Abstract
A new approach for investigating polynomial solutions of differential equations is proposed. It is based on elementary linear algebra. Any differential operator of the form L ( y ) = ∑ k = 0 k = N a k ( x ) y ( k ) , where a k is a polynomial of degree ≤ k, over an infinite field F has all eigenvalues in F in the space of polynomials of degree at most n, for all n. If these eigenvalues are distinct, then there is a unique monic polynomial of degree n which is an eigenfunction of the operator L, for every non-negative integer n. Specializing to the real field, the potential of the method is illustrated by recovering Bochner's classification of second order ODEs with polynomial coefficients and polynomial solutions, as well as cases missed by him - namely that of Romanovski polynomials, which are of recent interest in theoretical physics, and some Jacobi type polynomials. An important feature of this approach is the simplicity with which the eigenfunctions and their orthogonality and norms can be determined, resulting in significant reduction in computational complexity of such problems. 2000 MSC: 33C45; 34A05; 34A30; 34B24.
Highlights
Polynomial solutions of differential equations is a classical subject, going back to Routh [1], Bochner [2] and Brenke [3] and it continues to be of interest in applications, as in, e.g., [4,5]
The idea we wish to present in this article is to conduct the discussion of differential equations with polynomial coefficients in a linear algebraic context
N k=0 ak(x)y(k), where ak is a polynomial of degree ≤ k, with coefficients in an infinite field F
Summary
Polynomial solutions of differential equations is a classical subject, going back to Routh [1], Bochner [2] and Brenke [3] and it continues to be of interest in applications, as in, e.g., [4,5]. It is surprising that by such a change of view point, one can add more than what is available in the classical literature and, at the same time, recover classical results efficiently and in a unified manner We take this opportunity to correct a common misconception regarding Brenke’s contributions in the classification of second-order ODEs that have polynomial solutions [6, p. Specializing to second-order equations because of their importance in applications– and leaving in this article the higher-order case because of its technical complexity–the canonical forms of second-order equations, their eigenvalues, and multiplicities are investigated This includes the family of Romanovski polynomials and some Jacobitype polynomials, which are missing in the classification of Brenke and Bochner as well as in the latest books on the subject; the Romanovski polynomials are the main subject of some recent physics literature [5,7]. L is said to be simple if its algebraic multiplicity is equal to 1, semisimple if its algebraic and geometric multiplicities coincide, and defective if its algebraic multiplicity is greater than its geometric multiplicity
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