SIEGEL’S PROBLEM FOR E-FUNCTIONS OF ORDER 2
T. RIVOAL AND J. ROQUES
Abstract. E-functions are entire functions with algebraic Taylor coefficients at the origin satisfying certain arithmetic conditions, and solutions of linear differential equations
with coefficients in Q(z); they naturally generalize the exponential function. Siegel andShidlovsky proved a deep transcendence result for their values at algebraic points. Since
then, a lot of work has been devoted to apply their theorem to special E-functions, inparticular the hypergeometric ones. In fact, Siegel asked whether any E-function can be
expressed as a polynomial in z and generalized confluent hypergeometric series. As a first
positive step, Shidlovsky proved that E-functions with order of the differential equation
equal to 1 are in Q[z]eQz . In this paper, we give a new proof of a result of Gorelov thatany E-function (in the strict sense) with order ≤ 2 can be written in the form predicted
by Siegel with confluent hypergeometric functions 1 F1 [α; β; λz] for suitable α, β ∈ Q andλ ∈ Q. Gorelov’s result is in fact more general as it holds for E-functions in the large
sense. Our proof makes use of André’s results on the singularities of the minimal differential equations satisfied by E-functions, together with a rigidity criterion for (irregular)differential systems recently obtained by Bloch-Esnault and Arinkin. An ad-hoc versionof this criterion had already been used by Katz in his study of confluent hypergeometricseries. Siegel’s question remains unanswered for orders ≥ 3.

1. Introduction
We fix an embedding of Q into C. An E-function (in the strict sense) is a power seriesf (z) =

∞
X
an

n!
n=0

z n ∈ Q[[z]]such that:(1) f (z) satisfies a non-zero linear differential equation with coefficients in Q(z);(2) there exists C > 0 such that for any σ ∈ Gal(Q/Q), we have |σ(an )| ≤ C n+1 ; andthere exists a sequence of positive integers dn such that dn ≤ C n+1 and dn am is analgebraic integer for all m ≤ n.
If an ∈ Z, the conditions in (2) reduce to |an | ≤ C n+1 . This class of arithmetic power
series was defined by Siegel [18] (in a slightly more general way) to mimic the Diophantine properties of the exponential function, and his program was later completeddby Shidlovsky [20]. Throughout the paper, we set θ = z dz
and by “solution of a differend
tial operator L ∈ Q(z)[ dz ]”, it must be understood “solution of the differential equation
Date: May 1, 2020.
1

2

Ly(z) = 0”. More recently, André [2] and Beukers [4] gave a new impulse to the Diophantine theory of E-functions, whose prototypical example is the generalized hypergeometricfunction X∞(α1 )n · · · (αp )n (q−p+1)nα1 , . . . , αp q−p+1
=
;z
z
p Fq
β1 , . . . , βqn!(β1 )n · · · (βq )nn=0with q ≥ p ≥ 1, αj , βj ∈ Q, and none of the β’s is a negative integer. If p = q = 1, it is asolution of the differential operator θ(θ + β − 1) − z(θ + α) and when α = β, this is simply
exp(z).
The present paper is concerned with the following classical questions: What are Efunctions? Are they related to generalized hypergeometric functions? In fact, Siegel [19,p. 58] asked the following question: can any E-function be represented as a multivariate
polynomial, with coefficients in Q[z], in finitely many confluent hypergeometric series ofpq
the form p Fq [a; b; λz q−p+1 ], for various q ≥ p ≥ 1, a ∈ Q , b ∈ Q and λ ∈ Q? See also [20,p. 84]. (1)In the recent papers [16, 17], we studied the structural properties of differential equations
satisfied by strict E-functions, in the light of [2]. In particular, as a consequence of themain result of [16], we proved that any strict E-function f (z) solution of an inhomogeneous
linear equation
f 0 (z) = u(z)f (z) + v(z)(1.1)
×
of order 1 is essentially hypergeometric, where u(z) ∈ Q(z) and v(z) ∈ Q(z). Moreprecisely, there exist some a(z), b(z) ∈ Q[z, z −1 ], β ∈ {1} ∪ Q \ Z and λ ∈ Q such thatf (z) = a(z)1 F1 (1; β; λz) + b(z).

(1.2)

This solved a problem, raised by Shidlovsky [20, p. 184], and already partially solved byAndré in [2, p. 724]. If v(z) = 0, then in fact b(z) = 0, β = 1 and a(z) ∈ Q[z], so thatf (z) = a(z)eλz , a result due to Shidlovsky [20, p. 184]. When v(z) 6= 0, any solution ofd1 d(1.1) is also solution of the differential operator of order 2 given by dz( v(z)− u(z)
). We
dz
v(z)
were not aware at that time that Gorelov had already solved Shidlovsky’s problem in [11,p. 139, Theorem 2] for E-functions in Siegel’s original sense.
It is thus natural to wonder if something similar to (1.2) can be said of E-functions
solutions of differential equations of order 2. Actually, a first case was considered in [16].
Indeed, the following result is a direct consequence of [16, Theorem 4]. Again, this result
has been proved by Gorelov in [12, p. 515, Theorem 2] for E-functions in Siegel’s original
sense.
Theorem 1. Let f (z) be a strict E-function solution of a non-zero linear differential
operator of order 2 with coefficients in Q(z) and reducible over Q(z). Then, there exist
a(z), b(z) ∈ Q[z, z −1 ], β ∈ {1} ∪ Q \ Z and λ, µ ∈ Q such thatf (z) = a(z)eµz 1 F1 (1; β; λz) + b(z)eµz .

(1.3)

1In this problem, Siegel referred to his original definition of E-functions, which are slightly more generalthan the strict ones used in this paper, which are themselves called E ∗ -functions in [20]. It is widelybelieved that both class are identical, but our proof holds only in the strict sense.

3

We now state the result if we remove the reducibility hypothesis in the previous result.
We emphasize that Theorem 2 below is a particular case of a result of Gorelov [12, p. 514,
Theorem 1], who in fact did not make the distinction between the reducible and irreducible
case, which is in accordance with the remarks made after the theorem.
Theorem 2. Let f (z) be a strict E-function solution of a non-zero linear differential
operator of order 2 with coefficients in Q(z) and irreducible over Q(z). Then, we havef (z) = a(z)eµz 1 F1 (α; β; λz) + b(z)eµz 1 F10 (α; β; λz)

(1.4)

×

where a(z), b(z) ∈ Q(z), λ ∈ Q , µ ∈ Q, and α ∈ Q, β ∈ Q \ Z≤0 are such that α − β 6∈ Z.
Note that (1.3) is of the form (1.4) because we have the relation z 1 F10 (1; β; z) = (z − β +1)1 F1 (1; β; z)+β −1. Moreover, 1 F10 (α; β; z) = αβ 1 F1 (α+1; β +1; z). This explains why such
results give a positive solution to Siegel’s problem for orders ≤ 2. Gorelov proved a stronger
version of Theorem 2: he showed that f (z) can be assumed to be an E-function in Siegel’s
original sense and moreover that the conclusion holds with some a(z), b(z) ∈ Q[z]. On theother hand, he did not state that α − β 6∈ Z. We observe that in fact a strict E-functionof order 2 may have a representation of the form (1.4) with a(z) and b(z) not necessarily2+ 16z+ . . . ispolynomials, for instance ϕ(z) := z2 1 F1 (1; 1/2; z) − z1 1 F10 (1; 1/2; z) = 43 + 16z
1535
such an E-function. This does not contradict Gorelov’s stronger “polynomial coefficients”
version, because a strict E-function does not necessarily admit a unique representation of
the form (1.4); indeed, it is readily proved that ϕ(z) = 21 F10 (1; 3/2; z).
The main contribution of this paper is our proof of Theorem 2, which is quite different
from that of Gorelov, even though he also used André’s theory at some point. We did
not try to reprove his version in full. We hope our point of view will be useful for further
studies in this field.
We now illustrate Theorem 2 with the non-hypergeometric E-function∞n  
X
1 X n n + k  na(z) =z .
n! k=0 k
n
n=0

(1.5)

It was brought to our attention by F. Beukers during a lecture he gave in June 2016 at
the conference Automates and Number Theory held at Porquerolles, where he asked if a(z)was related in some way to hypergeometric series. Since it is solution of the irreducibled 2ddifferential operator z( dz) − (6z − 1) dz+ (z − 3), Theorem 2 applies to it and the answer

 n
P
Pn
n n+k
1
z = √1−6z+z
is yes. Let us give (1.4) in this case. Since ∞2 (see [15,
n=0
k=0 k
n
§3]), we have
Z
1
ezx
√
dx
a(z) =
2iπ L 1 − 6x + x2
where L is a “vertical” straight line leaving the roots of 1 − 6x + x2 to its left; see [10,§§4.2-4.3]. With the change of variable x = t + 3 and with L0 a “vertical” straight line

4leaving the roots of t2 − 8 to its left, we obtainZ∞X(2z 2 )n1ez(t+3)3z√a(z) =
dt = e= e3z · 0 F1 (· ; 1; 2z 2 )22iπ L0 t2 − 8
n!
n=0
√
√
(3−2 2)z· 1 F1 (1/2 ; 1; 4 2z).
=e

(1.6)
(1.7)

The hypergeometric function on the right of (1.6) is not formally of the form suggested by
Theorem 2, but this is the case of (1.7). The equality of both expressions is a consequenceof an hypergeometric identity between Kummer M function and Bessel I0 function [1, p.
509, 13.6.1]. In passing, we obtain the binomial identity Xn  
n  
X√n n+kn 2k √ k=
2 (3 − 2 2)n−k , n ≥ 0,
(1.8)
k
n
k
k
k=0
k=0
after multiplication of the two (implicit) power series in (1.7). Conversely, (1.8) could be
proved first (by means of Zeilberger’s algorithm) and then (1.7) would follow again.
We don’t know if results similar to Theorems 1 and 2 could be obtained for E-functionsof higher order, even for the order 3; the methods of this paper do not give enough informations to conclude. In fact, Siegel’s question might have a negative answer in generaland as the order increases, one needs to add more and more special functions to the hyPnPn 2 n+k1
pergeometric ones. For instance, ∞)z n is solution of z 2 y 000 (z) + (3z −k=0 k
n=0 n! (n
11z 2 )y 00 (z) + (1 − 22z − z 2 )y 0 (z) − (3 + z)y(z) = 0. Although not hypergeometric, can it beexpressed using confluent hypergeometric series as requested in Siegel’s problem, like a(z)above, or is it of a different nature? We refer to [13, 14] for further results on this problem.
The paper is organized as follows. In Sections 2 (hypergeometric operators), 3 (Fuchs
relation) and 4 (rigidity), we collect some results from the literature that we need for the
proof of Theorem 2, which is given in Section 5. Some aspects of the proof are similar withcertain results of Katz in [8], though the methods are not written in the same language;
we include these computations for the sake of completeness. Finally, in Section 6, we
make some remarks on E-operators and G-operators. From now on, any E-function isunderstood to be in the strict sense.
Acknowledgments. We thanks the referees for their comments that helped us to remove
some inaccuracies.
2. The differential operators Hα;β,γ and Lα,β,γ,λ,µIn this section, we gather some results on two explicit hypergeometric operators. They
will be used in the proof of Theorem 2 later on.
d2.1. The confluent hypergeometric operator Hα;β,γ . We recall that θ = z dz
. The
confluent hypergeometric operator with parameters α, β, γ ∈ C is the linear differential
operator given by
Hα;β,γ = (θ + β − 1)(θ + γ − 1) − z(θ + α).

5It has at most two singularities on P1 (C), namely 0 and ∞. The former is a regular singularpoint, the latter is an irregular singular point. More precisely, the slopes of the Newton
polygon of Hα;β,γ at ∞ are 0 and 1, both with multiplicity 1. We denote by
Y 0 = Aα;β,γ Y(2.1)

the differential system associated to Hα;β,γ .
For later use, we shall now describe the general form of the formal solutions of thedifferential system (2.1) as predicted by Turittin’s theorem [21, Theorem 3.54].
We first assume that β − γ 6∈ Z. Then, at 0, the differential system (2.1) admits afundamental matrix of formal solutions of the form F0 (z)z Γ0 where


1−β
0
F0 (z) ∈ GL2 (C((z))) and Γ0 =.
01−γ
At ∞, the differential system (2.1) admits a fundamental matrix of formal solutions of theform F∞ (z)z Γ∞ e∆z where
α−β−γ+1 0
1 0−1
F∞ (z) ∈ GL2 (C((z ))), Γ∞ =and ∆ =.
0−α
0 0
We now assume that β − γ ∈ Z. If α − β 6∈ Z, then, at 0, the differential system (2.1)
admits a fundamental matrix of formal solutions of the form F0 (z)z Γ0 where


1−β
1
F0 (z) ∈ GL2 (C((z))) and Γ0 =.
0
1−βRemark. If the condition α−β 6∈ Z is not satisfied, then Γ0 may be diagonalizable. Consider
for instance the case α = γ − 1 and β − 1 = α − 1.
We shall now consider a special case with α − β ∈ Z, namely α = β − 1 = γ − 1. In thiscase, at 0, the differential system (2.1) admits a fundamental matrix of formal solutions of
the form F0 (z)z Γ0 where


1−β
1
.
F0 (z) ∈ GL2 (C((z))) and Γ0 =01−βThis can be seen by direct calculation using that, in the present case, we have Hα;β,γ =(θ + α − z)(θ + α). At ∞, the differential system (2.1) admits a fundamental matrix offormal solutions of the form F∞ (z)z Γ∞ e∆z where




1−β
0
1 0
−1
F∞ (z) ∈ GL2 (C((z ))), Γ∞ =and ∆ =.
01−β
0 02.2. The operator Lα,β,γ,λ,µ . For any α, β, γ, λ, µ ∈ C, with λ 6= 0, we consider the linear
differential operator given by
Lα;β,γ;λ,µ = (θ + β − 1 − µz)(θ + γ − 1 − µz) − λz(θ + α − µz).
There is a simple relation between the operators Lα;β,γ;λ,µ and Hα;β,γ : we haveHα;β,γ y(z) = 0 ⇐⇒ Lα;β,γ;λ,µ y(λz)eµz = 0.

6

We denote by
Y 0 = Aα;β,γ;λ,µ Y
the differential system associated to Lα;β,γ;λ,µ .

(2.2)

The results of Section 2.1 imply the following facts.
We first assume that β − γ 6∈ Z. Then, at 0, the differential system (2.2) admits afundamental matrix of formal solutions of the form F0 (z)z Γ0 where


1−β
0
F0 (z) ∈ GL2 (C((z))) and Γ0 =.
01−γ
At ∞, the differential system (2.2) admits a fundamental matrix of formal solutions of theform F∞ (z)z Γ∞ e∆z where
α−β−γ+1 0
λ+µ 0−1
F∞ (z) ∈ GL2 (C((z ))), Γ∞ =and ∆ =.
0−α0µWe now assume that β − γ ∈ Z. If α − β 6∈ Z, then, at 0, the differential system (2.2)
admits a fundamental matrix of formal solutions of the form F0 (z)z Γ0 where


1−β
1
F0 (z) ∈ GL2 (C((z))) and Γ0 =.
01−β
If α = β − 1 = γ − 1, then, at 0, the differential system (2.2) admits a fundamental
matrix of formal solutions of the form F0 (z)z Γ0 where


1−β
1
F0 (z) ∈ GL2 (C((z))) and Γ0 =.
01−β
At ∞, the differential system (2.2) admits a fundamental matrix of formal solutions of theform F∞ (z)z Γ∞ e∆z where
α−β−γ+1 0
λ+µ 0−1
F∞ (z) ∈ GL2 (C((z ))), Γ∞ =and ∆ =.
0−α
0µ3. Fuchs’ relation
In this section, we state a “Fuchs relation” between the exponents of certain (possibly
irregular) differential systems.
Proposition 1. Let us consider a differential system Y 0 = AY with A ∈ M2 (C(z)). We
assume that the following properties are satisfied :(1) Y 0 = AY has only apparent singularities on C× ;(2) Y 0 = AY has a basis of solutions at 0 of the form F0 (z)z Γ0 where F0 (z) ∈ GL2 (C((z)))
and Γ0 ∈ M2 (C) is upper-triangular.
(3) Y 0 = AY admits a basis of formal solutions at ∞ of the form F∞ (z)z Γ∞ e∆z whereF∞ (z) ∈ GL2 (C((z −1 ))), Γ∞ ∈ M2 (C) is upper-triangular, ∆ = diag(θ1 , θ2 ) ∈M2 (C) is diagonal, and Γ∞ and ∆ commute.
Then, the trace of Γ0 − Γ∞ belongs to Z.

7

Proof. The monodromy of Y 0 = AY at 0 with respect to the fundamental matrix of solutions F0 (z)z Γ0 is given by M0 = e2πiΓ0 . Its monodromy at ∞ with respect to the same
fundamental matrix of solutions is of the form M∞ = P e−2πiΓ∞ S∞ P −1 where P ∈ GL2 (C)and where S∞ ∈ GL2 (C) is a product of Stokes matrices (see [21, Proposition 8.12]). In
particular, S∞ is unipotent and, hence, det(S∞ ) = 1. But, we have M0 M∞ = I2 because
Y 0 = AY has only apparent singularities on C× and, hence, trivial monodromies aroundeach point of C× . It follows that det(M0 M∞ ) = det(I2 ) = 1 i.e. e2πi tr(Γ0 −Γ∞ ) = 1. Whence
the result.

4. A Reminder on rigidity
This section is a reminder about the notion of rigidity of (possibly irregular) differential
systems. We start by recalling the notions of formal and rational equivalences.
4.1. Formal equivalence. Recall that two differential systems
Y 0 = AY and Y 0 = BY with A, B ∈ Mn (C(z))(4.1)

are formally equivalent at 0 if there exists R ∈ GLn (C((z))) such that
B = R−1 AR − R−1 R0 .

(4.2)

This means that one gets Y 0 = BY from Y 0 = AY by replacing Y by RY .
More generally, there is a notion of formal equivalence at any s ∈ P1 (C). More precisely,
cs the field of formal Laurent series at s ∈ P1 (C). We say that the differential
we denote by K
cs ) such that equation
systems (4.1) are formally equivalent at s if there exists R ∈ GLn (K(4.2) holds.
4.2. Rational equivalence. The differential systems (4.1) are called rationally equivalent
if there exists R ∈ GLn (C(z)) such that equation (4.2) holds.
Of course, “rationally equivalent” implies “formally equivalent at any s ∈ P1 (C)”, but
the converse is not true in general. This is where rigidity comes into play.
4.3. Rigidity. We say that a given differential system
Y 0 = AY with A ∈ Mn (C(z))(4.3)

is rigid if, for any differential system Y 0 = BY with B ∈ Mn (C(z)), the fact that Y 0 = AY
is formally equivalent to Y 0 = BY at each s ∈ P1 (C) implies that Y 0 = AY and Y 0 = BY
are rationally equivalent.
If Y 0 = AY is irreducible over C(z), there is a numerical rigidity criterion, which canbe stated as follows. We denote by ⊗ the Kronecker tensor product on Mn (C), i.e., for
C, D ∈ Mn (C(z)), the tensor product C ⊗ D ∈ Mn2 (C) is defined by
c1,1 D · · · c1,n D..  .
..
C ⊗ D =  ...
.
.
cn,1 D · · · cn,n D

8

We define the “internal End” of Y 0 = AY as the differential system Y 0 = E(A)Y with
E(A) = A ⊗ In − In ⊗ At ∈ Mn2 (C).
We setrig(A) = 2n2 −X

irr(E(A), s) + n2 − dimC sol(E(A), s)s∈P1 (C)where irr(E(A), s) is Malgrange irregularity of Y 0 = E(A)Y at s, and where sol(E(A), s)cs ) of Y 0 = E(A)Y . We recall that theis the C-vector space of the solutions in Mn,1 (KMalgrange irregularity irr(B, s) at s of a given differential system Y 0 = BY , with B ∈Mn (C(z)), is the height of its Newton polygon. Equivalently, it is equal to the sum of theslopes (counted with multiplicities) of the Newton polygon at s of Y 0 = BY .
Theorem 3 ([3, Proposition 3.4], [5, Theorems 4.7 and 4.10]). Assume that Y 0 = AY isirreducible. Then, it is rigid if and only if rig(A) = 2.
We will also use the following inequality.
Theorem 4 ([3, Remark following Proposition 3.4]). Assume that Y 0 = AY is irreducible.
Then, we have rig(A) ≤ 2.
5. Proof of Theorem 2
We are now in position to prove our main result. For simplicity, we split the proof into
two steps.
5.1. First step. We first prove the following result.
Theorem 5. Let us consider a differential system Y 0 = AY with A ∈ M2 (C(z)). We
assume that this differential system is irreducible and that the following properties are
satisfied :(1) Y 0 = AY has at most apparent singularities on C× ;(2) Y 0 = AY has at most a regular singularity at 0;(3) the slopes at ∞ of Y 0 = AY are included in {0, 1}.
Then, the differential system Y 0 = AY is rationally equivalent to Y 0 = Aα;β,γ;λ,µ Y for someα, β, γ, λ, µ ∈ C with λ 6= 0.
Proof. Turittin’s theorem yields the following facts :(1) Y 0 = AY has a basis of solutions at 0 of the form F0 (z)z Γ0 where F0 (z) ∈ GL2 (C((z)))
and Γ0 ∈ M2 (C) is upper-triangular;(2) Y 0 = AY admits a basis of formal solutions at ∞ of the form F∞ (z)z Γ∞ e∆z whereF∞ (z) ∈ GL2 (C((z −1 ))), Γ∞ ∈ M2 (C) is upper-triangular, ∆ = diag(θ1 , θ2 ) ∈M2 (C) is diagonal, and Γ∞ and ∆ commute.
Hence, we have:• at 0, the differential system Y 0 = AY is formally equivalent to Y 0 = Γz0 Y ;
• at ∞, the differential system Y 0 = AY is formally equivalent to Y 0 = Γz∞ + ∆ Y.

9Therefore, setting B = E(A), we see that• at 0, the differential system Y 0 = BY is formally equivalent to Y 0 = B0 Y with
Γ0 ⊗ I2 − I2 ⊗ Γt0.
z• at ∞, the differential system Y 0 = BY is formally equivalent to Y 0 = B∞ Y with


t

Γ∞
Γ∞
+ ∆ ⊗ I2 − I2 ⊗
+∆ .
B∞ =
z
z
B0 =

Note that θ1 6= θ2 . Indeed, assume at the contrary that θ1 = θ2 . Then, the differential
system Y 0 = AY is rationally equivalent to Y 0 = Γz0 + θ1 I2 Y (because the differential
system satisfied by F0 (z)z Γ0 e−θ1 z is regular singular on P1 (C), with at most apparent singularities on C× and, hence, is of the form R(z)z Γ0 for some R(z) ∈ GL2 (C(z))). Therefore,
the differential system Y 0 = AY is reducible. This is a contradiction.
Since ∆ and Γ∞ commute, the fact that θ1 6= θ2 implies that Γ∞ is diagonal :


γ∞,1
0
Γ∞ =
.
0 γ∞,2
We now split our study in several cases, but the idea of the proof will be the same in any
cases : we will prove that the differential system Y 0 = AY is rigid and formally equivalent
to Y 0 = Aα;β,γ;λ,µ Y for some α, β, γ, λ, µ ∈ C, with λ 6= 0, at any s ∈ P1 (C). We willconclude that these systems are actually rationally equivalent by rigidity.
The case Γ0 diagonal non resonant. We assume that Γ0 is a diagonal matrix
γ0,1 0Γ0 =0 γ0,2such that γ0,2 − γ0,1 6∈ Z. It follows that00
0 γ0,1 −γ0,2z
B0 = 
0
0
0
0

0
0
γ0,2 −γ0,1
z

0


0
0

0
0

and


0
0
B∞ = 
0
0

00
γ∞,1 −γ∞,2+ θ1 − θ20z
γ∞,1 −γ∞,20−(+ θ1 − θ2 )
z
0
0


0
0
.
0
0

Then, we haveirr(B, 0) = irr(B0 , 0) = 0 and dimC sol(B, 0) = dimC sol(B0 , 0) = 2.
Moreover, we haveirr(B, ∞) = irr(B∞ , ∞) = 2 and dimC sol(B, ∞) = dimC sol(B∞ , ∞) = 2.

10

So, we getrig(A) = 2 · 4 − (0 + 4 − 2) − (2 + 4 − 2) = 2.
Therefore, Y 0 = AY is rigid in virtue of Theorem 3.
We now consider α, β, γ, λ, µ ∈ C such that1−β≡ γ0,1 mod Z



1
−
γ
≡ γ0,2 mod Z
α − β − γ + 1 ≡ γ∞,1 mod Z.
−α≡ γ∞,2 mod Z




= θ1

 λ+µ
µ
= θ2
We can indeed solve this system of equations because, in virtue of Proposition 1, we have
γ0,1 + γ0,2 − γ∞,1 − γ∞,2 ≡ 0 mod Z. Note that λ 6= 0 because θ1 6= θ2 . Note also thatβ −γ ≡ γ0,2 −γ0,1 6≡ 0 mod Z. Using Section 2, we see that Y 0 = AY is formally equivalent
at 0 and ∞ to Y 0 = Aα;β,γ;λ,µ Y . Therefore, by rigidity, these two systems are rationallyequivalent.
The case Γ0 diagonal and resonant is impossible. We assume that Γ0 is a diagonal
matrix
γ0,1 0Γ0 =0 γ0,2such that γ0,2 − γ0,2 ∈ Z. Then, the equality rig(A) ≤ 2 (see Theorem 4 above) reads asfollows :8 − (0 + 4 − 4) − (2 + 4 − dimC sol(B, ∞)) ≤ 2
i.e. dimC sol(B, ∞) ≤ 0. This is a contradiction because dimC sol(B, ∞) ≥ 2.
The case Γ0 non diagonal. Up to conjugation, we can assume that
γ0,1 1
Γ0 =.
0 γ0,1
Then, we have


0
−1/z
B0 = 
 0
0


0 1/z
0
0
0
1/z 
,
0
0
0 
0 −1/z 0

whose Jordan normal form is given by
0 1/z 0 0
0 0 1/z 0

.
0 0
0 0
0 0
0 0

11

Moreover, we have00
B∞ = 
0
0

00
γ∞,1 −γ∞,2+ θ1 − θ20z
γ∞,1 −γ∞,20−(+ θ1 − θ2 )
z
0
0


0
0
.
0
0

Then, we haveirr(B, 0) = irr(B0 , 0) = 0 and dimC sol(B, 0) = dimC sol(B0 , 0) = 2.
Moreover, we haveirr(B, ∞) = irr(B∞ , ∞) = 2 and dimC sol(B, ∞) = dimC sol(B∞ , ∞) = 2.
So, we haverig(A) = 2 · 4 − (0 + 4 − 2) − (2 + 4 − 2) = 2.
Therefore, Y = AY is rigid in virtue of Theorem 3.
We consider α, β, γ, λ, µ ∈ C such that1−β≡ γ0,1 mod Z



1
−
γ
≡
γ0,1 mod Z
α − β − γ + 1 ≡ γ∞,1 mod Z.
−α≡ γ∞,2 mod Z




= θ1

 λ+µ
µ
= θ2
0

We can indeed solve this system of equations because, in virtue of Proposition 1, we have
γ0,1 + γ0,1 − γ∞,1 − γ∞,2 ≡ 0 mod Z. Note that λ 6= 0 because θ1 6= θ2 . Note also thatβ − γ ∈ Z. If α − β ∈ Z, then the congruence α − β − γ + 1 ≡ γ∞,1 mod Z implies
that β ≡ γ ≡ −γ∞,1 mod Z but β ≡ γ ≡ −γ0,1 mod Z so γ0,1 ≡ γ∞,1 mod Z. Now, thecongruence γ0,1 + γ0,1 − γ∞,1 − γ∞,2 ≡ 0 mod Z implies that γ∞,2 ≡ γ∞,1 mod Z henceα ≡ β ≡ γ mod Z. Therefore, we can and will assume that α = β − 1 = γ − 1. Using
Section 2, we see that Y 0 = AY is formally equivalent at 0 and ∞ to Y 0 = Aα;β,γ;λ,µ Y . By
rigidity, these two systems are rationally equivalent.

5.2. Second step. Let f (z) be an E-function as in Theorem 2. By hypothesis, f (z)satisfies a non-zero linear differential equation of order ≤ 2 with coefficients in Q(z).
If f (z) satisfies a non-zero differential equation of order 1 with coefficients in Q(z), thenit is well-known that f (z) = a(z)eαz with a(z) ∈ Q(z) and α ∈ Q. Whence the result.
We shall now assume that f (z) does not satisfy a differential equation of order 1 withcoefficients in Q(z). Then, according to [2, Theorem 4.3], the vector F (z) = (f (z), f 0 (z))tsatisfies some linear differential system F 0 (z) = A(z)F (z) for some A(z) ∈ M2 (Q(z)), withthe following properties :(1) Y 0 = AY has only apparent singularities on C× ;(2) Y 0 = AY has a basis of solutions at 0 of the form F0 (z)z Γ0 where F0 (z) ∈ GL2 (Q((z)))
and Γ0 ∈ M2 (Q) is upper-triangular;

12(3) Y 0 = AY admits a basis of formal solutions at ∞ of the form F∞ (z)z Γ∞ e∆z whereF∞ (z) ∈ GL2 (Q((z))), Γ∞ ∈ M2 (Q) is upper-triangular, ∆ ∈ M2 (Q) is diagonal
and such that Γ∞ ∆ = ∆Γ∞ .
0
If Y = AY is reducible over C(z) or, equivalently, over Q(z), then we are in the situationof Theorem 1, which is a consequence of [16, Theorem 4]. We observe here that the series
1 F1 (1; γ; λz) is a solution of L1;1,γ;λ,0 .
We shall now assume that Y 0 = AY is irreducible over C(z) or, equivalently, over Q(z).
Then, according to Theorem 5, there exists R(z) ∈ GL2 (C(z)) such that
A = R−1 Aα;β,γ;λ,µ R − R−1 R0 .
But both β and γ are congruent to one of the eigenvalues of Γ0 modulo Z. Therefore, β and×
γ belong to Q. A similar argument yields that α ∈ Q, λ ∈ Q and µ ∈ Q. In particular,
Aα;β,γ;λ,µ has coefficients in Q(z) and, hence, one can assume that R(z) has coefficients inQ(z).
We observe that RF is a vector solution of Y 0 = Aα;β,γ;λ,µ Y , and any such solutions
are of the form (h(z), h0 (z))t where h(z) is a solution of Lα;β,γ;λ,µ . Hence, there exist
a(z), b(z) ∈ Q(z) such thatf (z) = a(z)h(z) + b(z)h0 (z).
Since the entries of RF belong to Q((z)), we see that h(z) also belongs to Q((z)). As notedat the beginning of Section 2.2, there exist a solution g(z) ∈ Q((z)) of Hα;β,γ such thath(z) = g(λz)eµz .
By assumption, the differential system Y 0 = AY is irreducible over Q(z), so that thedifferential operators Lα;β,γ;λ,µ and, hence, Hα;β,γ are irreducible over Q(z). This implies
that α − β 6∈ Z and α − γ 6∈ Z. Using Section 2.1, we see that Hα;β,γ has a nonzero solutionin Q((z)) if and only if β ∈ Z or γ ∈ Z. Assume for instance that β ∈ Z. Moreover, if
β − γ ∈ Z, we can assume that γ ≥ β. Then, the Q-vector space of solutions of Hα;β,γ in
Q((z)) is generated by∞X(α − β + 1)n n
1−βz = z 1−β 1 F1 (α − β + 1; γ − β + 1; z).
z
n!(γ
−
β
+
1)
n
n=0
×

Hence, up to a multiplicative constant in Q , we have g(z) = z 1−β 1 F1 (α−β +1; γ −β +1; z).
This completes the proof of Theorem 2.
6. Some remarks on E-operators and G-operators
Finally, though this is not the subject of this paper, we mention that similar problems
have been formulated for G-functions.
P
P∞ an nn
A G-function at z = 0 is a power series f (z) = ∞n=0 n! z
n=0 an z ∈ Q[[z]] such thatis an E-function. Both classes of functions have been first introduced by Siegel, and recently
André [2] showed the deep relations that exist between E and G-functions. A non-zerodminimal equation in Q[z, dz] satisfied by a G-function is called a G-operator. From results

13of André, Chudnovky and Katz (see [2, 6, 7]), we know that a G-operator is Fuchsian withrational exponents at its singularities, and that all its solutions at any point α ∈ Q ∪ {∞}are essentially G-functions of the variable z − α or 1/z if α = ∞. André [2] defined andE-operator as a differential operator in Q[z, dz] such that its Fourier-Laplace transform isd] of an operatora G-operator. We recall that the Fourier-Laplace transform Lb ∈ Q[z, dz
dd] defined by
L ∈ Q[z, dz ] is the image of L by the automorphism of the Weyl algebra Q[z, dz
ddz 7→ − dz and dz 7→ z. Any E-function is solution of an E-operator, which is not necessarilydbut is minimal for the degree in z. André proved that theminimal for the degree in dz
leading polynomial of an E-operator is z m for some integer m ≥ 0, i.e, that 0 is its only
possible finite singularity. It follows that the minimal non-zero differential equation of agiven E-function has only apparent finite non-zero singularities. That property was crucial
for the results proved here.
Using the André-Chudnovky-Katz Theorem, it is easy to prove
Q that G-functions ofdifferential order 1 are algebraic functions over Q(z) of the form z m dj=1 (z − αj )ej , m ∈ N,×

αj ∈ Q , ej ∈ Z. Dwork conjectured that a globally nilpotent differential operator indQ(z)[ dz
] of order 2 either has algebraic solutions or there exists an algebraic pullback to
Gauss’s hypergeometric equation with rational parameters. This conjecture was disprovedby Krammer [9]. We shall not define the notion of “global nilpotence” here. Let us simply
say that G-operators are conjectured to be exactly the globally nilpotent operators ind], and that G-operators coming from geometry are known to be globally nilpotent
Q(z)[ dz
(Katz [7]). Hence, Krammer’s result rules out the possibility to describe all G-functions oforder 2 with algebraic functions and algebraic pullbacks of Gauss’s hypergeometric series
only. This is in clear contrast with Gorelov’s results.

Bibliography[1] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th edition, 1970.
[2] Y. André, Séries Gevrey de type arithmétique I. Théorèmes de pureté et de dualité, Ann. of Math. 151(2000), 705–740.
[3] D. Arinkin, Rigid irregular connections on P1 , Compositio Math. 146 (2010), 1323–1338.
[4] F. Beukers, A refined version of the Siegel-Shidlovskii theorem, Annals of Math. 163.1 (2006), 369–379.
[5] S. Bloch, H. Esnault, Local Fourier Transforms and Rigidity, Asian J. Math., Vol. 8, no. 4 (2004),
587–606.
[6] G. V. Chudnovsky, On applications of diophantine approximations, Proc. Natl. Acad. Sci. USA 81(1984), 7261–7265.
[7] N. Katz, Nilpotent connections and the monodromy theorem: applications of a result of Turritin, Publ.
Math. IHES, 32 (1970), 232–355.
[8] N. Katz, Exponential sums and differential equations, Annals of Mathematical Studies, Princeton 1990.
[9] D. Krammer, An example of an arithmetic Fuchsian group, J. reine angew. Math. 473 (1996), 69–85.
[10] S. Fischler, T. Rivoal, Arithmetic theory of E-operators, Journal de l’École polytechnique - Mathématiques 3 (2016), 31–65.
[11] V. A. Gorelov, On the algebraic independence of values of E-functions at singular points and the Siegel
conjecture, Mat. Notes 67.2 (2000), 174—190.

14[12] V. A. Gorelov, On the Siegel conjecture for second-order homogeneous linear differential equations,Math. Notes 75.4 (2004), 513–529.
[13] V. A. Gorelov, On the structure of the set of E-functions satisfying linear differential equations ofsecond order, Math. Notes 78.3 (2005), 304–319.
[14] V. A. Gorelov, On the weakened Siegel conjecture, Prikl. Mat. 11.6 (2005), 33–39, in russian; translation in J. Math. Sci. (N.Y.) 146.2 (2007), 5649–5654.
[15] É. Reyssat, Irrationalité de ζ(3) selon Apéry, Séminaire Delange-Pisot-Poitou, Théorie des nombres,
20 (1978-1979), Exposé No. 6, 6 p. Available at http://archive.numdam.org[16] T. Rivoal, J. Roques, On the algebraic dependence of E-functions, Bull. London Math. Soc. 48.2
(2016), 271–279.
[17] T. Rivoal, J. Roques, Holomorphic solutions of E-operators, Israel J. Math., 6 pages, to appear.
[18] C. L. Siegel, Über einige Anwendungen Diophantischer Approximationen, Abh. Preuss. Akad.Wiss.,
Phys.-Math. Kl. (1929-30), no. 1, 1–70.
[19] C. L. Siegel, Transcendental Numbers, Annals of Mathematical Studies, Princeton 1949.
[20] A. B. Shidlovsky, Transcendental Numbers, W. de Gruyter Studies in Mathematics 12, 1989.
[21] M. van der Put, M. F. Singer, Galois Theory of Linear Differential Equations, Grundlehren dermathematischen Wissenschaften, Vol. 328, Springer, 2003.
T. Rivoal, Institut Fourier, CNRS et Université Grenoble Alpes, CS 40700, 38058
Grenoble cedex 9, France.
Tanguy.Rivoal (@) univ-grenoble-alpes.fr
J. Roques, Université de Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne, France.
Julien.Roques (@) univ-lyon1.fr