A new approach to the construction of generalized classical polynomials - Makarov, V. L. (Ukrains’kyi Matematychnyi Zhurnal, 2021)
UDC 517.587In this paper, we develop a new method for constructing generalized classical polynomials, primarily Hermite polynomials in the sense of A. Krall, J. Koekoek, R. Koekoek, H. Bavinck, L. Littlejohn, et al. We construct a differential operator of infinite order whose eigenfunctions are such polynomials. For generalized Hermite polynomials, we investigate a number of properties inherent in classical orthogonal polynomials (orthogonality, generalized Rodrigues formula, three-term recurrence relation forming a function). The versatility of the method is revealed in constructing generalized Legendre and Chebyshev polynomials of the first kind.