WHAT IS...A Multiple Orthogonal Polynomial?

Abstract

Multiple orthogonal polynomials (MOPs) are polynomials of one variable which satisfy orthogonality conditions with respect to several measures. They should not be confused with multi-variate of multivariable orthogonal polynomials, which are polynomials of several variables. Other terminology is also used, e.g., Hermite-Padé polynomials, polyorthogonal polynomials (Nikishin and Sorokin [15]), d-orthogonal polynomials (the latter primarily by the French-Tunisian school of Maroni, Doauk, Ben Cheikh and collaborators). They are a very useful extension of orthogonal polynomials, and recently received renewed interest because tools have become available to investigate their asymptotic behavior and they do appear in a number of interesting applications. Various families of special MOPs have been found, extending the classical orthogonal polynomials but also giving completely new special functions [1], [8, Ch. 23]. MSC Codes 33C45, 42C05, 41A21, 60B20, 11J72,