#### Title

Regular Singular Differential Equations Whose Conjugate Equation Has Polynomial Solutions

#### Editor(s)

Miller Jr., W.

#### Abstract

Consider the n -dimensional singular differential system defined by the operator $L:(Ly)(z) = z^p y'(z) + A(z)y(z)$, where z is a complex variable and p is a positive integer. The solvability of the nonhomogeneous system $Ly = g$ depends on the solutions of the homogeneous conjugate system, $L^ * f = 0$, where $L^ * $ is the operator conjugate to L. We show that $L^ * f = 0$ has polynomial solutions if the constant matrix in the series expansion of $A(z)$ has at least one nonpositive integer eigenvalue. Also, we show that if $L^ * f = 0$ has a polynomial solution, then a finite number of the coefficients of $A(z)$ must satisfy certain properties. These results are then used to obtain a solvability condition for the nonhomogeneous Bessel equation of integer order.

#### Recommended Citation

L.
M.
Hall,
"Regular Singular Differential Equations Whose Conjugate Equation Has Polynomial Solutions," *SIAM Journal on Mathematical Analysis*, Society for Industrial and Applied Mathematics (SIAM), Jan 1977.

The definitive version is available at https://doi.org/10.1137/0508060

#### Department(s)

Mathematics and Statistics

#### Document Type

Article - Journal

#### Document Version

Citation

#### File Type

text

#### Language(s)

English

#### Rights

© 1977 Society for Industrial and Applied Mathematics (SIAM), All rights reserved.