Voros Coefficients at the Origin and at the Infinity of the Generalized Hypergeometric Differential Equations with a Large Parameter
Voros coefficients of the generalized hypergeometric differential equations with a large parameter are defined, and their explicit forms are given for the origin and for infinity. It is shown that they are Borel summable in some specified regions in the space of parameters, and their Borel sums in t...
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| citation_txt | Voros Coefficients at the Origin and at the Infinity of the Generalized Hypergeometric Differential Equations with a Large Parameter. Takashi Aoki and Shofu Uchida. SIGMA 18 (2022), 002, 23 pages |
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| description | Voros coefficients of the generalized hypergeometric differential equations with a large parameter are defined, and their explicit forms are given for the origin and for infinity. It is shown that they are Borel summable in some specified regions in the space of parameters, and their Borel sums in the areas are given.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 002, 23 pages
Voros Coefficients at the Origin and at the Infinity
of the Generalized Hypergeometric Differential
Equations with a Large Parameter
Takashi AOKI a and Shofu UCHIDA b
a) Department of Mathematics, Kindai University, Higashi–Osaka 577-8502, Japan
E-mail: aoki@math.kindai.ac.jp
b) Graduate School of Science and Engineering, Kindai University,
Higashi–Osaka 577-8502, Japan
E-mail: 1944310104r@kindai.ac.jp
Received July 20, 2021, in final form December 30, 2021; Published online January 03, 2022
https://doi.org/10.3842/SIGMA.2022.002
Abstract. Voros coefficients of the generalized hypergeometric differential equations with
a large parameter are defined and their explicit forms are given for the origin and for the
infinity. It is shown that they are Borel summable in some specified regions in the space of
parameters and their Borel sums in the regions are given.
Key words: exact WKB analysis; Voros coefficients; generalized hypergeometric differential
equations
2020 Mathematics Subject Classification: 33C20; 34E20; 34M60
1 Introduction
The notion of the Voros coefficients is one of the keys in the exact WKB analysis of differential
equations with a large parameter. It has been introduced mainly for second-order ordinary dif-
ferential equations and used effectively in the descriptions of the parametric Stokes phenomena,
of calculation of the monodromies and of the relations between Borel resummed WKB solutions
and classical special functions [4, 5, 6, 7, 8, 11, 17, 22, 25, 27, 28]. Recently, new insights into the
relationship between cluster algebra and Voros coefficients have been obtained in [19]. Further-
more, it is known that Voros coefficient of the Gauss hypergeometric differential equation can be
described using the free energy in the topological recursion [18]. Besides, the Voros coefficients
have also appeared in the physics literature as the spectral coordinates [15].
Our purpose is to define the Voros coefficients at the origin and at the infinity for the gen-
eralized hypergeometric differential equation with a large parameter and to give the explicit
forms of them. Here the generalized hypergeometric differential equation means the differential
equation which characterizes the generalized hypergeometric series (or function) NFN−1. It is
well known [12] that this equation is of order N and obtained as a natural generalization of
the Gauss hypergeometric differential equation. The monodromy group of the generalized hy-
pergeometric differential equation plays a role in studying the solution of the equation. The
classification of the finite hypergeometric groups [23] and the differential Galois group of the
generalized hypergeometric equation have been obtained in [10].
Basic notions and tools in the exact WKB analysis of higher-order or of infinite-order dif-
ferential equations with a large parameter are established in [3]. Our discussions are based on
the theory developed in this paper. For example, WKB solutions of a higher-order differential
equation are defined by using the characteristic roots of the equation, which are branches of an
algebraic function. Once WKB solutions are introduced, one can define the Voros coefficients in
mailto:aoki@math.kindai.ac.jp
mailto:1944310104r@kindai.ac.jp
https://doi.org/10.3842/SIGMA.2022.002
2 T. Aoki and S. Uchida
a similar manner to the second-order case. Fortunately, our equation has ladder operators for
parameters as in the case of the Gauss hypergeometric differential equation. Therefore the basic
idea of computation of the explicit forms of the Voros coefficients is the same as the second-order
case [7, 8, 22, 27]. But in our case, we have to consider the integral of the algebraic function.
Although this fact causes some difficulties, some of them can be overcomed by using an idea
invented by Iwaki and Koike [17]. There is another difficulty, or complexity, coming from the
number of parameters. Our equation contains (2N − 1) parameters and we have to manage
some combinations of them. Thus our descriptions become somewhat complicated.
The plan of this article is as follows. In Section 2, we define WKB solutions of the generalized
hypergeometric differential equation with a large parameter and investigate the local behavior
of the solutions at the singularities of the equation. In Section 3, we define the Voros coefficients
at the origin and at the infinity for our equation. The explicit forms of the Voros coefficients
are given (Theorem 3.9).
2 WKB solutions
2.1 The generalized hypergeometric differential equation
with a large parameter
Let N be an integer greater than 1, let a denote an N -tuple of parameters ai ∈ C, i =
1, 2, . . . , N ,) and let b denote an (N − 1)-tuple of parameters bj ∈ C \ {0,−1,−2, . . . }, j =
1, 2, . . . , N − 1. We set
NFN−1
(
a
b
;x
)
=
∞∑
k=0
(a1)k(a2)k · · · (aN )k
(b1)k(b2)k · · · (bN−1)kk!
xk (2.1)
and call this the generalized hypergeometric series [12]. Here (α)k denotes the Pochhammer
symbol (= Γ(α+ k)/Γ(α)). As is well known, the radius of convergence of (2.1) equals 1 and
the right-hand side of (2.1) defines a holomorphic function on the universal covering of C\{0, 1},
which is also denoted by NFN−1. This series or function satisfy the following N -th order ordinary
differential equation:N−1∏
j=1
(ϑx + bj)
∂x −
(
N∏
i=1
(ϑx + ai)
)w = 0. (2.2)
Here we set ∂x = d
dx and ϑx = x d
dx . We call (2.2) the generalized hypergeometric differential
equation. This equation has regular singular points at x = 0, 1,∞. We set bN = 1, b̃ =
(b1, b2, . . . , bN−1, 1) and 1 = (1, 1, 1, . . . , 1). We suppose that ai − aj , bi − bj ̸∈ Z, 1 ≤ i, j ≤ N ,
i ̸= j, and
∑N
i=1 (bi − ai) ̸∈ Z. Then there exists a fundamental system of solutions around the
singular point
{
w
[ϱ]
1 , . . . , w
[ϱ]
N
}
of (2.2). If ϱ = 0,∞, they are given by
w
[0]
1 = NFN−1
(
a
b
;x
)
,
w
[0]
j+1 = x1−bj
NFN−1
(
a+ (1− bj)1(
b̃+ (1− bj)1
)∨;x) ,
w
[∞]
i = (−x)−ai
NFN−1
(
(1 + ai)1− b̃
((1− ai)1+ a)∨
;
1
x
)
for j = 1, . . . , N − 1, i = 1, . . . , N , where ∗∨ indicates that the entry 1 + bj − bj or 1 − ai + ai
omitted in ∗ [24]. Connection problem of (2.2) between x = 0 and x = ∞ had already been
studied well [9, 21, 26].
Voros Coefficients at the Origin and at the Infinity 3
We introduce a positive large parameter η in a and b by setting
ai = ai,0 + ai,1η, i = 1, 2, . . . , N,
bj = bj,0 + bj,1η, j = 1, 2, . . . , N − 1,
where ai,k, bj,k ∈ C for k = 0, 1. We consider the following equation which contains the large
parameter:
NPN−1ψ = 0, (2.3)
where we set
NPN−1 = η−N
N−1∏
j=1
(ϑx + bj)
∂x −
(
N∏
i=1
(ϑx + ai)
) . (2.4)
We call (2.4) the generalized hypergeometric differential equation with the large parameter. The
singular points of the equation (2.3) (or (2.2)) are 0, 1 and ∞.
The differential operator (2.4) is a WKB type operator defined on C in the sense of [3,
Definition 2.1]. Let ξ denote the dual variable of x. Then ξ is regarded as the symbol of the
differential operator ∂x (cf. [1, 2, 3]). We introduce a new variable ζ by setting ζ = η−1ξ. Since
the large parameter η is regarded as the dual variable of the variable y of the Borel plane [20],
η indicates the symbol of ∂y = ∂/∂y. Hence ζ designates the symbol of the microdifferential
operator ∂−1
y ∂x. Then the total symbol σ(NPN−1)(x, ζ, η) in the sense of [3] can be considered as
a function x, ζ and η−1. We write σk(NPN−1)(x, ζ) the coefficient of η−k of σ(NPN−1)(x, ζ, η).
Then the total symbol can be expressed in the form
σ(NPN−1)(x, ζ, η) =
N∑
k=0
η−kσk(NPN−1)(x, ζ). (2.5)
We call the leading term σ0(NPN−1)(x, ζ) of (2.5) the principal symbol of NPN−1. For anm-tuple
c = (c1, c2, . . . , cm) of parameters, we denote by sℓ(c) the elementary symmetric polynomial of
degree ℓ of c1, c2, . . . , cm. If ℓ > m or ℓ ≤ −1, we set sℓ(c) = 0.
Lemma 2.1. The total symbol σ(NPN−1)(x, ζ, η) is written in the form:
σ(NPN−1)(x, ζ, η)
=
N∑
k=1
N∑
j=k
({
j − 1
k − 1
}
sN−j(b)−
{
j
k
}
sN−j(a)x
)xk−1η−N+kζk − η−NsN (a).(2.6)
Here
{
j
k
}
denotes the Stirling number of the second kind [14, 24].
Proof. By the definition of the Stirling number of the second kind, the j-th power of the Euler
operator ϑx becomes
ϑjx =
j∑
k=0
{
j
k
}
xk∂kx , j ≥ 0. (2.7)
Expanding the right-hand side of (2.4) and using (2.7), we have
ηNNPN−1 =
∑
0≤j≤N−1
0≤k≤j
{
j
k
}
sN−1−j(b)x
k∂k+1
x −
∑
0≤j≤N
0≤k≤j
{
j
k
}
sN−j(a)x
k∂kx .
Hence, we obtain (2.6). ■
4 T. Aoki and S. Uchida
Remark 2.2. The principal symbol σ0(NPN−1)(x, ζ) is written in the form:
σ0(NPN−1)(x, ζ) = ζ
N−1∏
j=1
(xζ + bj,1)−
N∏
i=1
(xζ + ai,1)
=
N∑
k=1
(
sN−k(b1)− sN−k(a1)x
)
xk−1ζk − sN (a1). (2.8)
Here we set a1 = (a1,1, a2,1, . . . , aN,1) and b1 = (b1,1, b2,1, . . . , bN−1,1).
2.2 Turning points and WKB solutions
A point x∗ ∈ C is called a turning point of NPN−1 with the characteristic value ζ∗ if
σ0(NPN−1)(x∗, ζ∗) = ∂ζσ0(NPN−1)(x∗, ζ∗) = 0
and σ0(NPN−1)(x, ζ) does not vanish identically as a function ζ (see [3, Definition 3.3]). The
turning point x∗ with the characteristic value ζ∗ is said to be simple if
∂xσ0(NPN−1)(x∗, ζ∗) ̸= 0, ∂2ζσ0(NPN−1)(x∗, ζ∗) ̸= 0
(see [3, Definition 3.6]).
Let P(NPN−1) denote the set
{(x∗, ζ∗) |σ0(NPN−1)(x∗, ζ∗) = ∂ζσ0(NPN−1)(x∗, ζ∗) = 0}.
This set can be regarded as a subset of P2
C. Let Ptp(NPN−1) be the projection of P(NPN−1) to
the x-space. Note that the singular points 0, 1, ∞ do not belong to Ptp(NPN−1) for generic a1
and b1. By the definition, the turning points of (2.3) should satisfy the following equation:
resζ (σ0(NPN−1)(x, ζ), ∂ζσ0(NPN−1)(x, ζ)) = 0. (2.9)
Here resζ(F,G) denotes the resultant of F and G with respect to ζ.
Lemma 2.3. The resultant (2.9) is rewritten in the form
resζ
(
σ0(NPN−1)(x, ζ), ∂ζσ0(NPN−1)(x, ζ)
)
=
(−1)N−1
NN−2
x(N−1)2(1− x) resζ
(
f(x, ζ), g(x, ζ)
)
. (2.10)
Here we set
f(x, ζ) =
N−1∑
k=0
(k + 1)(sN−k−1(b1)− sN−k−1(a1)x)ζ
k, (2.11)
g(x, ζ) =
N−1∑
k=1
(N − k)(sN−k(b1)− sN−k(a1)x)ζ
k −NsN (a1)x. (2.12)
Proof. For the sake of simplicity, we show (2.10) for the case N = 3. General case can be
proved in a similar way. For N = 3, the principal symbol is written in the form
σ0(3P2)(x, ζ) = x2q3ζ
3 + xq2ζ
2 + q1ζ + q0,
Voros Coefficients at the Origin and at the Infinity 5
Hence the left-hand side of (2.10) has the form∣∣∣∣∣∣∣∣∣∣
q3x
2 q2x q1 q0 0
0 q3x
2 q2x q1 q0
3q3x
2 2q2x q1 0 0
0 3q3x
2 2q2x q1 0
0 0 3q3x
2 2q2x q1
∣∣∣∣∣∣∣∣∣∣
. (2.13)
Firstly we eliminate the (3, 1)-element by using the first row. Next we eliminate the (4, 2)-
element by the second row. Then, expanding the determinant by the first column, we have
(−1)2q3x
2
∣∣∣∣∣∣∣∣
q3x
2 q2x q1 q0
q2x 2q1 3q0 0
0 q2x 2q1 3q0
0 3q3x
2 2q2x q1
∣∣∣∣∣∣∣∣ .
We eliminate the (1, 4)-element by using the third row and multiply the first row by 3. Then,
factoring x out from the first column, multiplying the third and the fourth columns by x and x2,
respectively, exchanging the second row and the fourth row and finally, exchanging the third
row and the fourth row, we have
(−1)2
3
q3
∣∣∣∣∣∣∣∣
3q3x 2q2x q1x 0
0 3q3x
2 2q2x
2 q1x
2
q2 2q1 3q0x 0
0 q2x 2q1x 3q0x
2
∣∣∣∣∣∣∣∣ .
We factor x, x2 and x out from the first, the second and the fourth row, respectively. Then we
obtain the expression of (2.13) of the form
(−1)2
3
q3x
4
∣∣∣∣∣∣∣∣
3q3 2q2 q1 0
0 3q3 2q2 q1
q2 2q1 3q0x 0
0 q2 2q1 3q0x
∣∣∣∣∣∣∣∣ .
Thus we have
resζ(σ0(3P2)(x, ζ), ∂ζσ0(3P2)(x, ζ)) =
(−1)2
3
(1− x)x4 resζ (f(x, ζ), g(x, ζ))
with
f(x, ζ) = 3q3ζ
2 + 2q2ζ + q1,
g(x, ζ) = q2ζ
2 + 2q1ζ + 3q0x.
This proves Lemma 2.3 for N = 3. ■
We assume that the following conditions are satisfied.
Assumption 2.4.
(i) All turning points of the equation (2.3) are simple.
(ii) s1(a1) ̸= s1(b1).
(iii) The leading coefficient of resζ (f(x, ζ), g(x, ζ)) with respect to x does not vanish.
6 T. Aoki and S. Uchida
(iv) Disx (resζ (f(x, ζ), g(x, ζ))) ̸= 0. Here Disx(F ) denotes the discriminant of F with respect
to x.
Lemma 2.5. The number of elements in the set P(NPN−1) equals 2(N − 1).
Proof. From Lemma 2.3, a point (x, ζ) belongs to P(NPN−1) if and only if (x, ζ) satisfies
f(x, ζ) = 0 and g(x, ζ) = 0. Here f(x, ζ) and g(x, ζ) denote (2.11) and (2.12), respectively.
Eliminating x from f(x, ζ) = 0 and g(x, ζ) = 0, we get the equation h(ζ) = 0 for ζ, where
h(ζ) =
2(N−1)∑
k=N
(
2N−k∑
j=1
j(N − k − j + 1)
(
sj−1(b1)s2N−k−j(a1)− sj−1(a1)s2N−k−j(b1)
))
ζk
+
N−1∑
k=0
(
k+1∑
j=0
N(k + j + 1)
(
sN−k+j−1(a1)sN−j(b1)− sN−k+j−1(b1)sN−j(a1)
))
ζk.
We can see that the coefficient of the highest degree part of h(ζ) does not vanish if s1(a1) ̸=
s1(b1). The coefficient of ζk of h(ζ) is a homogeneous polynomial of degree 2N − k − 1 of a1
and b1 which does not vanish identically. Thus, we can show that there are distinct 2(N − 1)
roots of h(ζ) = 0 by using Assumption 2.4. ■
Remark 2.6. If N = 3,
Disx (resζ (f(x, ζ), g(x, ζ))) = 256
∏
i=1,2,3
ai,1
∏
i=1,2,3
j=1,2
(ai,1 − bj,1)h
′(a1, b1)
3
holds. Here h′(a1, b1) is a homogeneous polynomial of degree 9 with respect to a1, b1. From
h′(a1, b1) = 0, we have the following conditions:
(i) bj,1 ̸= 0 for any j, 1 ≤ j ≤ 2.
(ii) bj,1 ̸= bj′,1 for j ̸= j′, 1 ≤ j, j′ ≤ 2.
The leading coefficient of resζ(f(x, ζ), g(x, ζ)) with respect to x does not vanish. Then we have∏
i ̸=i′
(ai,1 − ai′,1) ̸= 0.
Outside the turning points, there are N distinct roots of the algebraic equation
σ0(NPN−1)(x, ζ) = 0
in ζ of degree N . We call these roots the characteristic roots of NPN−1. We consider the Laurent
expansion at the singular point of each characteristic root. For the case of ϱ = 0, we substitute
ζ =
∑∞
k=m ckx
k for (2.8). Taking note of the degree of the leading term of (2.8) with respect
to x, we find m = −1 or m = 0. If m = −1, there are N − 1 choices for the leading coefficient:
c−1 = −bℓ,1, ℓ = 1, 2, . . . , N − 1.
If m = 0, we can see
c0 =
a1,1a2,1 · · · aN,1
b1,1b2,1 · · · bN−1,1
.
The coefficients of higher-order terms cm+ℓ, ℓ > 0, are determined recursively. Similarly, we can
find the Laurent expansions of the characteristic roots at x = ∞.
Voros Coefficients at the Origin and at the Infinity 7
Definition 2.7. We take local numbering of the characteristic roots of NPN−1 as follows:
(i) For ϱ = 0, the roots are denoted by ζ
(0)
m , m = 1, 2, . . . , N , so that
ζ
(0)
ℓ = −
bℓ,1
x
+O(1), ℓ = 1, 2, . . . , N − 1,
ζ
(0)
N =
a1,1a2,1 · · · aN,1
b1,1 · · · bN−1,1
+O(x)
hold.
(ii) For ϱ = ∞, the roots are denoted by ζ
(∞)
m , m = 1, 2, . . . , N , so that
ζ(∞)
m = −am,1
x
+O
(
1
x2
)
hold.
Taking suitable branch cuts connecting simple turning points, we can regard this numbering
is defined globally. For a simple turning point x∗ and a singular point ϱ, there are two numbers
j, k ∈ {1, 2, . . . , N}, j ̸= k such that ζ
(ϱ)
j (x∗) = ζ
(ϱ)
k (x∗) holds. We consider mainly the case where
ϱ = 0. Then we say that x∗ is a simple turning point of type (j, k) (see [16, Definition 1.2.1]).
The characteristic variety Ch(NPN−1) of NPN−1 is, by definition, an algebraic curve
Ch(NPN−1) = {(x, ζ) |σ0(NPN−1)(x, ζ) = 0}.
This can be regarded as a compact Riemann surface Σ. There is a natural projection
π : Σ → P1
C,
which is an N -covering map.
Lemma 2.8. The genus of Σ equals 0.
Proof. By using the Riemann–Hurwitz formula [13, Section 17], we have the following relation:
2− 2g(Σ) = N
(
2− 2g
(
P1
C
))
−
2(N−1)∑
i=1
(2− 1).
Here g(X) denotes the genus of the compact Riemann surface X. Hence we get g(Σ) = 0. ■
A WKB solution ψ of (2.3) is a formal solution of the form
ψ = exp
(∫
S dx
)
,
S = ηS−1 + S0 + η−1S1 + · · · =
∞∑
ℓ=−1
η−ℓSℓ
(see [3, Definition 3.2]). By inserting ψ = exp
( ∫
S dx
)
into (2.3), we have the following nonlinear
differential equation for S:
Ri(NPN−1)(S) = 0. (2.14)
Here we set
Ri(NPN−1)(S) = exp
(
−
∫
S dx
)
NPN−1 exp
(∫
S dx
)
.
8 T. Aoki and S. Uchida
The leading term of the equation (2.14) with respect to η−1 determines S−1. Therefore S−1
should satisfy σ0(NPN−1)(x, S−1) = 0. Hence we can take one of the characteristic roots as S−1.
The higher-order terms Sj , j = 0, 1, 2, . . . , can be determined recursively and uniquely outside
the turning points if S−1 is chosen. The following Lemma is obtained by the same way as [13,
Proposition 3.6]:
Lemma 2.9. Let S(ϱ,m), m = 1, 2, . . . , N , be the formal solutions of (2.3) such that the num-
bering are consistent with that of the leading terms given in Definition 2.7. Then the formal
solutions S(ϱ,m), m = 1, 2, . . . , N , have the following local behaviors near x = ϱ, ϱ = 0,∞:
S(0,ℓ) =
1− bℓ
x
+
N∏
m=1
(1 + am − bℓ)
(bℓ − 2)
N−1∏
m=1
m ̸=ℓ
(bℓ − bm − 1)
+O(x), ℓ = 1, 2, . . . , N − 1, (2.15)
S(0,N) =
a1a2 · · · aN
b1b2 · · · bN−1
+O(x), (2.16)
S(∞,ℓ) = −aℓ
x
−
aℓ
N−1∏
m=1
(1 + aℓ − bm)
N∏
m=1
m̸=ℓ
(1 + aℓ − am)
1
x2
+O
(
1
x3
)
, ℓ = 1, 2, . . . , N. (2.17)
Here we use the notation O(x) in (2.15) and (2.16) in the sense that these parts can be written
as
x
∞∑
j=−1
η−jfj(x)
with some holomorphic functions fj(x), j = −1, 0, 1, . . . , near x = 0. The notation O
(
1/x3
)
in (2.17) should be understood similarly.
2.3 Factorization
We assume that there is a simple turning point x∗ of NPN−1 with the characteristic value ζ∗.
Then σ0(NPN−1)(x, ζ) is uniquely decomposed holomorphically in a neighborhood of (x∗, ζ∗) in
the form
σ0(NPN−1)(x, ζ) = l(x, ζ)r(x, ζ),
where r(x, ζ) is a Weierstrass polynomial of degree 2 in ζ with the center at (x∗, ζ∗) and l(x∗, ζ∗)
̸= 0. By the definition, r(x, ζ) has the form
r(x, ζ) = (ζ − ζ∗)
2 + f1(x)(ζ − ζ∗) + f2(x),
where fj(x) vanishes at x = x∗ for j = 1, 2. We may assume ζ
(ϱ)
j and ζ
(ϱ)
k are the roots of
r(x, ζ) = 0. We set
N P̃N−1 =
1
xN−1(1− x)
NPN−1.
Then σ
(
N P̃N−1
)
(x, ζ, η) becomes a monic polynomial with respect to ζ:
σ
(
N P̃N−1
)
(x, ζ, η) = ζN +
N−1∑
k=0
p̃k(x, η)ζ
k.
Voros Coefficients at the Origin and at the Infinity 9
It follows from [3, Theorem 5.1] that there uniquely exist differential operators L and R of WKB
type near x∗ which satisfy
N P̃N−1 = LR (2.18)
and
(i) The principal symbol σ0(R)(x, ζ) of R coincides with r(x, ζ).
(ii) For each j > 0, the coefficient σj(R)(x, ζ) of η
−j of the symbol of R is of degree at most
one in ζ.
(iii) The principal symbol σ0(L)(x, ζ) of L does not vanish at (x∗, ζ∗).
Thus constructed operator R has the form
R =
(
η−1∂x
)2
+A(x, η)η−1∂x +B(x, η)
with
A(x, η) = A0(x) +A1(x)η
−1 +A2(x)η
−2 + · · · ,
B(x, η) = B0(x) +B1(x)η
−1 +B2(x)η
−2 + · · ·
and L is an (N − 2)-th order operator
L =
(
η−1∂x
)N−2
+
N−3∑
k=0
Lk(x, η)
(
η−1∂x
)k
with
Lk(x, η) = Lk,0(x) + Lk,1(x)η
−1 + Lk,2(x)η
−2 + · · · .
The operators L and R are constructed as follows. Relation (2.18) yields
p̃2 = A+ L0, p̃1 = B +AL0 + η−1A′, p̃0 = BL0 + η−1B′
for N = 3 and
p̃N−1(x, η) = A+ LN−3,
p̃k(x, η) = η−N+k+2
((
N − 2
k − 1
)
η−1A(N−k−1) +
(
N − 2
k
)
B(N−k−2)
)
+
N−3∑
j=k
η−j+k
((
j
k − 1
)
η−1A(j−k+1) +
(
j
k
)
B(j−k)
)
Lj
+ Lk−2 +ALk−1, 2 ≤ k ≤ N − 2,
p̃1(x, η) = η−N+3
(
η−1A(N−2) + (N − 2)B(N−3)
)
+
N−3∑
j=1
η−j+1
(
η−1A(j) + jB(j−1)
)
Lj +AL0,
p̃0(x, η) = η−N+2B(N−2) +
N−3∑
j=0
η−jB(j)Lj
10 T. Aoki and S. Uchida
for N ≥ 4. These relations determine A, B and Lk, 0 ≤ k ≤ N − 3, if we choose the leading
terms of them, which solve a system of algebraic equations. For example, if N = 3, we obtain A
by solving the following equation:
A3 − 2p̃2A
2 −
(
3η−1A′ − η−1p̃2 − p̃1 − p̃22
)
A
+ 2η−1p̃2A
′ + η−1A′′ + p̃0 − p̃1p̃2 − η−1p̃′1 = 0.
We consider a WKB solution
ϕ = exp
(∫
T dx
)
,
T = ηT−1 + T0 + η−1T1 + · · · =
∞∑
ℓ=−1
η−ℓTℓ
of Rϕ = 0. Here T is determined by the Riccati equation associated with this equation:
Ri(R)(T ) = η−2
(
T 2 + T ′)+Aη−1T +B = 0. (2.19)
Since we have (2.18), ϕ is a formal solution of (2.18). The following lemma shows that we may
regard it as a WKB solution of (2.18).
Lemma 2.10. We have the following relation for N ≥ 3:
Ri
(
N P̃N−1
)
(S) =
N−2∑
k=0
η−k
k!
Ri
(
:∂kζ σ(L)(x, ζ, η):
)
(S)∂kx Ri(R)(S).
Here :σ(P )(x, ζ, η): designates the differential operator defined by the total symbol σ(P )(x, ζ, η)
(see [1, Definition 4.6] and [3]). Hence Ri(R)(T ) = 0 implies Ri
(
N P̃N−1
)
(T ) = 0.
Remark 2.11. The differential operator :∂kζ σ(L)(x, ζ, η): is recovered from the total symbol
∂kζ σ(L)(x, ζ, η) by replacing ζ by η−1∂x after moving all powers of ζ in each term in the symbol
to the rightmost part.
Proof. Since we have (2.18), Ri
(
N P̃N−1
)
(S) is computed in the following form
Ri
(
N P̃N−1
)
(S) = e−
∫
S dxLRe
∫
S dx
= e−
∫
S dxLe
∫
S dxe−
∫
S dxRe
∫
S dx.
By the definition of Ri(R)(S), this equals
e−
∫
S dxLe
∫
S dxRi(R)(S),
which can be written as
e−
∫
S dxLRi(R)(S)e
∫
S dx.
The product of the differential operator L of order N−2 and the multiplying operator Ri(R)(S)
has the symbol
N−2∑
k=0
η−k
k!
∂kζ σ(L)(x, ζ, η)∂
k
x Ri(R)(S).
Voros Coefficients at the Origin and at the Infinity 11
Since ∂kx Ri(R)(S) does not contain ζ for each k, we can write
LRi(R)(S) = :
N−2∑
k=0
η−k
k!
∂kζ σ(L)(x, ζ, η)∂
k
x Ri(R)(S):
=
N−2∑
k=0
η−k
k!
∂kx Ri(R)(S):∂
k
ζ σ(L)(x, ζ, η):.
Thus we have
e−
∫
S dxLRi(R)(S)e
∫
S dx =
N−2∑
k=0
η−k
k!
Ri
(
:∂kζ σ(L)(x, ζ, η):
)
(S)∂kx Ri(R)(S).
This proves the lemma. ■
2.4 Definition of Voros coefficients
Recall that the characteristic roots ζ
(ϱ)
j and ζ
(ϱ)
k introduced in the preceding section satisfy the
quadratic equation
ζ2 +A0ζ +B0 = 0.
We may assume
S
(ϱ,j)
−1 = ζj =
−A0 +
√
A2
0 − 4B0
2
, S
(ϱ,k)
−1 = ζk =
−A0 −
√
A2
0 − 4B0
2
for suitable choice of the branch of the square root. We set
S(ϱ,j,k)
even =
1
2
(
S(ϱ,j) + S(ϱ,k)
)
, S
(ϱ,j,k)
odd =
1
2
(
S(ϱ,j) − S(ϱ,k)
)
. (2.20)
Then we have
S(ϱ,j,k)
even +
1
2
ηA = −1
2
∂xS
(ϱ,j,k)
odd
S
(ϱ,j,k)
odd
= −1
2
d
dx
logS
(ϱ,j,k)
odd
by using (2.19). Thus we can take the following normalization of integration:∫
S(ϱ,j,k)
even dx = −1
2
logS
(ϱ,j,k)
odd − 1
2
η
∫
Adx.
Here we choose a primitive function
∫
Adx.
Definition 2.12. The WKB solutions ψ
(τ)
± of (2.3) normalized at the turning point τ of type
(j, k) are defined by
ψ
(τ)
± =
1√
S
(ϱ,j,k)
odd
exp
(
−1
2
η
∫
Adx
)
exp
(
±
∫ x
τ
S
(ϱ,j,k)
odd dx
)
.
Here the integration of S
(ϱ,j,k)
odd from τ to x is understood as a half of the contour integral of it
on the path starting from x on the k-th sheet, going around τ counterclockwise and back to x
on the j-th sheet.
12 T. Aoki and S. Uchida
We denote by S
(ϱ,j,k)
odd,ℓ the coefficient of η−ℓ of S
(ϱ,j,k)
odd and set S
(ϱ,j,k)
odd,≤0 = ηS
(ϱ,j,k)
odd,−1 + S
(ϱ,j,k)
odd,0 . It
follows from Lemma 2.9 and (2.20) that
Res
x=ϱ
S
(ϱ,j,k)
odd dx = Res
x=ϱ
S
(ϱ,j,k)
odd,≤0 dx
holds.
Definition 2.13. The WKB solutions ψ
(ϱ)
± of (2.3) normalized at the singular point ϱ are defined
by
ψ
(ϱ)
± =
1√
S
(ϱ,j,k)
odd
exp
(
−1
2
η
∫
Adx
)
exp
(
±
∫ x
ϱ
(
S
(ϱ,j,k)
odd − S
(ϱ,j,k)
odd,≤0
)
dx±
∫ x
τ
S
(ϱ,j,k)
odd,≤0 dx
)
.
Definition 2.14. The Voros coefficient V
(j,k)
ϱ at x = ϱ of type (j, k) is defined by
V (j,k)
ϱ =
∫ τ
ϱ
(
S
(ϱ,j,k)
odd − S
(ϱ,j,k)
odd,≤0
)
dx.
The Voros coefficient relates two kinds of the normalization of WKB solutions defined in the
preceding subsection. We have the following formal relations:
ψ
(ϱ)
± = exp
(
V (j,k)
ϱ
)
ψ
(τ)
± .
The Voros coefficient is rewritten as
V (j,k)
ϱ =
1
2
∫
γj,k
(S − ηS−1 − S0) dx.
Here we consider that S is defined on the Riemann surface Σ of S−1 and γj,k is a path on Σ
starting from the singular point ϱ on the j-th sheet, going to and detouring τ counterclockwise
on the base space and back to the singular point ϱ on the k-th sheet.
k-th sheet
j-th sheet
τ
ϱ
γj,k
Figure 1. The path γj,k.
Remark 2.15. For any 1 ≤ j, k, ℓ ≤ N , the following relation holds from Lemma 2.8:
V (j,k)
ϱ + V (k,ℓ)
ϱ = V (j,ℓ)
ϱ .
Especially, V
(j,k)
ϱ = −V (k,j)
ϱ holds.
Voros Coefficients at the Origin and at the Infinity 13
3 Explicit forms of the Voros coefficients
3.1 Ladder operators
Lemma 3.1. Let S ( ab ) be the linear space of all solutions of NPN−1ψ = 0. The operators
H(ai) = ϑx + ai, i = 1, 2, . . . , N,
B(bj) = ϑx + bj , j = 1, 2, . . . , N − 1
induce homomorphisms
H(ai) : S
(
a
b
)
→ S
(
a+ ei
b
)
, i = 1, 2, . . . , N,
B(bj) : S
(
a
b+ ej
)
→ S
(
a
b
)
, j = 1, 2, . . . , N − 1, (3.1)
respectively. Here em denotes the m-th unit vector.
Proof. We set
N P̂N−1 = NPN−1
∣∣
ai,1 7→ai,1+η−1 (3.2)
for 1 ≤ i ≤ N . Then we have the following relation for 1 ≤ i ≤ N :
H(ai)xNPN−1 = xN P̂N−1H(ai). (3.3)
Hence ψ ∈ S ( ab ) implies H(ai)ψ ∈ S
(
a+ei
b
)
. Similarly, (3.1) can be proved. ■
Lemma 3.2. Let S be a formal solution of Ri(NPN−1)(S) = 0. Then
Ri
(
N P̂N−1
)(
∂x log(xS + ai) + S
)
= 0, i = 1, 2, . . . , N, (3.4)
Ri(NPN−1)
(
∂x log
(
xŜ + bj
)
+ Ŝ
)
= 0, j = 1, 2, . . . , N − 1 (3.5)
hold. Here we set (3.2) and
Ŝ = S
∣∣
bj,1 7→bj,1+η−1 (3.6)
for j = 1, 2, . . . , N − 1.
Proof. We fix the index i, 1 ≤ i ≤ N . By the definition of Ri(∗)(⋆), we have the following
relations:
Ri
(
N P̂N−1
)(
∂x log(xS + ai) + S
)
=
1
xS + ai
e−
∫
S dx
N P̂N−1(xS + ai)e
∫
S dx, (3.7)
Ri
(
N P̂N−1H(ai)
)
(S) = e−
∫
S dx
N P̂N−1(xS + ai)e
∫
S dx. (3.8)
Combining (3.7) and (3.8), we have
Ri
(
N P̂N−1
)(
∂x log(xS + ai) + S
)
=
1
xS + ai
Ri
(
N P̂N−1H(ai)
)
(S). (3.9)
From (3.3), we get
Ri(H(ai)xNPN−1)(S) = Ri
(
xN P̂N−1H(ai)
)
(S). (3.10)
14 T. Aoki and S. Uchida
By using a similar argument employed in the proof of Lemma 2.10, the left-hand side of (3.10)
can be written in the form:
Ri(H(ai)xNPN−1)(S) =
∑
k≥0
η−k
k!
Ri
(
:∂kζ (xηζ + ai):
)
(S)∂kx Ri(xNPN−1)(S). (3.11)
Then the right-hand side of (3.11) is equal to
(xS + ai)xRi(NPN−1)(S) + x
(
Ri(NPN−1)(S) + x∂xRi(NPN−1)(S)
)
.
Here we used the relation
Ri(xNPN−1)(S) = xRi(NPN−1)(S).
Since Ri(NPN−1)(S) = 0, the right-hand side of (3.11) vanishes. Combining (3.9), (3.10)
and (3.11), we obtain (3.4). The relation (3.5) can be proved as well. ■
Lemma 3.3. The formal solution S satisfies the following relations:
∆ai,1S = ∂x log(xS + ai), i = 1, 2, . . . , N,
∆bj,1S = −∂x log
(
xŜ + bj
)
, j = 1, 2, . . . , N − 1,
where we set (3.6) and
∆ρS = S
∣∣
ρ 7→ρ+η−1 − S
for ρ = a1,1, a2,1, . . . , aN,1, b1,1, b2,1, . . . , bN−1,1.
Remark 3.4. Note that we distinguish ρ from ϱ.
Remark 3.5. The difference operator ∆ρ in ρ by η−1can be written in the following form by
using a formal differential operator of infinite order:
∆ρ = eη
−1∂ρ − 1.
Proof. We denote ψ̂ = H(ai)ψ, which belongs to S
(
a+ei
b
)
. Then
∂x log ψ̂ = ∂x log(xS + ai) + S
holds. It follows from Lemma 3.2 that Ŝ satisfies the equation obtained from Ri(NPN−1)(S) = 0
by replacing ai by ai + 1. Hence we have
∂x log(xS + ai) + S =
(ηS−1 + S0 + · · · ) + ∂x(ηS−1 + S0 + · · · )
x(ηS−1 + S0 + · · · ) + ai
+ (ηS−1 + S0 + · · · ). (3.12)
The leading term with respect to η−1 of the right-hand side of (3.12) equals S−1. We can
choose the leading term of Ŝ as S−1
∣∣
ai,1 7→ai,1+η−1 . Thus, we conclude that Ŝ = S
∣∣
ai,1 7→ai,1+η−1 .
Similarly, Lemma 3.3 can be proved for b1,1, b2,1, . . . , bN−1,1. ■
Lemma 3.6. Let (ρ, ρ0) denote one of (a1,1, a1,0), . . . , (bN−1,1, bN−1,0). The Voros coefficient
V
(j,k)
ϱ , j < k, satisfies the following differential-difference equations:
∂ρ∆ρV
(j,k)
ϱ = f (j,k)ρ +∆ρg
(j,k)
ρ . (3.13)
Here f
(j,k)
ρ are given as follows and here g
(j,k)
ρ is a linear function of η:
Voros Coefficients at the Origin and at the Infinity 15
(i) If ϱ = 0,
2f (j,k)ai,1 =
η
1 + ai − bk
− η
1 + ai − bj
, k ̸= N,
2f (j,N)
ai,1 =
η
ai
− η
1 + ai − bj
,
2f
(j,k)
bm,1
=
η
1 + bm − bj
− η
1 + bm − bk
, k ̸= N,m ̸= j, k,
2f
(j,k)
bj,1
= − η
1 + bj − bk
− η
bj − 1
−
N−1∑
ℓ=1
ℓ̸=j
η
bj − bℓ
−
N∑
ℓ=1
η
aℓ − bj
, k ̸= N,
2f
(j,k)
bk,1
=
η
1 + bk − bj
+
η
bk − 1
+
N−1∑
ℓ=1
ℓ ̸=k
η
bk − bℓ
+
N∑
ℓ=1
η
aℓ − bk
, k ̸= N,
2f
(j,N)
bm,1
=
η
1 + bm − bj
− η
bm
, m ̸= j,
2f
(j,N)
bj,1
=
N∑
ℓ=1
η
bj − aℓ
− η
bj
− η
bj − 1
−
N−1∑
ℓ=1
ℓ̸=j
η
bj − bℓ
,
i = 1, 2, . . . , N and m = 1, 2, . . . , N − 1.
(ii) If ϱ = ∞,
2f (j,k)ai,1 =
η
ai − ak
− η
ai − aj
, i ̸= j, k,
2f (j,k)aj,1 = − η
ak − aj
− η
aj
+
N∑
ℓ=1
ℓ̸=j
η
1 + aj − aℓ
−
N−1∑
ℓ=1
η
1 + aj − bℓ
,
2f (j,k)ak,1
=
η
aj − ak
+
η
ak
−
N∑
ℓ=1
ℓ̸=k
η
1 + ak − aℓ
+
N−1∑
ℓ=1
η
1 + ak − bℓ
,
2f
(j,k)
bm,1
=
η
bm − aj
− η
bm − ak
,
i = 1, 2, . . . , N and m = 1, 2, . . . , N − 1.
The system of differential-difference equations (3.13) satisfy the compatibility conditions:
∂τ∆τ∂ρ∆ρV
(j,k)
ϱ = ∂ρ∆ρ∂τ∆τV
(j,k)
ϱ , ∀ρ, τ ∈ {a1,1, . . . , aN,1, b1,1, . . . , bN−1,1}.
Proof. We consider the integral∫
γ̃j,k
∆ρS dx
for ρ ∈ {a1,1, . . . , aN,1, b1,1, . . . , bN−1,1}. Here the path γ̃j,k of integration is taken as follows:
Let x be a point near the origin and x(m) the point on the m-th sheet of Riemann surface Σ
such that π(x(m)) = x. We take a path on Σ starting from x(j), going to and detouring τ
counterclockwise on the base space and back to x(k) and denote this by γ̃j,k. It follows from
Lemma 2.9 that this integral can be written in the form
c
(j,k)
−1 log x+ c
(j,k)
0 +O(x)
16 T. Aoki and S. Uchida
k-th sheet
j-th sheet
τ
x
γ̃j,k
Figure 2. The path γ̃j,k.
with some constant series c
(j,k)
−1 and c
(j,k)
0 of η−1. Here we use the notation O(x) in the same
sense as in Lemma 2.9. Let d
(j,k)
m denote the coefficient of η−m of c
(j,k)
0 :
c
(j,k)
0 =
∞∑
m=0
d(j,k)m η−m. (3.14)
On the other hand, by using Lemma 3.3, we have
∫
γ̃j,k
∆ρS dx =
log
(
xS(0,k) + ai
xS(0,j) + ai
)
, ρ = ai,1, 1 ≤ i ≤ N,
− log
(
xŜ(0,k) + bm
xŜ(0,j) + bm
)
, ρ = bm,1, 1 ≤ m ≤ N − 1.
(3.15)
Expanding the right-hand side of (3.15) with respect to x, we obtain the coefficient c
(j,k)
0 . More-
over, Lemma 2.9 implies that c
(j,k)
0 is written as the logarithm of a product of several linear
functions in ρ, whose coefficients of ρ are −1 or 1.
Let F
(m)
ℓ be a primitive function of S
(0,m)
ℓ with respect to x and let
∫ x
dx denote the termwise
integration in x of the Laurent expansion of the integrand at x = 0, i.e.,∫ x (p−1
x
+ p0 + p1x+ · · ·
)
dx = p−1 log x+ p0x+
1
2
p1x
2 + · · · .
We set
r
(j,k)
ℓ = F
(k)
ℓ (x)− F
(j)
ℓ (x)−
(∫ x
S
(0,k)
ℓ dx−
∫ x
S
(0,j)
ℓ dx
)
. (3.16)
Note that r
(j,k)
ℓ does not depend on x and η. We let apply ∆ρ on both sides of (3.16). Then we
have ∫
γ̃j,k
∆ρSℓ dx =
∫ x
∆ρS
(0,k)
ℓ dx−
∫ x
∆ρS
(0,j)
ℓ dx+∆ρr
(j,k)
ℓ . (3.17)
Since ∆ρ = eη
−1∂ρ − 1, the right most term of (3.17) can be written in the form
∆ρr
(j,k)
ℓ =
(
η−1∂ρ +
1
2
η−2∂2ρ + · · ·
)
r
(j,k)
ℓ .
Voros Coefficients at the Origin and at the Infinity 17
Multiplying both members of (3.17) by η−ℓ and summing them up in ℓ, we have∫
γ̃j,k
∆ρS dx =
∞∑
ℓ=−1
η−ℓ
(∫ x
∆ρS
(0,k)
ℓ dx−
∫ x
∆ρS
(0,j)
ℓ dx+∆ρr
(j,k)
ℓ
)
. (3.18)
The constant term of the right-hand side of (3.18) at x = 0 is equal to
η
(
η−1∂ρ +
1
2
η−2∂2ρ + · · ·
)
r
(j,k)
−1 +
(
η−1∂ρ +
1
2
η−2∂2ρ + · · ·
)
r
(j,k)
0 + · · ·
= ∂ρr
(j,k)
−1 +
(
1
2
∂2ρr
(j,k)
−1 + ∂ρr
(j,k)
0
)
η−1 + · · · . (3.19)
Comparing the coefficients of η−m, m = 0, 1, of (3.14) and the right-hand side of (3.19), we have
d
(j,k)
0 = ∂ρr
(j,k)
−1 ,
d
(j,k)
1 =
1
2
∂2ρr
(j,k)
−1 + ∂ρr
(j,k)
0 . (3.20)
By the definition of V
(j,k)
0 , we have
∆ρV
(j,k)
0 =
1
2
lim
x→0
(∫
γ̃j,k
∆ρS dx− η
∫
γ̃j,k
∆ρS−1 dx−
∫
γ̃j,k
∆ρS0 dx
)
. (3.21)
By the definition of c
(j,k)
0 and r
(j,k)
ℓ , ℓ = −1, 0, the right-hand side of (3.21) is written in the
form:
∆ρV
(j,k)
0 =
1
2
(
c
(j,k)
0 − η∆ρr
(j,k)
−1 −∆ρr
(j,k)
0
)
. (3.22)
Differentiating (3.22) with respect to ρ and using (3.20), we have
∂ρ∆ρV
(j,k)
0 =
1
2
(
∂ρc
(j,k)
0 −∆ρ
(
ηd
(j,k)
0 + d
(j,k)
1 − 1
2
∂ρd
(j,k)
0
))
.
As we remarked above, c
(j,k)
0 is written as the logarithm of a product of several linear functions
in ρ. Setting
g(j,k)ρ = −1
2
(
ηd
(j,k)
0 + d
(j,k)
1 − 1
2
∂ρd
(j,k)
0
)
,
we have the lemma for ϱ = 0. Similarly, we get the differential-difference equations for Voros
coefficients at the infinity. ■
Remark 3.7. If N = 3, we have
2∂ai,1∆ai,1V
(1,2)
0 = − η
ai − b1 + 1
+
η
ai − b2 + 1
+∆ai,1
(
−η log
(
ai,1 − b2,1
ai,1 − b1,1
)
+
2ai,0 − 2b1,0 + 1
2ai,1 − 2b1,1
− 2ai,0 − 2b2,0 + 1
2ai,1 − 2b2,1
)
,
2∂ai,1∆ai,1V
(j,3)
0 =
η
ai − bj + 1
+
η
ai
+∆ai,1
(
−η log
(
ai,1
ai,1 − bj,1
)
− bj,0 − 1
ai,1 − bj,1
+
(1− 2ai,0) bj,1
2ai,1(ai,1 − bj,1)
)
,
18 T. Aoki and S. Uchida
2∂bm,1∆bm,1V
(1,2)
0
= (−1)m−1
3∑
i=1
η
bm − ai
−
∑
ℓ∈{1,2}\{m}
(
η
bm − 1
− η
bm − bℓ
− η
bm − bℓ + 1
)
+∆bm,1(−1)m−1
(
−η log
(
(a1,1 − bm,1) (a2,1 − bm,1) (a3,1 − bm,1)
bm,1 (b1,1 − b2,1) 2
)
− 1
2
(
3∑
i=1
2(ai,0 − bm,0) + 1
ai,1 − bm,1
+
3− 2bm,0
bm,1
− 4 (b1,0 − b2,0)
b1,1 − b2,1
))
,
2∂bm,1∆bm,1V
(j,3)
0 =
η
bm − bj + 1
− η
bm
+∆bm,1(−1)m−1
(
−η log
(
1− bj,1
bm,1
)
− bj,0 − 1
bj,1 − bm,1
+
(2bm,0 − 1)bj,1
2bm,1 (bj,1 − bm,1)
)
,
2∂bm,1∆bm,1V
(m,3)
0 =
3∑
i=1
η
bm − ai
− η
bm − 1
− η
bm
− η
bm − bj
+∆bm,1
(
−η log
(
(a1,1 − bm,1) (a2,1 − bm,1) (a3,1 − bm,1)
b2m,1 (bm,1 − bj,1)
)
− 1
2
(
3∑
i=1
2(ai,0 − bm,0) + 1
ai,1 − bm,1
+
4(1− bm,0)
bm,1
+
2(bj,0 − bm,0) + 1
bm,1 − bj,1
))
for i = 1, 2, 3, m = 1, 2 and j ∈ {1, 2} \ {m}.
The system of differential-difference equations given in Lemma 3.6 characterizes V
(j,k)
ϱ , for
we have the following lemma:
Lemma 3.8. Let W =
∑∞
n=−1wnη
−n be a formal solution of the differential-difference equation
∂ρ∆ρW = 0 (3.23)
for ρ = a1,1, . . . , aN,1, b1,1, . . . , bN−1,1 and suppose that wn is a homogeneous function of deg-
ree (−n) with respect to a1,1, . . . , aN,1, b1,1, . . . , bN−1,1 and w−1 = w0 = 0. Then, we have
W = 0.
Proof. By the definition of ∆ρ, we have
∂ρ∆ρW =
∞∑
ℓ=0
(
ℓ−1∑
n=−1
1
(ℓ− n)!
∂ℓ−n+1
ρ wn
)
η−ℓ
for all ρ ∈ {a1,1, . . . , aN,1, b1,1, . . . , bN−1,1}. Comparing the coefficients for both sides of (3.23)
with respect to the powers of η, we have
ℓ−1∑
n=−1
1
(ℓ− n)!
∂ℓ−n+1
ρ wn = 0, n = 0, 1, 2, . . . .
Since w−1 = w0 = 0, we have ∂2ρwn = 0 for n = 1, 2, . . . recursively. By the assumption, wn is
a homogeneous function of degree (−n) with respect to a1,1, . . . , aN,1, b1,1, . . . , bN−1,1. Therefore
we obtain
wn = 0, n = 1, 2, . . . . ■
Voros Coefficients at the Origin and at the Infinity 19
3.2 Voros coefficients
We have the explicit forms of the Voros coefficients V
(j,k)
ϱ at ϱ, ϱ = 0,∞.
Theorem 3.9. Let ϱ be 0 or ∞. If there exists a simple turning point of type (j, k), j < k,
of (2.3), then the Voros coefficients V
(j,k)
ϱ are written in the form:
1
2
∞∑
ℓ=2
(−1)ℓ+1η1−ℓ
ℓ(ℓ− 1)
V
(j,k)
ϱ,ℓ ,
where
V
(j,k)
0,ℓ =
Bℓ(bj,0 − 1)
bℓ−1
j,1
−
Bℓ(bk,0 − 1)
bℓ−1
k,1
+
N∑
i=1
Bℓ(bk,0 − ai,0)
(bk,1 − ai,1)ℓ−1
−
N∑
i=1
Bℓ(bj,0 − ai,0)
(bj,1 − ai,1)ℓ−1
+
N−1∑
m=1
m̸=j
Bℓ(bj,0 − bm,0)
(bj,1 − bm,1)ℓ−1
−
N−1∑
m=1
m̸=k
Bℓ(bk,0 − bm,0)
(bk,1 − bm,1)ℓ−1
, k ̸= N,
V
(j,N)
0,ℓ =
Bℓ(bj,0 − 1)
bℓ−1
j,1
−
N∑
i=1
Bℓ(ai,0)
aℓ−1
i,1
−
N∑
i=1
Bℓ(bj,0 − ai,0)
(bj,1 − ai,1)ℓ−1
+
N−1∑
m=1
m ̸=j
Bℓ(bj,0 − bm,0)
(bj,1 − bm,1)ℓ−1
+
N−1∑
m=1
Bℓ(bm,0)
bℓ−1
m,1
,
V
(j,k)
∞,ℓ =
N−1∑
m=1
(
Bℓ(bm,0 − ak,0)
(bm,1 − ak,1)ℓ−1
− Bℓ(bm,0 − aj,0)
(bm,1 − aj,1)ℓ−1
)
+
Bℓ(aj,0)
aℓ−1
j,1
+
N−1∑
m=1
m̸=j
Bℓ(ai,0 − aj,0)
(ai,1 − aj,1)ℓ−1
−
Bℓ(ak,0)
aℓ−1
k,1
+
N−1∑
m=1
m ̸=k
Bℓ(ai,0 − ak,0)
(ai,1 − ak,1)ℓ−1
.
Here Bℓ(t) denotes the ℓ-th Bernoulli polynomial defined by
xext
ex − 1
=
∞∑
ℓ=0
Bℓ(t)
ℓ!
xℓ.
Proof. To obtain the explicit forms of V
(j,k)
ϱ , we will solve (3.13). We may write f
(j,k)
ρ in the
form
f (j,k)ρ =
1
2
∑
m
1
u
(j,k)
ρ,m + v
(j,k)
ρ,m η−1
, (3.24)
where u
(j,k)
ρ,m and v
(j,k)
ρ,m being linear functions of ρ and ρ0, respectively, which are independent
of η. The explicit forms of u
(j,k)
ρ,m and v
(j,k)
ρ,m are given by Lemma 3.6. The coefficient of ρ of u
(j,k)
ρ,m
is equal to 1 or −1, which denotes ϵ
(j,k)
ρ,m (= ±1). The right-hand side of (3.24) is obtained
from u
(j,k)
ρ,m by the shift:
1
2
∑
m
1
u
(j,k)
ρ,m + v
(j,k)
ρ,m η−1
=
1
2
∑
m
1
u
(j,k)
ρ,m
∣∣∣∣
ρ7→ρ+ϵ
(j,k)
ρ,m v
(j,k)
ρ,m η−1
.
20 T. Aoki and S. Uchida
Thus (3.13) is written in the following form:
∂ρ∆ρV
(j,k)
ϱ =
1
2
∑
m
eϵ
(j,k)
ρ,m v
(j,k)
ρ,m η−1∂ρ 1
u
(j,k)
ρ,m
+∆ρg
(j,k)
ρ .
Hence (3.13) can be solved by using a formal differential operator of infinite order:
V (j,k)
ϱ =
1
2
∑
m
eϵ
(j,k)
ρ,m v
(j,k)
ρ,m η−1∂ρ(
eη
−1∂ρ − 1
) ∂−1
ρ
1
u
(j,k)
ρ,m
+ ∂−1
ρ g(j,k)ρ
=
1
2
∞∑
ℓ=0
∑
m
Bℓ
(
ϵ
(j,k)
ρ,m v
(j,k)
ρ,m
)
ℓ!
η1−ℓ∂ℓ−2
ρ
1
u
(j,k)
ρ,m
+ ∂−1
ρ g(j,k)ρ .
Using Lemma 3.8, we have
V (j,k)
ϱ =
1
2
∞∑
ℓ=2
∑
m
Bℓ
(
ϵ
(j,k)
ρ,m v
(j,k)
ρ,m
)
ℓ!
η1−ℓ∂ℓ−2
ρ
1
u
(j,k)
ρ,m
+ Cρ(ρ̌),
where Cρ(ρ̌) is an arbitrary function of {a1,1, . . . , aN,1, b1,1, . . . , bN−1,1} \ {ρ}. We repeat the
above discussion for every ρ ∈ {a1,1, . . . , aN,1, b1,1, . . . , bN−1,1}. Then we can determine Cρ(ρ̌)
by adjustment. Thus we have the theorem. ■
Remark 3.10. If N = 3, we have
V
(1,2)
0,ℓ =
∑
i=1,2,3
j=1,2
(−1)j
Bℓ(bj,0 − ai,0)
(bj,1 − ai,1)ℓ−1
+
Bℓ(b1,0 − 1)
bℓ−1
1,1
− Bℓ(b2,0 − 1)
bℓ−1
2,1
+
Bℓ(b1,0 − b2,0 + 1) +Bℓ(b1,0 − b2,0)
(b1,1 − b2,1)ℓ−1
,
V
(1,3)
0,ℓ = −
∑
i=1,2,3
Bℓ(b1,0 − ai,0)
(b1,1 − ai,1)ℓ−1
−
∑
i=1,2,3
Bℓ(ai,0)
aℓ−1
i,1
+
Bℓ(b1,0) +Bℓ(b1,0 − 1)
bℓ−1
1,1
+
Bℓ(b2,0)
bℓ−1
2,1
+
Bℓ(b1,0 − b2,0)
(b1,1 − b2,1)ℓ−1
,
V
(2,3)
0,ℓ = −
∑
i=1,2,3
Bℓ(b2,0 − ai,0)
(b2,1 − ai,1)ℓ−1
−
∑
i=1,2,3
Bℓ(ai,0)
aℓ−1
i,1
+
Bℓ(b2,0) +Bℓ(b2,0 − 1)
bℓ−1
2,1
+
Bℓ(b1,0)
bℓ−1
1,1
+
Bℓ(b2,0 − b1,0)
(b2,1 − b1,1)ℓ−1
,
V
(j,k)
∞,ℓ =
∑
m=1,2
(
Bℓ(bm,0 − ak,0)
(bm,1 − ak,1)ℓ−1
− Bℓ(bm,0 − aj,0)
(bm,1 − aj,1)ℓ−1
)
+
Bℓ(aj,0)
aℓ−1
j,1
+
3∑
i=1
i ̸=j
Bℓ(ai,0 − aj,0)
(ai,1 − aj,1)ℓ−1
−
Bℓ(ak,0)
aℓ−1
k,1
+
3∑
i=1
i ̸=k
Bℓ(ai,0 − ak,0)
(ai,1 − ak,1)ℓ−1
.
We consider the Borel summability of V
(j,k)
ϱ . We set
Ṽ (κ, κ0) =
1
2
∞∑
ℓ=2
(−1)ℓ+1η1−ℓ
ℓ(ℓ− 1)
Bℓ(κ0)
κℓ−1
.
Voros Coefficients at the Origin and at the Infinity 21
As we saw in the preceding theorem, V
(j,k)
ϱ are expressed as a finite sum of the formal series
of the form Ṽ (κ, κ0) for suitable choice of κ, κ0. Hence it is sufficient to consider the Borel
summability of Ṽ (κ, κ0). Its summability is well known: the formal series Ṽ (κ, κ0) of η−1 is
Borel summable if Re(κ) ̸= 0 and the Borel sum of Ṽ (κ, κ0) equals
1
2
log
(√
2π(κη)κ0+κη− 1
2
eκηΓ(κ0 + κη)
)
, Re(κ) > 0,
1
2
log
(
(−κη)κ0+κη− 1
2Γ (1− (κ0 + κη))√
2πeκη
)
, Re(κ) < 0,
(see [6, Lemma 4.6]). This expression is obtained by the Binet formula [12]. Thus, we have
Corollary 3.11. Let ϱ be 0 or ∞. Also let
D(κ) = {(a1,1, . . . , aN,1, b1,1, . . . , bN−1,1) | Re(κ) ̸= 0},
where κ is a linear homogeneous function of a1,1, . . . , aN,1, b1,1, . . . , bN−1,1. The Voros coeffi-
cient V
(j,k)
ϱ of type (j, k), j < k, is Borel summable if (a1,1, . . . , aN,1, b1,1, . . . , bN−1,1) belongs
to D
(j,k)
ϱ . Here we set
D
(j,k)
0 = D(bj,1) ∩D(bk,1) ∩
N⋂
i=1
D(bj,1 − ai,1) ∩
N−1⋂
m=1
m ̸=j
D(bj,1 − bm,1)
∩
N⋂
i=1
D(bk,1 − ai,1) ∩
N−1⋂
m=1
m ̸=k
D(bk,1 − bm,1), k ̸= N,
D
(j,N)
0 =
N⋂
i=1
D(ai,1) ∩
N−1⋂
m=1
D(bm,1) ∩
N⋂
i=1
D(bj,1 − ai,1) ∩
N−1⋂
m=1
m ̸=j
D(bj,1 − bm,1),
D(j,k)
∞ = D(aj,1) ∩D(ak,1) ∩
N−1⋂
m=1
D(bm,1 − ak,1) ∩
N−1⋂
m=1
D(bm,1 − aj,1)
∩
N⋂
i=1
i ̸=j
D(ai,1 − aj,1) ∩
N⋂
i=1
i ̸=k
D(ai,1 − ak,1).
4 Concluding remarks and future problems
We have obtained explicit forms of the Voros coefficients at the origin and at the infinity (Theo-
rem 3.9). To give an explicit form of the Voros coefficient at x = 1 is our future problem. The
difficulty of the problem comes from the multiplicity of the characteristic exponents at x = 1.
As in the case of the Gauss hypergeometric differential equation [8] and of its confluent
families [22, 27], we may get the formulas describing the parametric Stokes phenomena for WKB
solutions of (2.3) by using Corollary 3.11 under the assumption that their formal solutions are
Borel summable in some region. The Borel summability of WKB solutions of our equation is
also a future problem.
Acknowledgements
The first author is supported by JSPS KAKENHI Grant No. 18K03385. The authors would like
to thank the referees for their careful readings and detailed comments on this article.
22 T. Aoki and S. Uchida
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https://doi.org/10.1090/S0002-9904-1938-06776-6
1 Introduction
2 WKB solutions
2.1 The generalized hypergeometric differential equation with a large parameter
2.2 Turning points and WKB solutions
2.3 Factorization
2.4 Definition of Voros coefficients
3 Explicit forms of the Voros coefficients
3.1 Ladder operators
3.2 Voros coefficients
4 Concluding remarks and future problems
References
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| id | nasplib_isofts_kiev_ua-123456789-211543 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T13:56:16Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Aoki, Takashi Uchida, Shofu 2026-01-05T12:30:45Z 2022 Voros Coefficients at the Origin and at the Infinity of the Generalized Hypergeometric Differential Equations with a Large Parameter. Takashi Aoki and Shofu Uchida. SIGMA 18 (2022), 002, 23 pages 1815-0659 2020 Mathematics Subject Classification: 33C20; 34E20; 34M60 arXiv:2104.13751 https://nasplib.isofts.kiev.ua/handle/123456789/211543 https://doi.org/10.3842/SIGMA.2022.002 Voros coefficients of the generalized hypergeometric differential equations with a large parameter are defined, and their explicit forms are given for the origin and for infinity. It is shown that they are Borel summable in some specified regions in the space of parameters, and their Borel sums in the areas are given. The first author is supported by JSPS KAKENHI Grant No. 18K03385. The authors would like to thank the referees for their careful reading and detailed comments on this article. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Voros Coefficients at the Origin and at the Infinity of the Generalized Hypergeometric Differential Equations with a Large Parameter Article published earlier |
| spellingShingle | Voros Coefficients at the Origin and at the Infinity of the Generalized Hypergeometric Differential Equations with a Large Parameter Aoki, Takashi Uchida, Shofu |
| title | Voros Coefficients at the Origin and at the Infinity of the Generalized Hypergeometric Differential Equations with a Large Parameter |
| title_full | Voros Coefficients at the Origin and at the Infinity of the Generalized Hypergeometric Differential Equations with a Large Parameter |
| title_fullStr | Voros Coefficients at the Origin and at the Infinity of the Generalized Hypergeometric Differential Equations with a Large Parameter |
| title_full_unstemmed | Voros Coefficients at the Origin and at the Infinity of the Generalized Hypergeometric Differential Equations with a Large Parameter |
| title_short | Voros Coefficients at the Origin and at the Infinity of the Generalized Hypergeometric Differential Equations with a Large Parameter |
| title_sort | voros coefficients at the origin and at the infinity of the generalized hypergeometric differential equations with a large parameter |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211543 |
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