Electron. J. Diff. Equ.,
Vol. 2015 (2015), No. 212, pp. 116.
A matrix formulation of Frobenius power series solutions using products
of
matrices
Jeremy Mandelkern
Abstract:
In Coddington and Levison [7, p. 119, Thm. 4.1] and
Balser [4, p. 1819, 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 quasiderivatives.
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, 3403, 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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