On some differential transformations of hypergeometric equations

Many algebraic transformations of the hypergeometric equation σ(x)z"(x) + τ(x)z'(x) + lz(x) = 0, where σ, τ, l are polynomial functions of degrees 2 (at most), 1, 0, respectively, are well known. Some of them involve x = x(t), a polynomial of degree r, in order to recover the Heun equation, extension of the hypergeometric equation by one more singularity. The case r = 2 was investigated by K. Kuiken (see 1979 SIAM J. Math. Anal. 10 (3) 655-657) and extended to r = 3,4, 5 by R. S. Maier (see 2005 J. Differ. Equat. 213 171 - 203). The transformations engendered by the function y(x) = A(x)z(x), also very popular in mathematics and physics, are used to get from the hypergeometric equation, for instance, the Schroedinger equation with appropriate potentials, as well as Heun and confluent Heun equations. This work addresses a generalization of Kimura's approach proposed in 1971, based on differential transformations of the hypergeometric equations involving y(x) = A(x)z(x) + B(x)z'(x). Appropriate choices of A(x) and B(x) permit to retrieve the Heun equations as well as equations for some exceptional polynomials. New relations are obtained for Laguerre and Hermite polynomials.


Introduction
In 1971, Kimura [1] investigated in detail all Fuchsian differential equations F {y(x)} = 0 reducible to hypergeometric equations H{z(x)} = 0 by a linear transformation y(x) = P 0 (x)z(x) + P 1 (x)z (x), with P 0 and P 1 rational functions of x. To compute P 0 (x) and P 1 (x), he assumed the same set of singularities and the same monodromy group for H{y(x)} = 0 and F {z(x)} = 0, giving information on the structures and properties of functions P 0 and P 1 . The equation F {y} = y + A 1 (x)y + A 2 (x)y = 0, (y = y(x)), being Fuchsian and written as a generalized Heun equation, contains regular singular points with many parameters, (not all free), in order to eliminate logarithmic situations and irreducibility (equation not factorizable).
Many theorems and properties are provided to compute P 0 (x) and P 1 (x) in particular situations, but without using the coupled differential equations satisfied by P 0 and P 1 .
The aim of this work is to reverse this approach, making it possible to eliminate the Fuchsian constraints. We proceed as follows: inject an arbitrary linear transformation y = A(x)z + B(x)z into the hypergeometric equation and build a second order differential equation for y, (not always Fuchsian), depending on the two differentiable functions A(x) and B(x), polynomial or not.
Exceptional orthogonal polynomials (EOPs) like exceptional Jacobi, exceptional Laguerre, and so on, (see [2] and references therein), can be retrieved from the same linear transformation given a long time ago in the above mentioned seminal paper by Kimura. Recall that the notion of the exceptional (X l ) orthogonal polynomials was introduced in 2008 by Gomez-Ullate, Kamran and Milson [3] in the framework of Sturm Liouville theory. These orthogonal polynomials are exceptional in the sense that they start at degree l(l ≥ 1) instead of degree 0 constant term, thus avoiding restrictions of Bochner's theorem and they satisfy secondorder differential equations. These authors constructed the lowest examples, the X 1 Laguerre and X 1 Jacobi polynomials explicitly. A quantum mechanical reformulation with shape-invariant potentials was given by Quesne and collaborators [4,5] within the framework of one-dimensional quantum mechanics. The merit of quantum mechanical reformulation resides in the fact that the orthogonality and completeness of the obtained eigenfunctions are guaranteed. Besides, the well established solution mechanism of shape invariance combined with the Crum's method [6], or the so-called factorization method [7], or the susy quantum mechanics [8] is available.
Then, two sets of infinitely many shape invariant potentials, the deformed radial oscillator potentials and the deformed trigonometric/hyperbolic Darboux-Pöschl-Teller (DPT) potentials, and the corresponding X l Laguerre and Jacobi polynomials (l = 1, 2, . . . ∞) were investigated by Sasaki et al [9,10,11] in 2009. This paper is organised as follows. In section 2, we provide a quick overview on hypergeometric equations and Heun's equations. Section 3 deals with our generalization of Kimura's approach, based on a differential transformation of hypergeometric equations. Known special functions and orthogonal polynomials are retrieved from the presented general formalism. Concluding remarks are presented in section 4.

Hypergeometric differential equation and Heun's equations
The hypergeometric differential equation is characterized as a Fuchsian differential equation of the second order which has three singularities at x = 0, 1, ∞ corresponding to the exponents (0, 1 − γ); (0, 1 − δ) and (α, β) respectively, satisfying the Fuchs relation This equation has been studied in detail and extended in several directions by many mathematicians. As one of those extensions we are led in a natural way to a second order equation of Fuchsian type whose singularities and exponents are given by the following Riemannian scheme where the exponents are connected by Fuchs's relation The equation defined by these data contains just k accessory parameters, denoted by ρ 1 , · · · , ρ k , and can be written in the form: In the case when k = 1, the equation is often called Heun's equation that we write in the following form [12,13]: with the Fuchs relation: This equation has four regular singularities at 0, 1, a, ∞ with corresponding exponents respectively. All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane C ∪ {∞}, can be transformed into (4). The parameters play different roles: • a is the singularity parameter • α, β, γ, δ, are exponent parameters • q is the accessory parameter.
The total number of free parameters is six. (4) can be transformed into the normal form: Confluent forms of Heun's differential equation (4) arise when two or more of the regular singularities merge to form an irregular singularity. This is analogous to the derivation of the confluent hypergeometric equation. There are four standard forms, as follows (see [14] and references therein): It has regular singularities at x = 0 and 1, and an irregular singularity of rank 1 at x = ∞. Mathieu functions, spheroidal wave functions and Coulomb spheroidal functions are special cases of solutions of the confluent Heun equation. (ii) Doubly-confluent Heun equation has irregular singularities at x = 0 and ∞, each of rank 1. (iii) Biconfluent Heun equation has a regular singularity at x = 0, and an irregular singularity at ∞ of rank 2.
(iv) Triconfluent Heun equation has one singularity, an irregular singularity of rank 3 at x = ∞.

Differential transformation of hypergeometric equations
In this section, we consider the following hypergeometric equation: and transformation: where σ ≡ σ(x) is a polynomial of degree less or equal to 2, τ ≡ τ (x) a polynomial of degree exactly equal to 1, λ is a constant; A = A(x) and B = B(x) are polynomials of degrees r and s, respectively. Then, we build a general linear second order ordinary differential equation L 2 [y(x)] = 0 satisfied by the function y, polynomial or not, and investigate a set of situations.
whereĀ = σĀ −Bλ,B = σĀ + σB − τB. • Rewrite the coefficientsĀ andB of (14) in terms of σ, τ, λ, A, B and their derivatives as follows:Ā • Then, using (13) and (14) we form the following determinantal equation giving the ordinary differential equation satisfied by y : or, equivalently,L The following observations are in order: . Such a simplified choice has been already considered in the work by Kimura [1] and in recent literature on exceptional polynomials (see [2] and references therein).
Let us investigate some relevant particular cases in the sequel.

Basic choice: A(x) is arbitrary and B(x) = σ(x)
For arbitrary A(x) and B(x) = σ(x), the equation (17) can be transformed into the following second order differential equation: Note that by using appropriate choices of the coefficients in the polynomial A, we can reduce the above equation into hypergeometric equations and generate known or unknown contiguous relations between the solutions y(x). For instance, setting:  n−1 (x), polynomial of degree n − 1, as solution. One can then use the well known contiguous relation [15]: to find the polynomial coefficient A(x) of P (α,β) n (x). Of course if λ is arbitrary, non polynomial solutions like the second kind solutions of (11) can be exploited to generate similar contiguous relations for the z(x) as for the polynomial solutions y(x).

Second choice: A and B are constant
Let us consider now the trivial case where A and B are constants. Then, the equation (17) can be reduced to a simpler second order differential equation as follows: This equation is also not hypergeometric in general, but suitable choices of the quantities σ, τ, λ and A, B can also permit to deduce relevant contiguous relations. For instance, we retrieve the following particular equations: • Hermite equation If σ = 1 and τ = −2x, the equation (28) takes the form: which can be further simplified to give: (i) For the particular case of B = 0 with arbitrary A, the well known Hermite equation: (ii) In the opposite case, when A = 0 and B is arbitrary, the equation (33) is transformed into a modified Hermite equation 3.3. Link with the Heun equations and exceptional Jacobi polynomials Besides, Heun general equation, its confluent forms as well as exceptional Jacobi polynomials can also be retrieved from the general approach developed here.

General Heun's equation
• With the choice A = ax + b and σ = x 2 − x, the degrees of P, Q, R in equation (21) are 2, 3 and 2, respectively. Then, the HEUN equation is recovered when the coefficients a, b and τ are chosen so that P (x) reduces to x − µ, and where Q 1 (x) and R 1 (x) are polynomials.
• Other appropriate choices also produce general Heun and confluent Heun equations [13] with explicit solutions y(x) = A(x)z(x) + B(x)z (x). Kimura gives [1] solutions in two relevant cases: (i) for = −1 with A(x) of degree 1 and B(x) = x(x − 1), and (ii) for = −2 with A(x) of degree 2 and B(x) of degree 3. The Kimura method is very nice, but complicated and restrictive, the confluent equations being excluded. Its intrinsic complexity resides in the step = −1 to −2 increasing the degree of polynomials A(x) and B(x). In our approach, the problem is entirely algebraic, although not excluding, of course, also some difficulties.

Concluding remarks
This work addressed a generalization of Kimura's approach based on differential transformation of hypergeometric equations. This transformation involving y(x) = A(x)z(x)+B(x)z (x), where z(x) is a solution of a hypergeometric equation, led to a general second order differential equation for y that encompasses various ODEs of mathematical physics as particular cases such as Heun equations and equations for exceptional polynomials. In particular, the equations (21) and (28), defined with the relations (22)-(24) and (29)-(31), permited us to generate new as well as known relations for Laguerre and Hermite polynomials.