In the first part of our paper we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painlevé I-VI equations. Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional nonlogarithmic singularity. We show that there is a direct relationship between deformed Heun class equations and all Painlevé equations. In particular, Painlevé equations can be also divided into 5 supertypes, and subdivided into 10 types. This relationship is not so easy to describe in a completely unified way, because the choice of the ''time variable'' may depend on the type. We describe unified treatments for several possible ''time variables''.
We show that the q-Heun equation and its variants appear in the linear q-difference equations associated to some q-Painlevé equations by considering the blow-up associated to their initial-value spaces. We obtain the firstly degenerated Ruijsenaars-van Diejen operator from the linear q-difference equation associated to the q-Painlevé equation of type $E_8$.
We relate two parameter solutions of the sixth Painlevé equation and finite-gap solutions of the Heun equation by considering monodromy on a certain class of Fuchsian differential equations. In the appendix, we present formulae on differentials of elliptic modular functions, and obtain the ellitic form of the sixth Painlevé equation directly.
We show that there is a full correspondence between the parameters space of the degenerate biconfluent Heun connection (BHC) and that of Painlevé IV that admits special solutions. The BHC degenerates when either the Stokes' data for the irregular singularity at $\infty$ degenerates or the regular singular point at the origin becomes an apparent singularity. We show that if the BHC is written as isomonodromy family of biconfluent Heun equations (BHE), then the BHE degenerates precisely when it admits eigen-solutions of the biconfluent Heun operators, after choosing appropriate accessory paramet…
It is known that the Painleve VI is obtained by connection preserving deformation of some linear differential equations, and the Heun equation is obtained by a specialization of the linear differential equations. We inverstigate degenerations of the Ruijsenaars-van Diejen difference opearators and show difference analogues of the Painleve-Heun correspondence.
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 Painlevé equations $P_{II}-P_{VI}$.
In this paper, we consider the monodromy and, in particularly, the isomonodromy sets of accessory parameters for the Heun class equations. We show that the Heun class equations can be obtained as limits of the linear systems associated with the Painlevé equations when the Painlevé transcendents go to one of the actual singular points of the linear systems. While the isomonodromy sets of accessory parameters for the Heun class equations are described by the Taylor or Laurent coefficients of the corresponding Painlevé functions, or the associated tau functions, at the positions of the critical v…
Heun's equation naturally appears as special cases of Fuchsian system of differential equations of rank two with four singularities by introducing the space of initial conditions of the sixth Painlevé equation. Middle convolutions of the Fuchsian system are related with an integral transformation of Heun's equation.
We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun's differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that corresponds to Picard's solution of the sixth Painlevé equation, and to Heun's equation.
We describe the close connection between the linear system for the sixth Painlevé equation and the general Heun equation, formulate the Riemann-Hilbert problem for the Heun functions and show how, in the case of reducible monodromy, the Riemann-Hilbert formalism can be used to construct explicit polynomial solutions of the Heun equation.
A $q$-deformation of the middle convolution was introduced by Sakai and Yamaguchi. We apply it to a linear $q$-difference equation associated with the $q$-Painlevé VI equation. Then we obtain integral transformations. We investigate the $q$-middle convolution in terms of the affine Weyl group symmetry of the $q$-Painlevé VI equation. We deduce an integral transformation on the $q$-Heun equation.
The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of ``classical" weights multiplied by suitable ``deformation factors", usually dependent on a ``time variable'' $t$. From ladder operators one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlevé and related functions appear as the residues of these rational functions. We will be interested in the situation when $n$, the order of the Hankel matrix and also t…
It is known that the Fuchsian differential equation which produces the sixth Painlevé equation corresponds to the Fuchsian differential equation with different parameters via Euler's integral transformation, and Heun's equation also corresponds to Heun's equation with different parameters, again via Euler's integral transformation. In this paper we study the correspondences in detail. After investigating correspondences with respect to monodromy, it is demonstrated that the existence of polynomial-type solutions corresponds to apparency (non-branching) of a singularity. For the elliptical repr…
Several results on Heun's equation are generalized to a certain class of Fuchsian differential equations. Namely, we obtain integral representations of solutions and develop Hermite-Krichever Ansatz on them. In particular, we investigate linear differential equations that produce Painlevé equation by monodromy preserving deformation and obtain solutions of the sixth Painlevé equation which include Hitchin's solution. The relationship with finite-gap potential is also discussed.
Several results including integral representation of solutions and Hermite-Krichever Ansatz on Heun's equation are generalized to a certain class of Fuchsian differential equations, and they are applied to equations which are related with physics. We investigate linear differential equations that produce Painlevé equation by monodromy preserving deformation and obtain solutions of the sixth Painlevé equation which include Hitchin's solution. The relationship with finite-gap potential is also discussed. We find new finite-gap potentials. Namely, we show that the potential which is written as th…
The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heun-to-hypergeometric reductions are similar to classical hypergeometric identities, but the conditions for the existence of a reduction involve features of the Heun equation that the hypergeometric equation does not possess; namely, its cross-ratio and accessory parameters. The reductions include quadratic and cubic transformations, which may be performed only if the singular points of the Heun equation form a harmonic or an equianharmoni…
We consider some properties of double confluent Heun equation related to the Josephson Effect. In particular, we prove that adjacency points of phased-locked areas on a parameter plane can be described via poles of Bessel solution of Painleve 3 equation.
We derive the most general families of differential operators of first and second degree semi-commuting with the differential operators of the Heun class. Among these families we classify all those families commuting with the Heun class. In particular, we discover that a certain generalized Heun equation commutes with the Heun differential operator allowing us to construct the general solution to a complicated fourth order linear differential equation with variable coefficients which Maple 16 cannot solve.
Classical Virasoro conformal blocks are believed to be directly related to accessory parameters of Floquet type in the Heun equation and some of its confluent versions. We extend this relation to another class of accessory parameter functions that are defined by inverting all-order Bohr-Sommerfeld periods for confluent and biconfluent Heun equation. The relevant conformal blocks involve Nagoya irregular vertex operators of rank 1 and 2 and conjecturally correspond to partition functions of a 4D $\mathcal{N}=2$, $N_f=3$ gauge theory at strong coupling and an Argyres-Douglas theory.
We consider the rational Heun operator defined as the most general second-order $q$-difference operator which sends any rational function of type $[(n-1)/n]$ to a rational function of type $[n/(n+1)]$. We shall take the poles to be located on the Askey-Wilson grid. It is shown that this operator is related to the one-dimensional degeneration of the Ruijsenaars-van Diejen Hamiltonians. The Wilson biorthogonal functions of type ${_{10}}Φ_9$ are found to be solutions of a generalized eigenvalue problem involving rational Heun operators of the special "classical" kind.
Since the advent of quantum mechanics different approaches to find analytical solutions of the Schrödinger equation have been successfully developed. Here we follow and generalize the approach pioneered by Natanzon and others by which the Schrödinger equations can be transformed into another well-known equation for transcendental function (e.g., the hypergeometric equation). This sets a class of potentials for which this transformation is possible. Our generalization consists in finding potentials allowing the transformation of the Schrödinger equation into a triconfluent Heun equation. We fin…
The well-known Heun equation has the form: Q(z)S''(z)+P(z)S'(z)+V(z)S(z)=0 where Q(z) is a cubic complex polynomial, P(z) and V(z) are polynomials of degrees at most 2 and 1 resp. One of the classical problems about the Heun equation is for a given positive integer N to find all possible linear polynomials V(z) such that the latter equation has a polynomial solution S(z) of degree N. Below we prove a conjecture of the 2nd author claiming that the union of roots of such V(z)'s for a given N tends when N->oo to a certain compact connecting the three roots of Q(z) and given by the condition that …
The classical Heun equation has the form {Q(z) d^2/dz^2 +P(z) d/dz +V(z)}S(z)=0 where Q(z) is a cubic, P(z) at most quadratic and V(z) linear polynomials resp. In the second half of the 19-th century E.Heine and T.STieltjes initiated the study of the set of all V(z) such that the above equation has a polynomial solution S(z) of a given degree n. The main goal of the present paper is to study the union of the roots of the latter set of V(z)*s when n->oo. We formulate an intriguing conjecture of K.Takemura describing the limiting set and give a substantial amount of additional information.
We perform a detailed study of a class of irregular correlators in Liouville Conformal Field Theory, of the related Virasoro conformal blocks with irregular singularities and of their connection formulae. Upon considering their semi-classical limit, we provide explicit expressions of the connection matrices for the Heun function and a class of its confluences. Their calculation is reduced to concrete combinatorial formulae from conformal block expansions.
The present article discusses the two point connection problem for Heun's differential equation. We employ a contour integral method to derive connection matrices for a sub-class of the Heun equation containing 3 free parameters. Explicit expressions for the connection coefficients are found.
We introduce a nine-parameter Heun-type differential equation and obtain three classes of its solutions as series of square integrable functions written in terms of the Jacobi polynomial. The expansion coefficients of the series satisfy three-term recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Some of these are well-known polynomials while others are either new or modified versions of known ones.
We express discrete Painlevé equations as discrete Hamiltonian systems. The discrete Hamiltonian systems here mean the canonical transformations defined by generating functions. Our construction relies on the classification of the discrete Painlevé equations based on the surface-type. The discrete Hamiltonians we obtain are written in the logarithm and dilogarithm functions.
Heun differential equations are the most general second order Fuchsian equations with four regular singularities. An explicit integral series representation of Heun functions involving only elementary integrands has hitherto been unknown and noted as an important open problem in a recent review. We provide explicit integral representations of the solutions of all equations of the Heun class: general, confluent, bi-confluent, doubly-confluent and triconfluent, with integrals involving only rational functions and exponential integrands. All the series are illustrated with concrete examples of us…
In this paper classical solutions of the degenerate fifth Painlevé equation are classified, which include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions. Solutions of the degenerate fifth Painlevé equation are known to expressible in terms of the third Painlevé equation. Two applications of these classical solutions are discussed, deriving exact solutions of the complex sine-Gordon equation and of the coefficients in the three-term recurrence relation associated with generalised Charlier polynomials.
We present a constructive procedure to obtain the critical behavior of Painleve' VI transcendents and solve the connection problem. This procedure yields two and one parameter families of solutions, including trigonometric and logarithmic behaviors, and three classes of solutions with Taylor expansion at a critical point.
The critical behavior of a three real parameter class of solutions of the sixth Painlevé equation is computed, and parametrized in terms of monodromy data of the associated $2\times 2$ matrix linear Fuchsian system of ODE. The class may contain solutions with poles accumulating at the critical point. The study of this class closes a gap in the description of the transcendents in one to one correspondence with the monodromy data. These transcendents are reviewed in the paper. Some formulas that relate the monodromy data to the critical behaviors of the four real (two complex) parameter class of…
We prove the meromorphy of solutions for a wide class of ordinary differential equations. These equations are given by invariant manifolds of non-linear partial differential equations integrable by the inverse scattering method. Some higher analogues of the Painlev\' e equations are considered as examples.
Pull-back transformations between Heun and Gauss hypergeometric equations give useful expressions of Heun functions in terms of better understood hypergeometric functions. This article classifies, up to Mobius automorphisms, the coverings P1-to-P1 that yield pull-back transformations from hypergeometric to Heun equations with at least one free parameter (excluding the cases when the involved hypergeometric equation has cyclic or dihedral monodromy). In all, 61 parametric hypergeometric-to-Heun transformations are found, of maximal degree 12. Among them, 28 pull-backs are compositions of smalle…
In this paper some open problems for Painlevé equations are discussed. In particular the following open problems are described: (i) the Painlevé equivalence problem; (ii) notation for solutions of the Painlevé equations; (iii) numerical solution of Painlevé equations; and (iv) the classification of properties of Painlevé equations.
We find a new class of algebraic geometric solutions of Heun's equation with the accessory parameter belonging to a hyperelliptic curve. Dependence of these solutions from the accessory parameter as well as their relation to Heun's polynomials is studied. Methods of calculating the algebraic genus of the curve, and its branching points, are suggested. Monodromy group is considered. Numerous examples are given.
In this paper we present a general scheme for how to relate differential equations for the recurrence coefficients of semi-classical orthogonal polynomials to the Painlevé equations using the geometric framework of the Okamoto Space of Initial Conditions. We demonstrate this procedure in two examples. For semi-classical Laguerre polynomials appearing in \cite{HC17}, we show how the recurrence coefficients are connected to the fourth Painlevé equation. For discrete orthogonal polynomials associated with the hypergeometric weight appearing in \cite{FVA18} we discuss the relation of the recurrenc…
The paper is an essentially extended version of the work math.CA/0601371, supplemented with an application. We present new results in the theory of classical $θ$-functions of Jacobi and $σ$-functions of Weierstrass: ordinary differential equations and series expansions. We also give the extension of canonical $θ$-functions and consider an application to the sixth Painlevé equation (P6). Picard--Hitchin's general solution of P6 is represented explicitly in a form of logarithmic derivative of a corresponding $τ$-function (Painlevé's form).
We show that the topological recursion for the (semi-classical) spectral curve of the first Painlevé equation $P_{\rm I}$ gives a WKB solution for the isomonodromy problem for $P_{\rm I}$. In other words, the isomonodromy system is a quantum curve in the sense of [Dumitrescu O., Mulase M., Lett. Math. Phys. 104 (2014), 635-671, arXiv:1310.6022] and [Dumitrescu O., Mulase M., arXiv:1411.1023].
A class of special solutions are constructed in an intuitive way for the ultradiscrete analog of $q$-Painlevé II ($q$-PII) equation. The solutions are classified into four groups depending on the function-type and the system parameter.
We introduce a notion of the divisor type for rational functions and show that it can be effectively used for the classification of the deformations of dessins d'enfants related with the construction of the algebraic solutions of the sixth Painlevé equation via the method of $RS$-transformations.
We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semi-classical Laguerre weight and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions which arise in the description of special function solutions of the fourth Painlevé equation.
One of the authors has recently introduced the concept of conjugate Hamiltonian systems: the solution of the equation $h=H(p,q,t),$ where $H$ is a given Hamiltonian containing $t$ explicitly, yields the function $t=T(p,q,h)$, which defines a new Hamiltonian system with Hamiltonian $T$ and independent variable $h.$ By employing this construction and by using the fact that the classical Painlevé equations are Hamiltonian systems, it is straightforward to associate with each Painlevé equation two new integrable ODEs. Here, we investigate the conjugate Painlevé II equations. In particular, for the…
We show that the coefficients of the three-term recurrence relation for orthogonal polynomials with respect to a semi-classical extension of the Laguerre weight satisfy the fourth Painlevé equation when viewed as functions of one of the parameters in the weight. We compare different approaches to derive this result, namely, the ladder operators approach, the isomonodromy deformations approach and combining the Toda system for the recurrence coefficients with a discrete equation. We also discuss a relation between the recurrence coefficients for the Freud weight and the semi-classical Laguerre …
A class of classical solutions to the $q$-Painlevé equation of type $(A_1+A_1')^{(1)}$ (a $q$-difference analog of the Painlevé II equation) is constructed in a determinantal form with basic hypergeometric function elements. The continuous limit of this $q$-Painlevé equation to the Painlevé II equation and its hypergeometric solutions are discussed. The continuous limit of these hypergeometric solutions to the Airy function is obtained through a uniform asymptotic expansion of their integral representation.
We will study special solutions of the fourth, fifth and sixth Painlevé equations with generic values of parameters whose linear monodromy can be calculated explicitly. We will show the relation between Umemura's classical solutions and our solutions.
We find a class of solutions of the sixth Painleve' equation appearing in the theory of WDVV equations. This class covers almost all the monodromy data associated to the equation, except one point in the space of the data. We find the critical behavior close to the critical points in terms of two parameters and solve the connection problem. We also study the critical behavior of Painleve' transcendents in the Elliptic representation.
We will study two types of special solutions of the sixth Painleve equation, which are invariant under the symmetries obtained from the Backlund transformations. In most cases, the fixed points of the Backlund transformations are classical solution, but our solutions are not classical for generic parameters. We will calculate the linear monodromy of these solutions exactly, and we will characterize them on Fricke's cubic surface of monodromy.
The Painlevé equations are here connected to other classes of equations with the Painlevé Property (Ince's equations) by the same degeneracy procedure that connects the Painlevé equations (coalescence). These Ince's equations here are also connected among themselves like in the traditional Painlevé's coalescence cascade. Such degeneracy is considered also for the special equations, symmetric equations and Bäcklund transformations.
We consider the $n{\times}n$ matrix linear differential systems in the complex plane. We find necessary and sufficient conditions under which these systems have meromorphic fundamental solutions. Using the operator identity method we construct a set of systems which have meromorphic solutions. We prove that the well known operator with the sine kernel generates a class of meromorphic Painleve type functions. The fifth Painleve function belongs to this class. Hence we obtain a new and simple proof that the fifth Painleve function is meromorphic.
Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function $w$ such that $w'/w$ is a rational function) are shown to be solutions of non linear differential equations with respect to a well-chosen parameter, according to principles established by D. G. Chudnovsky. Examples are given. For instance, the recurrence coefficients in $a_{n+1}p_{n+1}(x)=xp_n(x) -a_np_{n-1}(x)$ of the orthogonal polynomials related to the weight $\exp(-x^4/4-tx^2)$ on {\blackb R\/} satisfy $4a_n^3\ddot a_n = (3a_n^4+2ta_n^2-n)(a_n^4+2ta_n^2+n)$, and $a_n^2$ sa…
We consider orthogonal polynomials on the unit circle with respect to a weight which is a quotient of $q$-gamma functions. We show that the Verblunsky coefficients of these polynomials satisfy discrete Painlevé equations, in a Lax form, which correspond to an $A_3^{(1)}$ surface in Sakai's classification.
The first five classical Painlevé equations are known to have solutions described by divergent asymptotic power series near infinity. Here we prove that such solutions also exist for the infinite hierarchy of equations associated with the second Painlevé equation. Moreover we prove that these are unique in certain sectors near infinity.
The Painlevé classification is one of the central problems in analytics theory of differential equations rooted in the XIX century. Although it saw many significant advances in analyzing certain classes of equations, the classification still remains an open problem especially for the higher-order equations. One of the main classical methods of Painlevé analysis is based on considering the resonance numbers corresponding to the possible indices of arbitrary coefficients in the Laurent expansion of the general solution in a neighborhood of a movable singularity. Complex and non-integer values of…
