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
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
matrix recurrence relations of both [7] and
[4] 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 email: mandelkernj@easternflorida.edu |
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