Integrals of Legendre Polynomials and Solution of Some Partial Differential Equations

Abstract. We show a connection between the polynomials whose in-flection points coincide with their interior roots (let us write shorter PIPCIR), Legendre polynomials, and Jacobi polynomials, and study some properties of PIPCIRs (Part I). In addition, we give new formulas for some classical orthogonal polynomials. Then we use PIPCIRs to solve some partial differential equations (Part II). 1. Part I. Properties of PIPCIRs 1.1. Relation to classical polynomials. Since translating all the roots an equal amount or multiplying a polyno-mial by a constant will not affect the position of the roots relative to any critical or inflection points, we restricted our attention to a polynomial with the first and last roots at x = ±1, given by Qn(x) = (1 − x2)qn−2(x), n ≥ 2. (1)