First-Order Ode Systems Generating Confluent Heun Equations
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FIRST-ORDER ODE SYSTEMS GENERATING CONFLUENT HEUN EQUATIONS A. A. Salatich,∗ S. Yu. Slavyanov,† and O. L. Stesik∗
UDC 517.289, 517.923, 517.926
We study the relation between linear second-order equations that are confluent Heun equations, namely, the biconfluent and triconfluent Heun equations, and first-order linear systems of equations generating Painlev´e equations. The generation process is interpreted in physical terms as antiquantization. Technically, the study in volves manipulations with polynomials. The complexity of computations sometimes requires using computer algebra systems. Bibliography: 13 titles.
Introduction A procedure for generating nonlinear integrable Painlev´e equations from linear Heun class equations was developed by S. Yu. Slavyanov and his coauthors. At a heuristic level, this was done first in the paper [1], and then reproduced in the book [2]. Nonlinear integrable Painlev´e equations were obtained there from linear Heun class equations. The notion of “antiquantization” and more fundamental considerations appeared in the papers [3–5]. It turned out later that antiquantization is closely related to the isomonodromy condition for a first-order system of linear differential equations [6]. The derivation of the Painlev´e equation P 6 as the isomonodromy property for a first-order system of linear differential equations, as well as the history of this approach, can be found in the book [7]. The difficulty and ambiguity of this derivation is due to an “excessive” number of free coefficients of the system. Besides, it obscures the key role of the zeros of the off-diagonal matrix elements of the system, which appear in the second-order equations for components as apparent singularities. Passing from a linear system of ODEs with polynomial coefficients to second-order equations (the Schr¨ odinger equation) involves nontrivial matrix calculations, which often require using computer algebra systems. This publication continues the studies performed for the single confluent and double confluent Heun equations [8, 9]. The biconfluent Heun equation The biconfluent Heun equation in the canonical form (with polynomial equation symbol) reads as (see [2]) (1) (H(z, D) − h)w(z) := z 2 D 2 + (−z 2 − tz + c)D + (−az − h) w(z) = 0, where h is the accessory parameter and the parameters a, c characterize the local monodromy. The point z = 0 is a regular singular point, and the point z = ∞ is an irregular singular point with s-rank equal to 3. In parallel, we consider the linear system of first-order ODEs (z) = T (z) Y (z), (z) = A(z)Y Y z
(2)
A = A(1) /z + A(2) + A(3) z;
(3)
(z) is a vector with components y1 (z), y2 (z) and the matrix A(z) is chosen as where Y ∗ †
St.Petersburg State University, St.Petersburg, Russia, e-mail: [email protected], [email protected]. Deceased.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 485, 2019, pp. 187–194. Original article submitted October 3, 2019. 1072-3374/20/2513-0427 ©2020 Springer Science+Business Media, LLC 427
the determinants of the matrices A(1) , A(3) a
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