On the Derivatives of the Heun Functions
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ERENTIAL EQUATIONS
On the Derivatives of the Heun Functions 1*** ´ G. Filipuk1* , A. Ishkhanyan2** , and J. Derezinski 1 2
University of Warsaw, Warsaw, Poland
Russian-Armenian University, Yerevan, Armenia
Received July 26, 2019; revised January 21, 2020; accepted February 6, 2020
Abstract—The Heun functions satisfy linear ordinary differential equations of second order with certain singularities in the complex plane. The first order derivatives of the Heun functions satisfy linear second order differential equations with one more singularity. In this paper we compare these equations with linear differential equations isomonodromy deformations of which are described by the Painleve´ equations PII − PV I . MSC2010 numbers : 33E10, 34B30, 34M55, 34M56 DOI: 10.3103/S1068362320030036 Keywords: linear ordinary differential equation, Heun function, isomonodromy deformation.
1. INTRODUCTION The general Heun equation is the most general second order linear Fuchsian ordinary differential equation with four regular singular points in the complex plane [2–5]. Although it is a genaralization of the well-studied Gauss hypergeometric equation with three regular singularities, it is much more difficult to investigate properties of the Heun functions. The additional singularity causes many complications in comparison with the hypergeometric case (for instance, the solutions in general have no integral representations involving simpler mathematical functions). There also exist confluent Heun equations (see [3, 4]) which have irregular singularities. There are many studies on the properties of solutions of the Heun equations from different perspectives (see, for instance, [6–17] and the references therein). The Heun functions (and their confluent cases) appear extensively in many problems of mathematics, mathematical physics, physics and engineering (e.g., [18–20]). An extensive bibliography can be found at [1]. The general Heun equation is given by the following equation: γ δ ε du αβz − q d2 u + + + u = 0, (1.1) + dz 2 z z − 1 z − t dz z(z − 1)(z − t) where the parameters satisfy the Fuchsian relation 1 + α + β = γ + δ + ε.
(1.2)
This equation has four regular singular points at z = 0, 1, t and ∞. Its solutions, the Heun functions, are usually denoted by u = H(t, q; α, β, γ, δ; z) assuming that ε is obtained from (1.2). The parameter q is referred to as the accessory parameter. It is well-known that the derivative of the hypergeometric function 2 F1 is again a hypergeometric function with different values of the parameters. However, for the Heun function it is generally not the case. The first order derivative of the general Heun function satisfies a second order Fuchsian differential *
E-mail: [email protected] E-mail: [email protected] *** E-mail: [email protected] **
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ON THE DERIVATIVES OF THE HEUN FUNCTIONS
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equation with five regular singular points [7, 8, 12]. It can be verified by direct computations that the function v(z) = du/dz, where u = u(z) is a solution of (1.1), satisfies the following equation:
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