Symbolic Approach to 2-Orthogonal Polynomial Solutions of a Third Order Differential Equation

Abstract

In a recent work, a generic differential operator on the vectorial space of polynomial functions was presented and applied in the study of differential relations fulfilled by polynomial sequences either orthogonal or 2-orthogonal. Considering a third order differential operator that does not increase the degree of polynomials, we search for polynomial eigenfunctions with the help of symbolic computations, assuming that those polynomials constitute a 2-orthogonal polynomial sequence. Two examples are extensively described.

A governing equation of Rossby waves and its dynamics evolution by Bilinear neural network method

“…To derive the exact solutions of nonlinear equations, many methods have been raised, for example, the Hirota bilinear transformation [44][45][46][47], symbolic computation approach [48,49], Darboux transformation [50,51], Bäcklund transformation [52,53], and so on. A new method incorporates virtually all the above methods and is a general approach to search the exact solutions of NLPDEs.…”

On a General Family of 2-Orthogonal Polynomial Eigenfunctions of a Third Order Differential Equation via Symbolic Computation

“…

The search for 2-orthogonal polynomial eigenfunctions, with respect to a third order differential operator that does not increase the degree of polynomials, was recently developed in [23] by means of a symbolic approach. This work allowed us to establish some impossible cases as also to present a few families of such 2-orthogonal polynomial sequences.

…”

“…This work allowed us to establish some impossible cases as also to present a few families of such 2-orthogonal polynomial sequences. In this paper, we apply the symbolic setup proposed in [23] in order to enlighten us about further 2-orthogonal polynomial solutions of this problem. Concerning a general family inhere described, it is also proved its Hahn-classical character.…”

“…In this paper, as in [23], we have adopted a symbolic method that aims to identify the recurrence coefficients of 2-orthogonal polynomial sequences that are eigenfunctions of a given third-order differential operator of a generic form as described in [22]. The results presented along this paper are detailed for each operator and also, several functional identities are developed.…”

“…This paper has the following structure: in Sect. 2 we provide the basic concepts, the most common notations and we list the results given in [22] and [23] that are required for the sequel. Section 3 is dedicated to the description of the general family found by means of the symbolic approach, and further features of the 2-orthogonal polynomials found are depicted.…”