Mathematics. - A prop,erty ,of a DIRICHLET series, repr-esenting a function
satisfying an algebraic difre~ence~difflf!rential equation. By J.
POPKEN. (Communicated by Prof. J. G. VAN DER CORPUT.)
(Communicatcd at the meeting of April 23. 1949.)
By definition a function f (s) satisfies an algebraic difference~differential
equation if there exists a polynamial F(xo. Xl' ...• Xt) 'jÉ O. a set af t real
nu mb ers U1' U2 • ... , Ut and a set af t non~negative integers nl. n2 • .... nt.
such th at the t systems (u1' nd, (U2' n2)' .... (Ut. nt) are different and
In this paper we will deduce the following
00
Theorem: Let the convergent DIR1CHLET series I
Bh e-1h s ,
with non~
h=\
va,nishing coefficients ah, tlepr1,es.ent a function f (s) satisfying an al[JJf!braic
IequatiOln. Then there exists a positive number c,
such that
diffeknae~diffet'ential
(1)
is valid forall but a finite Tlumber of ,values of h.
This theorem can be used to derive the transcendental~transcendency
of certain functians. Let us take for example
00
C(s)= 1:
e-slollh,
h=\
=
th en it is clear. that the exponents )'h
log h in this series dOl not have
the property (1). hence ,( s ) cannot satisfy an algebraic difference~
diHerential equation; it fallaws in particular that ,(s) is transcendental~
transcendent 1).
aD
In this paper we confine ourselves to DIRICHLET series
I
ah e-1hS.
h=\
where the "exponents" )'h are real numbers with
An "empty" sum. i.e. a formal sum with no terms at all, will be equal
to zera by definition.
1) However this proof for HILBERT' s theorem on the transcendental-transcendency of
the C-function is not essentially new. but closely related to OSTROWSKl's proof in the
paper cited below,
500
A
well~known
theorem of OSTROWSKI 2) states: Let the convergent
CXI
DIRICHLET
series
ah e-).h s with
~
non~vanishing
coefficients ah, represent
h=1
a function, satisfying an algebraic difference~differential equation. Then
there exists for the set of all expcinents J.h a fini te linear integral basis.
In the same paper of OSTROWSKI we find another result, connected with
the foregoing theorem, which will serve us as a basis for the proof of our
own theorem:
CXI
Lemma 1 3): Let the conIJlergent DIRICHLET series 2: ah e-).h s , with all
h=1
c~fficients
ah diffel1e'nt from zero, represent a function, ;Satisfying ',aJn
algebraic diff'er~.nce~diffret1entiallequation. Then thete \exists a real number
A and a positive integ:er N, such that for sufficiently large values of h
every numb;er Ah + A is a sum of Nexponents at most, talCen from the
sequence J.! , A2' ... , Ah-I'
Now for sufficiently large values of h the number Ah + A is positive,
hence the sum mentioned in the lemma is not empty. Putting AO
-A
we can give th is lemma the following form: There !exists a positive integer
r, such that
=
(2)
where n = n(h) denotes a positive integer ;5 N and hl;:;;; h 2 ;:;;; ... ;:;;; h n
rrepresent positiv.e in&!gers < h.
Evidently
lh
> lh, ==- lh ==- , , , ==- lh n ==- ll'
2
We can take r so large, that
lh
=r 0
for h ==- r ,
(3)
In order to prove our theorem we first deduce the following
Lemma 2: Let th,e conditio,ns of lemma 1 be fulfilled, so that (2) holds,
Then ther.e 'exists a positi~e in~.eger H ;:;;; r, such that
l~ ==- l~,
Proo f.
+ l~, + . ,. + l t + ~
We put
then it follows
Cl ;:;;;
0 and
l~
Let h take the va lues r, r
lh
2) A.
Mathem,
3) A.
notations
for h ==- H.
+ N l~ -== c~,
+ L r + 2, .. " so that
= lh, + lh. + ' .. + lh n + lo .
OSTROWSKI, Ober Dirichletsche Reihen und algebraische Differentialgleichungen,
Zeitschrift 8, 241-298 (1921), Satz 6, p. 260.
OSTROWSKI, loc. cito p, 261-262; compare the proof of "Satz 7", OSTROWSKI's
for h. N and A are respectively i, n and Al.
501
Now we distinguish the following two cases:
n~2.
1°.
n=1 or
2 0•
n ~ 2. lh, > 3ct.
lh,-==3cl'
In the first case the expression
lh-lh, = lh,
+ lh. + ... + lh n + lo
can take only a finite number of different values; for th is is clear for
n
1. and. if n ~ 2. then the positive integers n. h 2 • h3' .... hl! satisfy
=
n
-== N.
h n -== h n-
-== •.. -== h 2•
lh, :s; 3c!. ).h ~ + <Xl
I
where h 2 is bounded on account of
for h ~ <Xl. Hence
the numbers Àh - )'h, • being positive. have a positive minimum. It follows
the existence of a positive integer H ~ r. such that
. (4)
Now
on account of
50
that
l~,
+ lt + ... + l~n + l~ -== l~, + ~ + (N-l) li + (N-l) l~,
-== l~, + ci + 9 (N-I) d
-== l~, + 9 Nci.
From (4) it follows for h ~ H
+ "'h. + ... + lh n + < "'h, + "'h-"'h, = "'h.
,2,2
"'h,
2,2,2,2,2,2
"'0
and the assertion of our lemma is true in the considered case.
H. on the other hand. n ~ 2. lh. > 3c!. th en we denote the positive
exponents in the right~hand member of Àh = lh,
lh.
lh n 10 by
lh,. lh,• ...• lh m (m ~ 2) and we write
+ + ... +
lh = lh,
+
+ lh, + ... + lh m + ~h,
where
1~h 1-== (n-m) 11 1 1+ 1lo 1-== N 1II I + 1lo I=
Ct·
Now
=- I
m
1'=1
= I
m
,.=1
2
lhl'
2
lhl'
+ 2 lh,
m
m
I lhl'-2 Cl I lhl'
1'=2
+ (21h,-2cl)
,..=1
m
I lhl' + lh, (lh.-2cl) + lh.(lh,-2ct).
1'=3
502
m
2
Z
Àh ::=-
2
Àhp.
1'=1
The exponents
m
+ Àh,
12
+
~
~
1
2
Ah y
2
Àhp.
1'=1
2hm+l' Àhm+2' •••• Àh n
"0
Z
Cl::=-
+ 3S.
are negative or zero. hence
-==
= "012 + N
2
1
AI
-== Cl2 •
=
It follows
=
12::=Ah
m
....
~
1'=1
+ . n.
12
Ah
12
~
/A
+,
2
Ahy
IL() •
y=m+1
This proves the lemma.
Proo f
0
f th e th e 0 rem: On account of lemma 1 we have
<
with 1 -== n -== N. h n -== hn-I -== ••• -== hl
h.
Now we consider r arbitrary numbers Xo. Xl' ...• Xr-I and the infinite
sequence Xo. Xl' ...• Xr-I. Xr. Xr+1 •.•.• defined by the recurrent relations
Xh
=
Xh,
+ Xh, + ... + Xh n + Xo
{or h = r. r
+ 1. r + 2. . . . •.
(5)
where n. hl' h 2 • •••• h n are the same int eg ers as in the foregoing formula.
It follows that thc sequence Xo. xl. X2 • •.. has an integral linear basis
consisting of the numbers Xo. Xl' ...• Xr-l ; we even have
Xh
= PhO Xo + Phl + ... + Ph.
XI
(h
r-I Xr-l
= O. 1. 2•... ).
• (6)
where PhO. Phl ••••• Ph. r-l denote integers ~ O.
In particular
Àh
= PhO )'0 + Phl Ä,l + ... + ph, r-l Ä,r-I
(h = O. 1. 2•... ).
Let
_ ~ 1 i{ Àe -=t- 0
Xe - .
{or
o r{ Àe=O
_
{! -
• (7)
O. 1. . . .• r-l
(so that at most two of the numbers X 0, X I'
being equal to unity). Hence
... ,
X r-I are zero, the others
so that
Àh
= PhO X o Ào + Phl Xl Àl + ... + ph, r-l X r - l Àr"':'l
(h
= O. 1. 2•... ).
Now all exponents 21 , À2 • 23' ... are different; it follows that all systems
(PhO X o' Phl
arè -different.
Xl' .... ph,r-I X
r- l )
{or h = 1. 2. 3 ....
503
Let P denote an arbitrary positive integer. The total number of different
systems
(Po X o• PI Xl' ...• pr-l X r- l).
=
where Po. Pl •...• Pr-l are integers with 0:;;;; PI! ~ P-I (e
O. I •.... r-I)
is pr-T • • denoting the number of zeros in the sequence X o• Xl' .... X r- l •
Hence there exists in thc sequence 1, 2 •...• pr + 1 at least one number k.
such that the system
(PkO Xo. Pkl Xl' ...• pk.r-l X r- I)
contains at least one integer Pke Xe::=- p, and therefore
PkO X o
+ Pkl Xl + ... + Pk. r-l X r- l
::=-
P.
We introduced already the numbers X o• Xl' .... X r - I in (7); let
, ••• satisfy the recurrent relations (5) with X h in stead of Xh.
It follows from (6)
Xr.X r+ l
+
=
+ ... + ph.r-I X r-
Xh
PhO X o Phl Xl
Hence the sequence 1. 2 •...• pr
that
+ 1 contains
(h = 0.1, ... ).
I
at least one number k. such
. (8)
Let H be the integer of lemma 2. By (3) we know ),h -=j:- 0 for h ~ r;
it follows from (7) that Àh
0 implies X h
0 (h
O. 1, 2 .... ). Hence
there exists a positive number C2. such that
=
=
Xh -= C2 À~ for h
=
= O. 1, ...• H-l.
We shall show by induction thé!t this inequality also holds for h
Let h ~ Hand let
~
H.
= O. 1, ...• h-l.
X, -= C2 À~ for 1
We have h ~ H
~
r. hence
Xh=Xh, +Xh.+ ... +Xhn +Xo•
where 1 ~ hy:;;;; h-l (v
= 1.2....• n). so that
12
12
X h -=
= C2 "h,
+ C2 "h,
+ ... + C2 "h12 n + C2 "0,2 •
Applying lemma 2 we find. on account of h
12
"h,
+ "h. + ... +
12
12
"'h n
~
H.
+ =-= "h •
,2
12
"0
hence
X h -=C2 À~
holds for any nu mb er h = O. 1. 2 •....
=
Taking h k. where kis the integer in formula (8). we deduce C2À~::=- P.
so that every sequence 1. 2 ..... pr + 1 contains at least one number k.
such that
C2
l~
::=-
P.
5M
Now it is easy to prove the assertion of our theorem. Let h be an
arbitrary integer G 4 r , hen ce
h7~ 4;
1
put P
=[h7]-1.
1
1
P>h r -2===-thr and P+l-=h r
hence pr
+ 1 ::;;; h.
th en
.
It follows. that every sequence 1. 2 ..... h contains at
1
least one number k, such that C2À.~ ~ P
> ih r.
For sufficiently large h
1
clearly À.%>À.~, hence
C3
=
1
Y2C2
.
number <
. Now h
J.k
~ k,
is positive and it follows
À.k
> cah2r
• wh ere
1
hence J. h G J. k
> cah2r.
If we take for c a positive
1
ï-';' then
).h
> he
for all but a finite number of values of h.
Mathematics. - Remark on my paper "On LAMBERT'S proof for the irrationality of n". By J. POPKEN.
In these Proceedings 1) I have given an elementary proof for the
irrationality of n. However Dr M. VAN VLAARDINGEN kindly informed me.
that the method I used nearly is the same as that applied by HERMITE
in the fourth edit ion of "Cours de la faculté des Sciences" (1891),
p. 74-75 2 ).
Vol. XLIII (1940) p. 712-714.
See also: A. PRINGSHEIM, Vorlesungen über Zahlen- und Punctionenlehre 11, 1.
p. 471-474; p. 613.
1)
2)