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

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.

An addendum was posted on May 11, 2017, It modifies two matrices from Section 7. See the last page of this article.

Show me the PDF file (284 KB), TEX file, and other files for this article.

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

Return to the EJDE web page