On q-deformations of Heun equation

ON q-DEFORMATIONS OF HEUN EQUATION

arXiv:1712.09564v1 [math.CA] 27 Dec 2017

KOUICHI TAKEMURA Abstract. The q-Heun equation and its variants were obtained by degenerations of Ruijsenaars-van Diejen operators with one particle. We investigate local properties of these equations. Especially we characterize the variants of q-Heun equation by using analysis of regular singularities. We also consider 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 the variants of q-Heun operator.

1. Introduction Accessory parameters on the 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 the 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, 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 dy δ ǫ αβz − q d2 y γ (1.1) + + + + y = 0, 2 dz z z − 1 z − t dz z(z − 1)(z − t) with the condition γ + δ + ǫ = α + β + 1. The parameter q is an accessory parameter, which is independent from the local exponents. In [8], q-difference deformations of Heun’s differential equation were obtained by degenerations of Ruijsenaars-van Diejen operators [10, 6], which were also obtained in connection with q-Painlevé equations [4, 11]. One of the q-deformed difference equations, which we call q-Heun equation, is written as (1.2) (x − q h1 +1/2 t1 )(x − q h2 +1/2 t2 )g(x/q) + q α1 +α2 (x − q l1 −1/2 t1 )(x − q l2 −1/2 t2 )g(qx) − {(q α1 + q α2 )x2 + Ex + q (h1 +h2 +l1 +l2 +α1 +α2 )/2 (q β/2 + q −β/2 )t1 t2 }g(x) = 0. Eq.(1.2) had been discovered by Hahn [3] in 1971, and Hahn recognized that the parameter E can be regarded as an accessory parameter. In [8], q-Heun equation 2010 Mathematics Subject Classification. 39A13,33E10. Key words and phrases. Heun equation, q-deformation, regular singularity, quasi-exact solvability, degeneration. 1

2

KOUICHI TAKEMURA

was introduced as an eigenfunction of the fourth degenerated Ruijsenaars-van Diejen operator (1.3) Ah4i =x−1 (x − q h1 +1/2 t1 )(x − q h2 +1/2 t2 )Tq−1 + q α1 +α2 x−1 (x − q l1 −1/2 t1 )(x − q l2 −1/2 t2 )Tq − {(q α1 + q α2 )x + q (h1 +h2 +l1 +l2 +α1 +α2 )/2 (q β/2 + q −β/2 )t1 t2 x−1 } with the eigenvalue E, where Tq−1 g(x) = g(x/q) and Tq g(x) = g(qx). Namely Eq.(1.2) is written as Ah4i g(x) = Eg(x).

(1.4)

Variants of q-Heun equation were also obtained in [8] by using the third and the second degenerated Ruijsenaars-van Diejen operators (1.5) 3 3 Y Y h3i −1 −1 hn +1/2 A =x (x − q tn )Tq−1 + x (x − q ln −1/2 tn )Tq n=1

n=1

− (q 1/2 + q −1/2 )x2 +

3 X

(q hn + q ln )tn x + q (l1 +l2 +l3 +h1 +h2 +h3 )/2 (q β/2 + q −β/2 )t1 t2 t3 x−1 ,

n=1

(1.6) h2i

A

−2

=x

4 Y

(x − q

hn +1/2

−2

tn )Tq−1 + x

+

n=1

(q

hn

(x − q ln −1/2 tn )Tq − (q 1/2 + q −1/2 )x2

n=1

n=1 4 X

4 Y

ln

+ q )tn x +

4 Y

n=1

q

(hn +ln )/2

h

tn · − (q

1/2

+q

−1/2

−2

)x

4 X 1 −1 i 1 x , + + q hn tn q ln tn n=1

and they are written as Ah3i g(x) = Eg(x) and Ah2i g(x) = Eg(x). In this paper, we investigate some properties of q-Heun equation and its variants. One of the results is a characterization of the equations. The q-Heun equation (Eq.(1.2)) is written as (1.7)

a(x)g(x/q) + b(x)g(x) + c(x)g(qx) = 0

such that a(x), b(x), c(x) are the polynomials whose degree is no more than two. On the other hand, if degx a(x) = degx c(x) = 2, a(0) 6= 0 6= c(0) and degx b(x) ≤ 2, then the difference equation can be written as Eq.(1.2) by choosing the parameters suitably. Hence the q-Heun equation is characterized as the difference equation in Eq.(1.7) such that degx a(x) = degx c(x) = 2, a(0) 6= 0 6= c(0) and degx b(x) ≤ 2. The equation Ah3i g(x) = Eg(x) (resp. Ah2i g(x) = Eg(x)) is written in the form of Eq.(1.7), where a(x), b(x), c(x) are polynomials such that degx a(x) = degx c(x) = 3 and degx b(x) ≤ 3 (resp. degx a(x) = degx c(x) = 4 and degx b(x) ≤ 4), although there are relations in the coefficients the polynomials a(x), b(x), c(x). In this paper we describe the relations by using a theory of the regular singularity to the difference equation, which is analogous to the case of the differential equation. See section 2 for the definitions. Let us consider Eq.(1.7) such that a(x), b(x), c(x) are polynomials, degx a(x) = degx c(x) = 3, degx b(x) ≤ 3 and a(0) 6= 0 6= c(0). We factorize the


3

polynomials as a(x) = (x − q h1 +1/2 t1 )(x − q h2 +1/2 t2 )(x − q h3 +1/2 t3 ) and c(x) = (x − q l1 −1/2 t1 )(x − q l2 −1/2 t2 )(x − q l3 −1/2 t3 ). Write the polynomial b(x) as b(x) = b3 x3 + b2 x2 + b1 x + b0 .

(1.8)

If we rewrite the equation Ah3i g(x) = Eg(x) in the form of Eq.(1.7),then the constants b3 , b2 , b1 and b0 are written as (1.9)

b3 = −(q

1/2

+q

b1 = E, b0 = q

−1/2

), b2 =

3 X

(q hn + q ln )tn ,

n=1 (l1 +l2 +l3 +h1 +h2 +h3 )/2

(q β/2 + q −β/2 )t1 t2 t3 .

Here we are going to find a characterization of the values b3 , b2 and b0 by imposing the conditions of local behavior of Eq.(1.7). The result is written as follows; Theorem 1.1. If the exponents of Eq.(1.7) about x = ∞ are −1/2, 1/2 and the singularity x = ∞ is apparent (non-logarithmic), then the values b3 and b2 are determined as Eq.(1.9). The value b0 is determined by the local exponents about x = 0. See Theorem 3.1 for the precise description. Note that the parameter E is independent from the local conditions about x = 0, ∞, and it is reasonable to regard E as an accessory parameter. A characterization for the operator Ah2i is described in Theorem 3.2. We can take a limit q → 1 on the equation Ah3i g(x) = Eg(x), and we obtain Heun’s differential equation by a linear fractional transformation (see section 3). Then the singularity x = ∞ corresponds to a regular (non-singular) point of Heun’s differential equation by the limit q → 1. In this paper we also try to investigate special solutions to q-Heun equation and the variants For this purpose, we consider quasi-exact solvability [9] of the equations. Namely, we investigate finite-dimensional invariant spaces which are invariant under the action of the degenerated Ruijsenaars-van Diejen operator Ah4i , Ah3i or Ah2i . Recall that quasi-exact solvability was applied to investigate the Inozemtsev system of type BCN [2, 7], which is a generalization of the Heun equation to multi-variable. On the Ruijsenaars-van Diejen system (or the Ruijsenaars system of type BCN ), Komori [5] obtained subspaces spanned by theta functions which are invariant under the action of the Ruijsenaars-van Diejen system. Our invariant subspaces of the operator Ah4i , Ah3i or Ah2i would be the degenerated ones of Komori, although they are obtained independently and straightforwardly. Here we present a simple example of an invariant subspace on the operator Ah4i in Eq.(1.3). Set λ1 = (h1 + h2 − l1 − l2 − α1 − α2 − β)/2 + 1 and assume that λ1 = −α1 . Then the operator Ah4i preserves the one-dimensional space spanned by the function xλ1 , and we have (1.10)

Ah4i xλ1 = −{(q h1 +1/2 t1 + q h2 +1/2 t2 )q α1 + (q l1 −1/2 t1 + q l2 −1/2 t2 )q α2 }xλ1 .

Therefore the function x−α1 is an eigenfunction of the operator Ah4i with the eigenvalue −{(q h1 +1/2 t1 + q h2 +1/2 t2 )q α1 + (q l1 −1/2 t1 + q l2 −1/2 t2 )q α2 } under the assumption h1 + h2 − l1 − l2 + α1 − α2 − β + 2 = 0.

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KOUICHI TAKEMURA

This paper is organized as follows. In section 2, we explain local properties of solutions to q-difference equation about x = 0 and x = ∞. In section 3, we investigate local behaviors of solutions about x = 0 and x = ∞ to q-Heun equation and its variants. We show theorems for characterization of q-difference equations to variants of 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 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 the parameters and the exponents are real numbers. 2. Regular singularity to the difference equation We investigate the linear difference equation (2.1)

u(x)g(x/q) + v(x)g(x) + w(x)g(qx) = 0

such that u(x), v(x), w(x) are Laurent polynomials which are written as (2.2)

u(x) =

N X

′′

′

k

uk x , v(x) =

N X

k

vk x , w(x) =

k=M ′

k=M

N X

wk xk

k=M ′′

where uM , vM ′ , wM ′′ , uN ′ , vN ′ , wN ′′ are all non-zero. Then the point x = 0 (resp. x = ∞) of Eq.(2.1) is a regular singularity, iff M = M ′′ ≤ M ′ (resp, N = N ′′ ≥ N ′ ). The power series solutions about the regular singularity has been investigated for a long time (see [1] and related papers). Assume that the point x = 0 of Eq.(2.1) is a regular singularity. Then M = M ′′ ≤ M ′ . We investigate local solutions about x = 0 in the form λ

(2.3)

g(x) = x

∞ X

cℓ xℓ , c0 6= 0.

ℓ=0

We substitute it into Eq.(2.1) and set k + ℓ = n + M. Then we have (2.4)

∞ X n X n=0 ℓ=0

q −λ−ℓ un+M −ℓ + vn+M −ℓ + q λ+ℓ wn+M −ℓ cℓ xn+M +λ = 0.

Hence (2.5)

n X q −λ−ℓ un+M −ℓ + vn+M −ℓ + q λ+ℓ wn+M −ℓ cℓ = 0, (n = 0, 1, 2, . . . ). ℓ=0

If n = 0, then it follows from c0 6= 0 that (2.6)

q −λ uM + vM + q λ wM = 0,

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


5

characteristic equation. Let λ′ be an exponent. If λ′ + n is not an exponent for positive integers n, then the coefficient cn is determined by (2.7) n−1 ′ X ′ ′ ′ q −λ −ℓ un+M −ℓ + vn+M −ℓ + q λ +ℓ wn+M −ℓ cℓ . q −λ −n uM + vM + q λ +n wM cn = − ℓ=0

If λ1 + n is an exponent for some positive integer n, then we need the equation n−1 X

(2.8)

ℓ=0

′ ′ q −λ −ℓ un+M −ℓ + vn+M −ℓ + q λ +ℓ wn+M −ℓ cℓ = 0

in order to have series solutions. Otherwise we need logarithmic terms for the solutions. If Eq.(2.8) is satisfied, then the singularity x = 0 is called apparent (or non-logarithmic). Next, we assume that the point x = ∞ of Eq.(2.1) is a regular singularity, i.e. N = N ′′ ≥ N ′ in Eq.(2.2), and investigate local solutions about x = ∞ in the form (2.9)

h(x) = (1/x)

λ

∞ X

c˜ℓ (1/x)ℓ , c˜0 6= 0.

ℓ=0

Then we have (2.10)

n X q λ+ℓ uN +ℓ−n + vN +ℓ−n + q −λ−ℓ wN +ℓ−n c˜ℓ = 0, (n = 0, 1, 2, . . . ). ℓ=0

By setting n = 0, we obtain the characteristic equation at the regular singularity x = ∞ as follows; q λ uN + vN + q −λ wN = 0.

(2.11)

The exponents about the regular singularity x = ∞ are the values λ which satisfy the characteristic equation, i.e. Eq.(2.11). 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 (2.12)

n−1 X q λ+ℓ uN +ℓ−n + vN +ℓ−n + q −λ−ℓ wN +ℓ−n c˜ℓ = 0. ℓ=0

3. Local behaviors of solutions to q-Heun equation and its variants We investigate local behaviors of solutions about x = 0 and x = ∞ to 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 q-Heun equation.

6

KOUICHI TAKEMURA

3.1. Let Ah4i be the fourth degenerated operator in Eq.(1.3). Then q-Heun equation is written as (Ah4i − E)g(x) = 0,

(3.1)

where E is a constant. The singularities x = 0 and x = ∞ of the q-Heun equation are regular, and the constants M and N in section 2 are determined as M = −1 and N = 1. The action of Ah4i to xµ is written as Ah4i xµ = dh4i,+ (µ)xµ+1 + dh4i,0 (µ)xµ + dh4i,− (µ)xµ−1 ,

(3.2) where

(3.3) dh4i,+ (µ) = q α1 +α2 q µ − (q α1 + q α2 ) + q −µ , dh4i,0 (µ) = −(q h1 +1/2 t1 + q h2 +1/2 t2 )q −µ − (q l1 −1/2 t1 + q l2 −1/2 t2 )q α1 +α2 +µ , dh4i,− (µ) = {q h1 +h2 +1 q −µ − q (h1 +h2 +l1 +l2 +α1 +α2 )/2 (q β/2 + q −β/2 ) + q l1 +l2 +α1 +α2 −1 q µ }t1 t2 . We investigate the singularity x = 0. Set g(x) = xλ

(3.4)

∞ X

cn xn , c0 6= 0,

n=0

and substitute it into Eq.(3.1). Then we have (3.5) c0 dh4i,− (λ) = 0,

c1 dh4i,− (λ + 1) + c0 (dh4i,0 (λ) − E) = 0,

cn dh4i,− (λ + n) + cn−1 (dh4i,0 (λ + n − 1) − E) + cn−2 dh4i,+ (λ + n − 2) = 0,

(n ≥ 2).

The characteristic equation about x = 0 is written as dh4i,− (λ) = 0, i.e. (3.6)

q h1 +h2 +1 q −λ − q (h1 +h2 +l1 +l2 +α1 +α2 )/2 (q β/2 + q −β/2 ) + q l1 +l2 +α1 +α2 −1 q λ = 0.

Hence the values (3.7) λ1 = (h1 +h2 −l1 −l2 −α1 −α2 −β+2)/2, λ2 = (h1 +h2 −l1 −l2 −α1 −α2 +β+2)/2 are exponents about x = 0. If λ2 − λ1 = β is not a positive integer, then we have dh4i,− (λ1 + n) 6= 0 for n ∈ Z>0 and the coefficients cn for λ = λ1 and n ≥ 1 are determined recursively. If λ1 − λ2 6∈ Z, then we have solutions to Eq.(3.1) written as follows; (3.8)

λ1

x {1 +

∞ X

n

λ2

cn x }, x {1 +

∞ X

c′n xn }.

n=1

n=1

We investigate Eq.(3.1) about x = ∞. Set (3.9)

g(x) = (1/x)

λ

∞ X n=0

n

cñ (1/x) =

∞ X n=0

cñ x−λ−n , c˜0 6= 0.


7

By substituting it into Eq.(3.1), we have (3.10) c˜0 dh4i,+ (−λ) = 0, c˜1 dh4i,+ (−λ − 1) + c˜0 (dh4i,0 (−λ) − E) = 0, cñ dh4i,+ (−λ − n) + cñ−1 (dh4i,0 (−λ − n + 1) − E) + cñ−2 dh4i,+ (−λ − n + 2) = 0. The characteristic equation at x = ∞ is written as dh4i,+ (−λ) = 0, i.e. q α1 +α2 q −λ − (q α1 + q α2 ) + q λ = 0.

(3.11)

Hence the values λ1 = α1 and λ2 = α2 are exponents about x = ∞. If λ2 − λ1 (= α2 − α1 ) is not a positive integer, then we have dh4i,+ (−λ1 − n) 6= 0 for n ∈ Z>0 and the coefficients cñ (n ≥ 1) are determined recursively. If α1 − α2 6∈ Z, then then we have solutions to Eq.(3.1) written as follows; (3.12)

−α1

x

{1 +

∞ X

−n

cñ x

−α2

}, x

{1 +

n=1

∞ X

c˜′n x−n }.

n=1

Eq.(3.1) is written as (3.13)

a(x)g(x/q) + b(x)g(x) + c(x)g(qx) = 0,

where (3.14)

a(x) = (x − q h1 +1/2 t1 )(x − q h2 +1/2 t2 ), c(x) = (x − q l1 −1/2 t1 )(x − q l2 −1/2 t2 ), b(x) = b2 x2 + Ex + b0

and (3.15)

b2 = −(q α1 + q α2 ), b0 = −q (h1 +h2 +l1 +l2 +α1 +α2 )/2 (q β/2 + q −β/2 )t1 t2 .

If the exponents of Eq.(3.1) about x = 0 (resp. x = ∞) are λ1 and λ2 in Eq.(3.7) (resp. α1 and α2 ), then b0 (resp. b2 ) is determined as Eq.(3.15). The parameter E is independent from the exponents about x = 0, ∞, and it is reasonable to regard E as an accessory parameter. It was shown in [8] that q-Heun equation has a limit to Heun’s differential equation by q → 1. 3.2. Let Ah3i be the thirdly degenerated operator in Eq.(1.5). Then the equation (3.16)

(Ah3i − E)g(x) = 0.

for a fixed constant E is a variant of q-Heun equation. The singularities x = 0 and x = ∞ of Eq.(3.16) are regular, and the constants M and N in section 2 are determined as M = −1 and N = 2. The action of Ah3i to xµ is written as (3.17)

Ah3i xµ = dh3i,++ (µ)xµ+2 + dh3i,+ (µ)xµ+1 + dh3i,0 (µ)xµ + dh3i,− (µ)xµ−1 ,

8

KOUICHI TAKEMURA

where (3.18)

dh3i,++ (µ) =q −µ + q µ − q 1/2 − q −1/2 , d

h3i,+

(µ) =

3 X

(q hi + q li − q hi +1/2−µ − q li −1/2+µ )ti

i=1

d

h3i,0

(µ) =

X

(q hi+hj +1−µ + q li +lj −1+µ )ti tj

1≤i0 and the coefficients cn for λ = λ1 and n ≥ 1 are determined recursively. We investigate the singularity x = ∞ of Eq.(3.16). We substitute the expansion about x = ∞ as Eq.(3.9) into Eq.(3.16). Then we have (3.22)

dh3i,++ (−λ) = 0,

c˜1 dh3i,++ (−λ − 1) = −˜ c0 dh3i,+ (−λ),

c˜2 dh3i,++ (−λ − 2) = −{˜ c1 dh3i,+ (−λ − 1) + c˜0 (dh3i,0 (−λ) − E)}, cñ dh3i,++ (−λ − n) = −{˜ cn−1 dh3i,+ (−λ − n + 1) + cñ−2 (dh3i,0 (−λ − n + 2) − E) + cñ−3 dh3i,+ (−λ − n + 3)},

(n ≥ 3).

The singularity x = ∞ is regular and the characteristic equation at x = ∞ is written as dh3i,++ (−λ) = 0, i.e. (3.23)

q λ + q −λ − q 1/2 − q −1/2 = 0.

Hence the values λ1 = −1/2 and λ2 = 1/2 are exponents about x = ∞. In the case λ1 = −1/2, we have dh3i,++ (−λ1 − 1) = 0 and the condition for apparentness of regular singularity x = ∞ is equivalent to dh3i,+ (−λ1 ) = 0. It follows from Eq.(3.18)


9

that the condition is satisfied. Hence the singularity x = ∞ is apparent, and we have two independent solutions written as (3.24)

1/2

x

{1 +

∞ X

−n

cñ x

−1/2

}, x

{1 +

n=2

∞ X

c˜′n x−n }.

n=1

We have observed that the regular singularity x = ∞ of a variant of q-Heun equation in Eq.(3.16) is apparent and the difference of the exponents is 1. Recall that Eq.(3.16) is written as (3.25)

a(x)g(x/q) + b(x)g(x) + c(x)g(qx) = 0,

where (3.26)

a(x) = (x − q h1 +1/2 t1 )(x − q h2 +1/2 t2 )(x − q h3 +1/2 t3 ), c(x) = (x − q l1 −1/2 t1 )(x − q l2 −1/2 t2 )(x − q l3 −1/2 t3 ), b(x) = b3 x3 + b2 x2 + Ex + b0

and (3.27)

b3 = −(q 1/2 + q −1/2 ), b2 = b0 = q

(l1 +l2 +l3 +h1 +h2 +h3 )/2

(q

3 X n=1 β/2

(q hn + q ln )tn ,

+ q −β/2 )t1 t2 t3 .

Note that the parameter E is independent from the exponents about x = 0, ∞ and apparentness of the regular singularity x = ∞. Therefore it is reasonable to regard E as an accessory parameter. Next we consider a characterization of the coefficients b3 , b2 and b0 by imposing the conditions of local behaviors of Eq.(3.25) with Eq.(3.26). Then we have the following theorem. Theorem 3.1. Assume that the functions a(x), b(x) and c(x) are given in the form of Eq.(3.26). If the difference of the exponents about x = 0 is β(6= 0), the difference of the exponents about x = ∞ is 1, and the regular singularity x = ∞ is apparent, then we obtain Eq.(3.27) and recover a variant of q-Heun equation in Eq.(3.16). Proof. Let λ and λ + β be the exponents about z = 0. Then we have (3.28)

− q h1 +h2 +h3 +3/2−λ t1 t2 t3 − q l1 +l2 +l3 −3/2+λ t1 t2 t3 + b0 = 0, − q h1 +h2 +h3 +3/2−λ−β t1 t2 t3 − q l1 +l2 +l3 −3/2+λ+β t1 t2 t3 + b0 = 0.

Hence we have (q h1 +h2 +h3 +3/2−λ − q l1 +l2 +l3 −3/2+λ+β )(1 − q −β ) = 0, λ = (l1 + l2 + l3 + h1 + h2 + h3 + 3 − β)/2 and b0 = q (l1 +l2 +l3 +h1 +h2 +h3 )/2 (q β/2 + q −β/2 )t1 t2 t3 . Similarly it follows from the condition of the exponents about x = ∞ that b3 = −(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 as −q −1/2 (q h1 +1/2 t1 + q h2 +1/2 t2 + q h3 +1/2 t3 ) + b2 − q 1/2 (q l1 −1/2 t1 + q l2 −1/2 t2 + q l3 −1/2 t3 ) = 0. Therefore we obtain Eq.(3.27).

10

KOUICHI TAKEMURA

We are going to obtain a differential equation from the difference equation in Eq.(3.16) by taking a suitable limit. We rewrite Eq.(3.16) as (3.29) (x − t1 q h1 +1/2 )(x − t2 q h2 +1/2 )(x − t3 q h3 +1/2 )g(x/q) + (x − t1 q l1 −1/2 )(x − t2 q l2 −1/2 )(x − t3 q l3 −1/2 )g(xq) + {−(q

1/2

+q

+ t1 t2 t3 q

−1/2

3

)x +

3 X

˜ tn (q hn + q ln )x2 − (2(t1 t2 + t2 t3 + t3 t1 ) + (q − 1)E1 + (q − 1)2 E)x

n=1 (l1 +l2 +l3 +h1 +h2 +h3 )/2

(q β/2 + q −β/2 )}g(x) = 0,

where (3.30) E1 = (h1 + h2 + l1 + l2 )t1 t2 + (h2 + h3 + l2 + l3 )t2 t3 + (h3 + h1 + l3 + l1 )t3 t1 . Set q = 1 + ε. We divide Eq.(3.29) by ε2 . By using Taylor’s expansion (3.31)

g(x/q) = g(x) + (−ε + ε2 )xg ′ (x) + ε2 x2 g ′′ (x)/2 + O(ε3), g(qx) = g(x) + εxg ′ (x) + ε2 x2 g ′′ (x)/2 + O(ε3),

we find the following limit as ε → 0: (3.32) x2 (x − t1 )(x − t2 )(x − t3 )g ′′ (x) # " ˜ h − l + 1 h − l + 1 2 l h − l + 1 2 2 3 3 1 1 g ′(x) + + − + x2 (x − t1 )(x − t2 )(x − t3 ) x − t1 x − t2 x − t3 x − [x3 /4 + {(2h1 − 2l1 + 1)t1 + (2h2 − 2l2 + 1)t2 + (2h3 − 2l3 + 1)t3 }x2 /4 ˜ + t1 t2 t3 (˜l + 1/2 + β/2)(˜l + 1/2 − β/2)]g(x) = 0, + Bx where (3.33)

˜l = (h1 + h2 + h3 − l1 − l2 − l3 + 2)/2,

˜ = E˜ + constant. B

This is a Fuchsian differential equation with five singularities {0, t1 , t2 , t3 , ∞} and the local exponents are given by the following Riemann scheme:   x=0 x = t1 x = t2 x = t3 x = ∞ ˜l + β/2 + 1/2 0 0 0 −1/2  (3.34) ˜l − β/2 + 1/2 l1 − h1 l2 − h2 l3 − h3 1/2 It is shown that the singularity x = ∞ is non-logarithmic (apparent). By setting z = t2 (1 − t1 /x)/(t2 − t1 ) and g(x) = x1/2 g˜(z), the function y = g˜(z) satisfies Heun’s differential equation; d2 y dy δ ǫ α′ β ′ z − q γ (3.35) + + + + y = 0, dz 2 z z − 1 z − t dz z(z − 1)(z − t)

where t = t2 (t1 − t3 )/{t3 (t1 − t2 )}, γ = 1 + h1 − l1 , δ = 1 + h2 − l2 , ǫ = 1 + h3 − l3 , ˜ 3 (t2 − t1 )) + constant. {α′ , β ′} = {˜l − β/2, ˜l + β/2}, q = E/(t


11

3.3. Let Ah2i be the secondly degenerated operator in Eq.(1.6). Then the equation (Ah2i − E)g(x) = 0.

(3.36)

for a fixed constant E is a variant of q-Heun equation. The singularities x = 0 and x = ∞ of Eq.(3.36) are regular, and the constants M and N in section 2 are determined as M = −2 and N = 2. The action of Ah2i to xµ is written as (3.37) Ah2i xµ = dh2i,++ (µ)xµ+2 + dh2i,+ (µ)xµ+1 + dh2i,0 (µ)xµ + dh2i,− (µ)xµ−1 + dh2i,−− (µ)xµ−2 , where (3.38)

dh2i,++ (µ) =q −µ + q µ − q 1/2 − q −1/2 , d

h2i,+

(µ) =

4 X

(q hi + q li − q hi +1/2−µ − q li −1/2+µ )ti

i=1

d

h2i,0

(µ) =

X

(q hi +hj +1−µ + q li +lj −1+µ )ti tj

1≤i