What is $\ldots\ $ a multiple orthogonal polynomial?

TL;DR

This paper explains the concept of multiple orthogonal polynomials and discusses their applications across various fields, providing a comprehensive overview of their theoretical foundations and practical relevance.

Contribution

It offers an extended, detailed explanation of multiple orthogonal polynomials and highlights their significance in different mathematical and applied contexts.

Findings

Clarifies the definition and properties of multiple orthogonal polynomials

Identifies key applications in approximation theory and random matrix models

Provides insights into the theoretical and practical importance of these polynomials

Abstract

This is an extended version of our note in the Notices of the American Mathematical Society 63 (2016), no. 9, in which we explain what multiple orthogonal polynomials are and where they appear in various applications.