VI.—Generating Functions of Certain Continuous Orthogonal Systems

Summary10. Generating functions and bilinear generating functions (of the type of Mehler's celebrated formula) are known to be of great importance in the formal theory of orthogonal sequences. The present paper contains analogous formulae for a number of continuous orthogonal systems as well as “mixed” systems (which have a point spectrum as well as a continuous one). Four systems of the hypergeometric type have been selected as examples which are thought to be of some importance because of their presenting themselves in certain problems of Mathematical Physics.My thanks are due to the Carnegie Trust for the Universities of Scotland for grants towards the printing of this paper and my paper in Proceedings, vol. lx, no. 26, 1940.

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Combinatorics of KP hierarchy structural constants

The European Physical Journal C ◽

10.1140/epjc/s10052-021-09899-8 ◽

2021 ◽

Vol 81 (12) ◽

Author(s):

A. Andreev ◽

A. Popolitov ◽

A. Sleptsov ◽

A. Zhabin

Keyword(s):

Initial Data ◽

Integral Representation ◽

Internal Structure ◽

Generating Functions ◽

Transport Coefficients ◽

Mathematical Physics ◽

Combinatorial Structure ◽

Kp Hierarchy ◽

Topological Recursion ◽

Structure Research

AbstractWe investigate the structural constants of the KP hierarchy, which appear as universal coefficients in the paper of Natanzon–Zabrodin arXiv:1509.04472. It turns out that these constants have a combinatorial description in terms of transport coefficients in the theory of flow networks. Considering its properties we want to point out three novel directions of KP combinatorial structure research: connection with topological recursion, eigenvalue model for the structural constants and its deformations, possible deformations of KP hierarchy in terms of the structural constants. Firstly, in this paper we study the internal structure of these coefficients which involves: (1) construction of generating functions that have interesting properties by themselves; (2) restrictions on topological recursion initial data; (3) construction of integral representation or matrix model for these coefficients with non-trivial Ward identities. This shows that these coefficients appear in various problems of mathematical physics, which increases their value and significance. Secondly, we discuss their role in integrability of KP hierarchy considering possible deformation of these coefficients without changing the equations on $$\tau $$ τ -function. We consider several plausible deformations. While most failed even very basic checks, one deformation (involving Macdonald polynomials) passes all the simple checks and requires more thorough study.

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