received: June 30, 2012

Revised: March 31, 2013

Published: October 2013

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Abstract

The efficient use of Taylor series depends, not on symbolic differentiation, but on a standard set of recurrence formulas for each of the elementary functions and operations. These relationships are often rediscovered and restated, usually in a piecemeal fashion. We seek to provide a fairly thorough and unified exposition of efficient recurrence relations in both univariate and multivariate settings. Explicit formulas all stem from the fact that multiplication of functions corresponds to a Cauchy product of series coefficients, which is more efficient than the Leibniz rule for nth-order derivatives. This principle is applied to function relationships of the form h'=v*u', where the prime indicates a derivative or partial derivative. Each standard (calculator button) function corresponds to an equation, or pair of equations, of this form. A geometric description of the multivariate operation helps clarify and streamline the computation for each desired multi-indexed coefficient. Several research communities use such recurrences including the Differential Transform Method to solve differential equations with initial conditions.

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Mathematics Subject Classification: Primary: 65D25, 41A58; Secondary: 41A63, 65L05, 65D15.

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