Orthogonal polynomial eigenfunctions of second-order partial differerential equations

Abstract

Abstract. In this paper, we show that for several second-order partial differential equationswhich have orthogonal polynomial eigenfunctions, these polynomials can be expressed as a product of two classical orthogonal polynomials in one variable. This is important since, otherwise, it is very difficult to explicitly find formulas for these polynomial solutions. From this observation and characterization, we are able to produce additional examples of such orthogonal polynomials together with their orthogonality that widens the class found by H. L. Krall and Sheffer in their seminal work in 1967. Moreover, from our approach, we can answer some open questions raised by Krall and Sheffer.

Two-variable -1 Jacobi polynomials

“…The defining formula of the two-variable Big −1 Jacobi polynomials (1) is reminiscent of the expressions found in [15] for the Krall-Sheffer polynomials [14], which, as shown in [11], are directly related to two-dimensional superintegrable systems on spaces with constants curvature (see [17] for a review of superintegrable systems). The polynomials J n,k (x, y) do not belong to the Krall-Sheffer classification, as they will be seen to obey first order differential equations with reflections.…”

Centrally symmetric orthogonal polynomials and second order partial differential equations

“…Allowing complex linear change of variables, Krall and Sheffer [5] found only the equations (2.11) and (2.13). We now give the explicit form of OPS {^n}^ of solutions to each of the equations (2.7) ~ (2.13)(see [2]).…”

Tridiagonal Operators and Zeros of Polynomials in Two Variables

“…Furthermore, the properties of these polynomials or these operators are investigated. This is done, for example, in [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In other studies, as, for example, in [23], weight functions are constructed for certain orthogonal polynomials.…”