Efficient recurrence relations for univariate and multivariate Taylor series coefficients

Pages: 587 - 596, Issue special, November 2013

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Richard D. Neidinger - Davidson College, Box 7002, Davidson, NC 28035-7002, United States (email)

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.

Keywords:  Series, automatic differentiation, recurrence, multivariable series, differential transform method.
Mathematics Subject Classification:  Primary: 65D25, 41A58; Secondary: 41A63, 65L05, 65D15.

Received: July 2012;      Revised: April 2013;      Published: November 2013.