Two theorems on meromorphic functions used as a principle for proofs on irrationality

T

HOMAS

N

OPPER

R

OLF

W

ALLISSER

Two theorems on meromorphic functions used as a

principle for proofs on irrationality

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{1 (2001),}

p. 253-261

<http://www.numdam.org/item?id=JTNB_2001__13_1_253_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Two theorems

on

meromorphic

functions

used

as a

principle

for

proofs

on

irrationality

par THOMAS NOPPER et ROLF WALLISSER

RÉSUMÉ. Dans cet article, nous nous intéressons à deux théorè-mes dûs à Nikishin et

_Chudnovsky

se _rapportant à des

fonc-tions

_{méromorphes.}

Notre _{propos est} ici de déduire

_simplement

de certaines

_propriétés

de fonctions

_{méromorphes,}

satisfaisant

à des conditions

_{arithmétiques,}

des résultats d’irrationalité bien

que

déjà

connus mais non triviaux. L’intérêt de cette

_approche

ne réside _pas dans les résultats

(obtenus

comme corollaires de nos

théorèmes)

sur l’irrationalité de

_nombres,

dont la

transcen-dance a été établie

_{depuis longtemps}

(cf.

[4], [9]

and

[10]).

Il

réside

_plutôt

dans l’intervention de théorèmes concernant les co-efficients de

_Taylor

de fonctions

_méromorphes

_qui ont une

ca-ractérisation

_{arithmétique.}

De la même manière _que Niven

_[6]

utilise la méthode d’Hermite pour donner tous les résultats connus sur l’irrationalité de valeurs de fonctions

_{trigonométriques,}

nous

utilisons les résultats de Nikishin et

_Chudnovsky

_{(cf. [2], [8]),}

pour

déduire l’irrationalité de valeurs de fonctions non-élémentaires.

ABSTRACT. In this _paper we discuss two theorems on

meromor-phic

functions of Nikishin and

_Chudnovsky.

Our _purpose is to

show,

how to derive some well-known but not obvious results on

irrationality

in a

_systematic

and

_simple

_way from _properties of

meromorphic

functions with arithmetic conditions. As far as it

stands,

we have no new results on

_{irrationality,}

to the _contrary some results on numbers of the corollaries are known

_already

since

a

_long

time to be transcendental

(cf.

[4], [9]

and

[10]).

Our main intention lies in theorems on

meromorphic

functions whose

Tay-lor coefficients are

arithmetically

characterized. Like Niven

[6]

used Hermite’s method to

_give

all known results on

_{irrationality}

of

_{trigonometric functions,}

we use methods

_going

back to Nikishin

[5]

and

_{Chudnovsky (cf.}

_[2]

and

_[8]),

to _give results on

_{irrationality}

of values of

_{non-elementary}

functions.

In

_[5]

E.M. Nikishin _gave a short

proof

of the

_following

theorem on entire

functions

_satisfying

certain arithmetic conditions:

Theorem 1. Let

_f

be a transcendental entire

function of

strict order

~ 2

1.

All derivatives

_{of f}

at the

_points

0

_0,

as well as A

shall be numbers

_of

the Gauss

_field

_Q(i).

Let

_Do(n)

and

_D,B(n)

be the least

common

_{multiple of}

the denominators

_of

the numbers and

f(k)(À),

k =

0, ... ,

n,

respectively.

If for

a

_given

n

for

D(n)

:=

Do(n)Dx (n)

the relation

holds,

where a does not

depend

on n.

Nikishin remarks that from this theorem the

_{irrationality}

of the numbers

exp(p/q)

and 7r can be deduced.

Obviously

one can extract much more out

of

_{it; sometimes, however,}

the condition

_(1.1)

causes some

difficulty.

That

is the reason

why

we have tried to weaken condition

_(1.1).

With a

_slight

generalization

of Nikishin’s method we can _prove the

following

result:

Theorem 1’. Let

_f

be a transcendental entire

function of

strict order Q

3/2.

All derivatives

of f

at the

_points

0

_0,

as well as A

shall be numbers

_of

the Gauss

_field

_Q(i).

Let

_Do(n)

and

_Da(n)

be the least

common

multiples

of

the denominators

of

the numbers and

k =

_{0, ... ,}

n,

respectively..

Then

for infinitely

many n E N

where

and C &#x3E; 0 does not

_depend

on n.

Proof of Theorem 1’. We shall not

_give

the whole

_proof

of Theorem I’

but refer for the first

_part

to the _paper of Nikishin

_[5].

To avoid condition

(1.1)

we

proceed

in the

following

_way:

We take the Pade

_{approximation}

for e’ as mentioned in

_~5~,

with

An(z)

and

_Bn(z)

are

polynomials

of degree n

with

_integer

coefficients and

the

_property

_{An(z) = -Bn(-z).}

If we substitute z in

(1.2)

by

the

_operator

Ad/dz

and then

_apply

the

_operator

_{Rn (Ad/dz)}

to the entire function

_{f (1)}

we obtain

This holds for all n E N and 1 E

_No.

For a verification of this

identity

compare

[5].

Since

(1.2)

also holds for n -

1,

it

immediately

follows that

(cf.

Siegel

[11],

pp.

6)

Substitution of z in

_(1.4)

_by

the

_operator

_Ad/dz

and

_application

of the

resulting

operators

to

_{f yields}

where

_{Fn(z) :=}

and

_{An(z) :=}

_E’

_{ak,nz k}

E Since

_by

assumption

f

is a transcendental entire

_function,

there are

infinitely

_many

n E N with

_{f ~2~-1~ (0) ~}

0. Therefore there exists an infinite subset M C N

with the

_property:

for all n E M there is a 0

_kn

_n, such that

Without loss of

_generality

it can be assumed that in

(1.5)

always

the first

or second case holds

(take

otherwise a

subsequence

of

M).

Lemma. For

_{sufficiently large}

n E I~Y we have

where

Cl

is a constant

independent of n

and as in

Theo-rerra 1 ’.

Choosing

r = and

taking

into _account the estimation

where the constant

_C2

does not

_depend

on _r, we obtain for

sufficiently large

values of n

There the constant

_C3

does not

_depend

on n. A

simple

calculation shows

that

and with

we

finally

_get

the

inequality

~F~~"~ (0) ~

Cïn-8(u)n.

Obviously

the same

estimation holds for

F(kn) (0) -

The

_proof

of the Lemma is

_completed.

Now we assume 0 for all n E M. If we take 1

= kn

in

(1.3)

and

use the

_identity

An(z) = -Bn(-z),

we

_get

at the

_{point z}

= 0

If

_C5

denotes the denominator of

_A,

we obtain

by

(1.7)

and the

assumptions

about the values of the derivatives of

_f

at the

_points

0 and A

and therefore

Taking

into account the estimation of the Lemma for

FÁkn) (0)

this leads for all n E M to

with a constant C &#x3E; 0 not

_depending

on n. Since the case

(0) ~

0

yields

the same

_estimation,

the

proof

of Theorem 1’ is

completed .

Remark. Instead of

_Q(I)

we can consider _any

imaginary quadratic

number

field.

Application.

As we don’t

_require

condition

(1.1),

Theorem 1’ can be

ap-plied

more

easily

than Theorem 1 in order to

gain

results on the arithmetic

nature of the values of

_{non-elementary}

functions. As an

example,

we

de-rive some results on

_{irrationality}

for certain

_{hypergeometric}

series and the

incomplete

gamma function.

Corollary

1. For

_{all x 0}

0 in an

imaginary

quadratic

number

field

K

and all a E

_{QB{O, -1, 20132,... }}

the

_{confluent hyper geometric function}

never has values in the same

_field

K.

In

_particular,

all real zeros

_{different from}

0

_of

the so-called

_incomplete

function

,(a,

~)

for

a E

_{QB{0, 20131,}

_{-2, ... }}

are irrational and the

complex

zeros

_different

_from

0 are in no

_{imaginary quadratic}

extension

of

Q.

Here

_{,(a, x)}

is

_{defined by}

for

real x &#x3E; 0 and

_for

_Re(a)

&#x3E; 0 and

_{by analytic}

continuation

_{for complexe}

values

_of

a and x.

Proof of

_Corollary

1. The entire function

_y(x)

:=

~(1, 1-f-a; x)

has order

of

_growth

1 and fulfills the differential

_equation

If we consider

y(x)

at the

_{point 0, then dk}

:=

Y(k)(0)

=

k!/(a+1) ...

(a+k)

for all k &#x3E; 0.

_{Following Siegel}

_[11],

pp.

54,

there is a constant

_Co

&#x3E;

_0,

independent

of

_k,

such that we have for the denominators of the numbers

do, ... , d~

Let us _suppose that there is a number w E

_KB101

with

_y(w)

_{= q} E K.

(~K

denoting

the

_ring

of

_integers

of K.

_{Setting f}

=

y in the notation of

The-orem l’ we therefore

_get

D(2n)

=

Do(2n)Dw(2n)

Cn for all

sufficiently

large

n. As this limitation of the

growth

of the denominators contradicts

Theorem

_1’,

the

_assumption

made above _{is wrong.}

_Hereby

the statement

about the arithmetic nature of values of + a;

x)

is

proved.

holds

_(cf.

_Erdélyi

_[3],

_p.

_{133). If -y(a, x)}

= 0 for _{an x E}

x)

would vanish. This contradicts the considerations above and so the claim

about the zeros of the

incomplete

_gamma function is

proved.

2. On a theorem of

_Chudnovsky

In connection with the work of Nikishin we mention a

far-reaching

the-orem on

_meromorphic

functions with arithmetic conditions

_by

G.V.

Chud-novsky

[2] (also

cf. E.

_Reyssat

_[8]),

which is

_proved

within modern

tran-scendence

_theory:

Theorem 2. Let

_f

be a transcendental

meromorphic

junction of

order p

2.

S denotes the set

of

all

algebraic

numbers w E

_Q,

such that all derivatives

_f(k)(w)

are rational

_integers.

Then the set S is

_{of cardinality}

at most _p.

Like Niven

_[6]

_applied

Hermite’s

_proof

of the transcendence of e to _prove

irrationality

of values of

_elementary

_functions,

we use a

_simple

general-ization of Theorem 2 to

_get

results on the arithmetic nature of values of

non-elementary

functions,

too. In our

case Q

is

replaced by

a certain

alge-braic number field K of

_degree

4 and the values

_f(k)(w),

w E

_K,

are certain

algebraic

numbers with a restricted

growth

of the denominators. With this

modification one

_gets

the

following

result:

Theorem 2’. Let

_f

be a transcendental

meromorphic

function of

order

p 2 and K a

_quadratic

extension

of

k,

where

k

= Q

or _any

_imaginary

quadratic field.

Let there be a _sequence

of

positive integers bk

and a constant

2 a meromorphic function is said to be of order p, if it can be expressed as a _quotient of two

C with

such one has

bkf(k) (Wi) E

_Ok

(rsng

of

_integers

of

K).

Then wl =

w2.

Remark. Theorem 2’ can be _proven with some minor

_changes

in the

proof

of Theorem 0.1 in

_Chudnovsky

_[2]

because of the nature of the denominators

bk.

Therefore we omit the

_{proof. Furthermore,}

one can

_easily

see

(compare

the _way how

_Chudnovsky

obtained the

_inequalities

of Lemma 0.10 in

_[2],

p.

391)

that if a function of order 1 is

taken,

then a

_stronger

growth

of

the

denominators bk

can be allowed:

Theorem 3. Let

_f

be

_of

order _p 1 and K as in Theorem 2’.

_If

at

the

_point

wl E K condition

(2. 1)

of

Theorem 2’ holds and

if

there exist

Co,

Cl

E

_{N‘ ,}

such that

_for

W2 E K

then wl =

W2-In order to show that Theorem 3 can be

_applied

in a much more

_general

setting

than Theorem

1,

we

_give

an

application

concerning

the

irrationality

of certain continued fractions

_(compare

_e.g. Bertrand

_[1]):

Corollary

2. For _any

_imaginary

_quadratic

a E

K,

_{a ~}

_0,

which is not a

zero

of

the value

_of

the

_logarithmic

derivation

_of

_~"

is never in K. In

_particular,

for

all such a iuith 0 and v E

No

the value

_of

the continued

fraction

is not in K.

is

_meromorphic

and of order

_1/2.

As

4bv

fulfills a linear differential

equation

of second order one obtains

and this

_equation

_{inductively yields}

Here + 2 for i =

1, 2, 3. Furthermore,

_a

simple

calculation shows that

Let us assume that there is an a E

K,

_0,

with 0 and

E

K.

With C :=

_equation

_(2.2)

yields

If we now

apply

Theorem 3 with

f = fv,

Co

=

Cl = v

₊

1,

wi = a and

w2 = 0

then a = 0 follows. Hence the

_assumption

made above _was wrong,

which _proves the statement about the values of the

_logarithmic

derivation of

In order to obtain the statement about the continued fractions we consider

the entire function

The

_identity

can

_easily

be verified for all a

e k

with 0.

Following

[7],

_p.

210,

the

_right

side of

_(2.3)

can be

expanded

into the continued fraction

References

[1] D. _BERTRAND, A Transcendence Criterion _{for Meromorphic} Functions. In: Transcendence

Theory: Advances and Applications, Chapter 12. Academic Press _London, 1977.

[2] G.V. CHUDNOVSKY, Contributions to the _{Theory of} Transcendental Numbers. _Chapter 9.

AMS, 1984.

[3] M. _ERDÉLYI, T. OBERHETTINGER, Higher Transcendental _Functions, Vol. II. Mc Graw-Hill

New York, 1953.

[4] W. MAIER, Potenzreihen irrationalen Grenzwertes. J. Reine _Angew. Math. 156 _(1927),

93-148.

[5] E.M. NIKISHIN, A Property of the Taylor Coefficients. Math. Notes 29 _(1981), 525-527.

[6] I. NIVEN, Irrational Numbers. The Mathematical Association of America. _Quinn and Boden

Company, 1956.

[7] A.N. PARSHIN, A.N. _SHAFAREVICH, Number Theory IV. _{Encyclopaedia} of Math. _Sciences, Vol. 44. _Springer, 1998.

[8] E. REYSSAT, Travaux récents de G. V. Chudnovsky. Séminaire Delange-Pisot-Poitou n° 29

(1977).

[9] A. _SHIDLOVSKY, Über die Transzendenz und _{algebraische Unabhängigkeit} von Werten

einiger Klassen ganzer Funktionen. Doklady Akad. Nauk SSSR 96 _(1954), 697-700.

[10] C.L. SIEGEL, Über einige Anwendungen Diophantischer Approximationen. Abh. Preuss. Akad. Wiss. 1929, Nr.1, 70 S.

[11] C.L. SIEGEL, Transcendental Numbers. Princeton University Press, 1949.

Thomas NOPPER, Rolf WALLISSER

Mathematisches Institut

Universitat Freiburg

D-79104 _Freiburg i. Br.

Germany

E-mail : ° nopper@sun7.mathematik.uni-freiburg.de rowaQnim. mathemat ik. uni -fre iburg. de