Adam Doliwa
Rational approximation, multiple orthogonal polynomials, and integrability
I plan to discuss the connections between the three issues above. At the level of two independent variables, the structural relationships between the theory of orthogonal polynomials and the Padé approximants can be reduced to the study of the Hankel type determinants. The Frobenius identities they satisfy are a special case of the integrable discrete-time Toda equations introduced by Ryogo Hirota.
It is difficult to overestimate the role of orthogonal polynomials in theoretical physics and applied mathematics. They appear in the separation of variables of the fundamental partial differential equations of theoretical physics, representations of Lie groups and algebras and their quantum versions, random walks, numerical integration, to name a few notable examples. The spectral approach to orthogonal polynomials can be treated as a bridge between them and the theory of integrable systems, and the classical three-term recurrence relation is the spectral problem for the Toda system with continuous or discrete time.
The Hermite–Padé approximants, introduced by Hermite in his proof of the transcendence of Euler's number e, are closely related to the theory of multiple orthogonal polynomials, which has been intensively studied in recent years. It turns out that the recurrence relations, studied by Mahler and Paszkowski within the theory of multiple rational approximation, provide integrable extension of the Frobenius identities to arbitrary number of discrete variables. Their quasi-determinant generalizations can be interpreted as new non-commutative integrable discrete systems.
