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 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  and  by its implementation of a new ``vectorization'' procedure that yields recurrence relations of an altogether different form: namely, it replaces the implicit matrix recurrence relations of both  and  by explicit matrix recurrence relations that are implemented by means only of 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.
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| Jeremy Mandelkern |
Department of Mathematics
Eastern Florida State College
Melbourne, FL 32935, USA
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