On q-Deformations of the Heun Equation

. The q -Heun equation and its variants arise as degenerations of Ruijsenaars–van Diejen operators with one particle. We investigate local properties of these equations. In particular we characterize the variants of the q -Heun equation by using analysis of regular singularities. We also consider the quasi-exact solvability of the q -Heun equation and its variants. Namely we investigate ﬁnite-dimensional subspaces which are invariant under the action of the q -Heun operator or variants of the q -Heun operator


Introduction
Accessory parameters of an ordinary linear differential equation are the parameters which are not governed by local data (or local exponents) of the differential equation, and absence or existence of accessory parameters affects the strategy of analyzing the differential equation. A typical example which does not have an accessory parameter is the hypergeometric equation of Gauss. On the other hand, the Heun equation is an example which has an accessory parameter. Heun's differential equation is a standard form of the second order linear differential equation with four regular singularities on the Riemann sphere, and it is written as with the condition γ + δ + = α + β + 1. The parameter B is an accessory parameter, which is independent of the local exponents.
Recall that the q-Heun equation was introduced in [10] as an eigenfunction of the fourth degeneration of the Ruijsenaars-van Diejen operator of one variable with eigenvalue E, where T q −1 g(x) = g(x/q) and T q g(x) = g(qx). Namely equation (1.2) is written as The derivation of the q-Heun equation in [10] was motivated by the relationship between the elliptic 4-parameter Heun differential operator and the 8-parameter elliptic difference operator (Ruijsenaars-van Diejen operator). The (non-degenerate) Ruijsenaars-van Diejen operator [2,7] of one variable is given by 1 − e −(2k−1)πa + e 2πiz 1 − e −(2k−1)πa + e −2πiz is a modified version of the theta function, and U (z) is an elliptic function whose definition is omitted here (see [10]). The operator A 0 has 8 parameters h 1 , . . . , h 8 apart from the parameter a + in the elliptic function and the difference interval ia − . The multi-variable Ruijsenaars-van Diejen system can be regarded as a relativistic deformation of the quantum system called the Inozemtsev system [2,7]. By the non-relativistic limit a − → 0 of the one-variable Ruijsenaars-van Diejen operator, we obtain the Heun operator in term of the elliptic function written as where ℘(x) is the Weierstrass elliptic function with basic periods (2ω 1 , 2ω 3 ) and {ω 0 (= 0), ω 1 , , ω 3 } is a set of half periods (see [7] for details). Here the 8 parameters in the relativistic system are reduced to the 4 parameters l 0 , l 1 , l 2 , l 3 in the non-relativistic limit. It is known that the Heun equation given in equation (1.1) is equivalent to the equation Hf (x) = Ef (x), E ∈ C (see [9] for details). Hence the Ruijsenaars-van Diejen system is a difference analogue of the elliptic form of the Heun equation. A motivation of the paper [10] was a construction of the difference analogue of the Heun equation without the elliptic expression, and this was realized by considering degenerations four times. After writing the first version of the manuscript of [10], the author was informed of Hahn's paper [4] by Professor Ohyama.
In [10], other degenerate operators were also obtained. The third and the second degenerate Ruijsenaars-van Diejen operators are written as The variants of the q-Heun equation were introduced as A 3 g(x) = Eg(x) and A 2 g(x) = Eg(x), E ∈ C.
In this paper, we investigate some properties of the q-Heun equation and its variants. One of the results is a characterization of variants of the q-Heun equation. Recall that the equation 2)) is written as equation (1.3) whose coefficients are generic quadratic polynomials.
The equation We factorize the polynomials as a(

In the equation
The value b 0 is determined as b 0 = q (l 1 +l 2 +l 3 +h 1 +h 2 +h 3 )/2 q β/2 + q −β/2 t 1 t 2 t 3 by imposing a condition of the local exponents about x = 0. See Theorem 3.1 for the precise description. Note that the value b 1 = E is independent of the local conditions about x = 0, ∞, and it is reasonable to regard E as an accessory parameter. An analogous result for the equation A 2 g(x) = Eg(x) is described in Theorem 3.2. By a suitable limit in the equation A 3 g(x) = Eg(x), we obtain a differential equation which is equivalent to Heun's differential equation by a linear fractional transformation (see Section 3). Then the singularity x = ∞ of the equation A 3 g(x) = Eg(x) corresponds to a regular (non-singular) point of the differential equation in the limit, and we may say that the equation In this paper we also investigate special solutions to the q-Heun equation and its variants. For this purpose, we consider quasi-exact solvability [11] of the equations. Namely, we investigate finite-dimensional invariant spaces which are invariant under the action of the degenerate Ruijsenaars-van Diejen operator A 4 , A 3 or A 2 . Recall that quasi-exact solvability was applied to investigate the Inozemtsev system of type BC N [3,8], which is a generalization of the Heun equation to multiple variables. For the Ruijsenaars-van Diejen system (the Ruijsenaars system of type BC N ), Komori [6] obtained finite-dimensional subspaces spanned by theta functions which are invariant under the action of the Ruijsenaars-van Diejen system. The author believes that our invariant subspaces of the operator A 4 , A 3 or A 2 are degenerations of those of Komori, although they are obtained independently and straightforwardly. We hope to develop further our understandings of these invariant subspaces and to find a path for analysis of the (non-degenerate) Ruijsenaars-van Diejen system in the near future.
Here we present a simple example of an invariant subspace for the operator A 4 in equation (1.5). Set λ 1 = (h 1 + h 2 − l 1 − l 2 − α 1 − α 2 − β)/2 + 1 and assume that λ 1 = −α 1 . Then the operator A 4 preserves the one-dimensional space spanned by the function x λ 1 , and we have (1.9) Therefore the function x λ 1 is an eigenfunction of the operator The reminder of the paper is organized as follows. In Section 2, we explain local properties of solutions to second-order q-difference equation about x = 0 and x = ∞. In Section 3, we investigate local behaviors of solutions about x = 0 and x = ∞ to the q-Heun equation and its variants. We obtain theorems for characterization of q-difference equations to variants of the q-Heun equation. We also investigate the limit q → 1 and observe relationships with Heun's differential equation. In Section 4, we investigate invariant subspaces related with the q-Heun equation and its variants.
In this paper, we assume that q is a positive real number which is not equal to 1, and all exponents are real numbers.
2 Regular singularity to the difference equation We investigate the linear difference equation are Laurent polynomials which are written as The power series solutions about the regular singularity is by now well-understood (see [1] and related papers).
Assume that the point x = 0 of equation ( If we substitute this into equation (2.1) and set k + = n + M , then we get where n is a non-negative integer. If n = 0, then it follows from c 0 = 0 that which we call the characteristic equation about the regular singularity x = 0. The exponents about the regular singularity x = 0 are the values λ which satisfy the characteristic equation.
If the quadratic equation u M + v M t + w M t 2 = 0 has two positive roots, then we have two realvalued exponents. Let λ be an exponent. If λ + n (n ∈ Z ≥1 ) is not an exponent, then the coefficient c n is determined by If λ + n is an exponent for some positive integer n, then we need the equation in order to have series solutions. Otherwise we need logarithmic terms for the solutions. If the difference of the exponents is a non-zero integer and equation (2.3) is satisfied, then the singularity x = 0 is called apparent (or non-logarithmic).
Next, we assume that the point x = ∞ of equation (2.1) is a regular singularity, i.e., N = N ≥ N in equation (2.2), and investigate local solutions about x = ∞ of the form where n is a non-negative integer. By setting n = 0, we obtain the characteristic equation at the regular singularity x = ∞ as follows The exponents about the regular singularity x = ∞ are the values λ which satisfy the characteristic equation, i.e., equation (2.4). Assume that λ is an exponent at x = ∞ and λ + n is also an exponent at x = ∞ for some positive integer n. Then the singularity is called apparent, iff

Local behaviors of solutions to the q-Heun equation and its variants
We investigate local behaviors of solutions about x = 0 and x = ∞ to the q-Heun equation and its variants. We obtain theorems that the local behaviors of solutions determine the coefficients of the q-difference equation for the cases of the variants of the q-Heun equation.
Hence the values are exponents about x = 0. If λ 2 −λ 1 = β is not a positive integer, then we have d 4 ,− (λ 1 +n) = 0 for n ∈ Z >0 and the coefficients c n for λ = λ 1 and n ≥ 1 are determined recursively. If λ 1 − λ 2 ∈ Z, then we have solutions to equation (3.1) written as We investigate equation ( The characteristic equation at x = ∞ is written as d 4 ,+ (−λ) = 0, i.e., Hence the values α 1 and α 2 are exponents about x = ∞. If α 2 − α 1 is not a positive integer, then we have d 4 ,+ (−α 1 − n) = 0 for n ∈ Z >0 and the coefficientsc n , n ≥ 1, are determined recursively. If α 1 − α 2 ∈ Z, then we have solutions to equation (3.1) written as Equation (3.1) is written as If the exponents of equation (3.1) about x = 0 (resp. x = ∞) are λ 1 and λ 2 in equation (3.4) (resp. α 1 and α 2 ), then b 0 (resp. b 2 ) is given by equation (3.6). The parameter E is independent of the exponents about x = 0, ∞, and it is reasonable to regard E as an accessory parameter. It was shown in [10] that the q-Heun equation reduces to Heun's differential equation in the q → 1 limit.

Recall that equation (3.7) is written as
Note that the parameter E is independent of the exponents about x = 0, ∞ and apparency of the regular singularity x = ∞. Therefore it is reasonable to regard E as an accessory parameter.  Proof . Let λ and λ + β be the exponents about x = 0. Then we have Similarly it follows from the condition of the exponents about x = ∞ that b 3 = − q 1/2 + q −1/2 and the exponents are 1/2 and −1/2. The condition that the singularity x = ∞ is apparent is written Therefore we obtain equation (3.12).
Next we consider a characterization of the coefficients b 4 , b 3 , b 1 and b 0 as an analogue of Theorem 3.1. Namely we have the following theorem, which is proved similarly to Theorem 3.1.
Theorem 3.2. Assume that the functions a(x), b(x) and c(x) take the form of equation (3.18). If the difference of the exponents about x = 0 is 1, the difference of the exponents about x = ∞ is 1, and the regular singularities x = 0, ∞ are apparent, then we obtain equation (3.19) and recover a variant of the q-Heun equation in equation (3.16).
We are going to obtain a differential equation from the difference equation (3.16) by taking a suitable limit. We rewrite equation (3.16) as Set q = 1 + ε. Then we obtain the following differential equation after taking the limit ε → 0.

Invariant subspaces related with the q-Heun equation and its variants
In order to find special solutions to the q-Heun equation or its variants, we consider quasiexact solvability [11], i.e., we investigate subspaces which are invariant under the action of the operators A 4 , A 3 or A 2 . We look for subspaces which are invariant under the action of the operator A 4 given in equation (1.5).
Then the operator A 3 preserves the space V 3 .
Proof . It follows from equation (3.8
Then the operator A 2 preserves the space V 2 .