Polynomial solutions to second order linear homogeneous ordinary differential equations. Properties and approximation

Accurate computation of the zeros of the generalized Bessel polynomials

“…The method we used to construct the algorithm presented in this paper is the one introduced in [35] and improved on in [36]. It approximates polynomial solutions to a second-order differential equation of the type…”

“…In the two following proofs we shall need results proved in [35] and [36]. They refer to the general case of the differential equation (4.1) and to the associated system of nonlinear equations (4.2).…”

“…Also conjectures based on tests we carried out by our algorithm are stated. Section 8 contains the proofs of the previously stated new results, along with some other results that we report from [35] and [36] and that we include for the sake of completeness. Conclusions are drawn in Sect.…”

“…Several methods and respective algorithms are applied to the problem and compared with each other. Finally, it is shown that the only algorithm able to yield accurate results is the one we derive from the method for finding polynomial solutions to differential equations presented in [35] and studied further in [36].…”

Polynomial solutions for a class of second-order linear differential equations

“…where y 0 (x) = 1, y 1 (x) = a 1,0 x + a 1,1 . In terms of the hypergeometric functions, the polynomial solutions in the case of a 1,0 a 2,1 − a 2,0 a 1,1 = 0 and a 2,1 > 0 are y n a 2,0 a 2,1 0 a 1,0 a 1,1 x = (−1) n a n 2,1 a 1,0 a 2,1 − a 2,0 a 1,1 a 2,0 a 2,1 n 2 F 1 (−n, n − 1 + a 1,0 a 2,0 ; a 1,0 a 2,1 − a 2,0 a 1,1 a 2,0 a 2,1 ; a 2,0 a 2,1 x + 1) (33) which can be easily obtained from equation (16) as a limit case of a 2,2 → 0, while for a 1,0 a 2,1 − a 2,0 a 1,1 = 0, the polynomial solution is simplified to…”