Electron. J. Diff. Equ., Vol. 2015 (2015), No. 212, pp. 1-16.

A matrix formulation of Frobenius power series solutions using products of $4 \times 4$ matrices

Jeremy Mandelkern

Abstract:
In Coddington and Levison [7, p. 119, Thm. 4.1] and Balser [4, p. 18-19, Thm. 5], matrix formulations of Frobenius theory, near a regular singular point, are given using $2\times 2$ matrix recurrence relations yielding fundamental matrices consisting of two linearly independent solutions together with their quasi-derivatives. In this article we apply a reformulation of these matrix methods to the Bessel equation of nonintegral order. The reformulated approach of this article differs from [7] and [4] by its implementation of a new ``vectorization'' procedure that yields recurrence relations of an altogether different form: namely, it replaces the implicit $2\times 2$ matrix recurrence relations of both [7] and [4] by explicit $4 \times 4$ matrix recurrence relations that are implemented by means only of $4 \times 4$ matrix products. This new idea of using a vectorization procedure may further enable the development of symbolic manipulator programs for matrix forms of the Frobenius theory.

Submitted January 12, 2015. Published August 17, 2015.
Math Subject Classifications: 34B30, 33C10, 68W30, 34-03, 01A55.
Key Words: Matrix power series; Frobenius theory; Bessel equation.

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Jeremy Mandelkern
Department of Mathematics
Eastern Florida State College
Melbourne, FL 32935, USA
email: mandelkernj@easternflorida.edu

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