Miller Jr., W.
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.
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
Mathematics and Statistics
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