On the zeros of recurrence sequences with non-constant coefficients

By

CORDELIA METHFESSEL

Abstract. The paper deals with the zeros of sequences, which satisfy linearrecurrences with non-constant coefficients. We prove a weaker analogon of theSkolem-Mahler-Lech theorem on ordinary recurrent sequences. The proof relies onSzemereÂdi�s theorem on arithmetic progressions in sets of positive density.

1. Introduction and result. Let F be a field and let f :� fg : N! Fg be the ring of allF-valued arithmetical functions with the usual pointwise addition and multiplication.Further let r be a subring of f with the properties

g 2 r; g�n� � 0 for infinitely many n 2 N) g�n� � 0 for all n 2 N�1�and

g 2 r) ga; q 2 r where ga; q : n 7! ga; q�n� :� g�qn� a� for all a; q 2 N:�2�Observe that r contains no zero divisors, as g�n�h�n� � 0 for all n 2 N implies eitherg�n� � 0 for infinitely many n 2 N or h�n� � 0 for infinitely many n 2 N. So we can define qas the field of fractions of r.

A function g 2 q with g : n 7! g�n� :� p�n�q�n� with p; q 2 r is well-defined for

n 2 N; n > n0�g� where n0�g� :� maxÿfn 2 N

��q�n� � 0g [ f0g�.Ex a mpl es. a) If the field F has characteristic 0, the ring F�n� of all polynomials with

coefficients in F has the above properties.b) For F � C consider the ring h�B�1; 1�� of all functions which are analytic on the disk

B�1; 1�, where B�c; r� :� fz 2 C��jzÿ cj < rg for c 2 C and r 2 R�. Define

r :� f : n 7! h1n

� �����h 2h�B�1; 1��� �

. Because of the identity theorem, (1) is fulfilled. To

see ha;q 2 r for all a; q 2 N, we have to show that hz

q� az

� �is analytic on B�1; 1�. But this

is quite clear, as the image of B�1; 1� under the transformation z 7! zq� az

is

B1

q� 2a;

1q� 2a

� �7 B�1; 1� for a; q ^ 1.

c) Let again F � C and let p be the ring of all analytic functions g : C! C withg�z� 2p� � g�z� for all z 2 C. Further let r :� fg : n 7! g�n���g 2 pg:Then (2) is, of course,

Arch. Math. 74 (2000) 201±2060003-889X/00/030201-06 $ 2.70/0� Birkhäuser Verlag, Basel, 2000 Archiv der Mathematik

Mathematics Subject Classification (1991): 11B37.

fulfilled, because g�z� 2p� � g�z� for all z 2 C implies g�a�z� 2p� � b� � g�az� b� for allz 2 C if a; b 2 N. If g�n� � 0 for infinitely many n 2 N, then g has infinitely many zeros inthe intervall �0; 2p�. Thus, by the identity theorem, g�z� � 0 for all z 2 Z.

In this paper, we are going to study the set of the zeros of functions, which ultimatelysatisfy a linear recurrence relation with coefficients in q, i.e. an equation of the type

g�n� k� � akÿ1�n�g�n� kÿ 1� � � � � � a0�n�g�n� � 0 �n 2 N; n ^ n0��3�of order k 2 N with coefficients ak 2 q �k � 1; 2; . . . ; kÿ 1�. We assume ak to be well-definedfor n ^ n0 and a0�n� �j 0 for n 2 N; n ^ n0. A sequence is called recurrent of order k in thiscontext, if it satisfies an equation (3) for n ^ n0.

For the case of constant coefficients ak 2 C �k � 0; 1; . . . ; kÿ 1� the well-known theoremof Skolem, Mahler and Lech states, that all zeros of g except for a finite set lie in a finiteunion of arithmetic progressions. The proof (compare for example Cassels, [1]) relies on theexplicit expression of the function g via Binet�s formula, for which we do not know ananalogon in these more general circumstances.

Here we can obtain the following:

Theorem 1. Let k 2 N, k ^ 2, and let g 2f satisfy a k-th order recurrence (3) withcoefficient functions in q for n 2 N; n > n0: Further assume that there does not exist a numberq 2 N such that the subsequences ga;q �a � 0; 1; . . . ; qÿ 1� are recurrent of order kÿ 1: Then

limn!1

#fn 2 Njn % n; g�n� � 0gn

� 0:

The assumption on the subsequences on arithmetic progressions corresponds with theassumption on ordinary complex recurrences not to be ªdegenerateº (compare, for example,Shorey and Tijdemann, [5], p. 38).

The following corollary is closer to the above stated Skolem-Mahler-Lech Theorem:

Corollary 1. Let g 2f satisfy a k-th order recurrence (3) with coefficient functions in q forn 2 N; n > n0. Then all zeros of g, with the exception of a set of density zero, lie in a finiteunion of arithmetic progressions.

2. Proof of Theorem 1. Consider the ring �q�z�;�; �� with the usual addition and themultiplication �, defined byPk

k�0ak�n�zk �

Pl

l�0bl�n�zl � Pk�l

m�0

� Pk�l�m

ak�n�bl�n� k��

zm

�n > max �fn0�ak�j k � 0; 1; . . . ; kg [ fn0�bl�jl � 0; 1; . . . ; lg��

forPkk�0

ak�n�zk,Pl

l�0bl�n�zl 2 q�z�:

If ak �j 0 and bl �j 0 then ak�n�bl�n� k� �j 0 for all but finitely many n 2 N because of (1).So �q�z�; �� contains no zero divisors. Thus �q�z�;�; �� is a non-commutative integral domainwith identity element 1. (Compare [2] and [3] for the algebraic treatment of such skewpolynomial rings.)

202 C. METHFESSEL ARCH. MATH.

Interprete the indefinite z as shift-operator

z : f 7!f, z : g 7! zg with zg�n� :� g�n� 1� �n 2 N�and let q�z� operate on f according to

�ak�n�zk � akÿ1�n�zkÿ1 � � � � � a0�n���g�n�� :�ak�n�g�n� k��akÿ1�n�g�n� kÿ 1� � � � � � a0�n�g�n�; �n 2 N; n ^ n0�;

where we assume ak to be well-defined for n ^ n0.Then (3) can be written as

f �z��g�n�� � 0 �n 2 N; n ^ n0�with f �z� � zk � akÿ1�n�zkÿ1 � � � � � a0�n� 2 q�z�. The polynomial f �z� is called thecompanion polynomial of (3).

Re ma r k 1 . If all coefficient functions of f �z�; d�z� 2 q�z� are well-defined for n ^ n0,then f �z��g�n�� � 0 for n 2 N; n ^ n0, implies �d�z� � f �z���g�n�� � d�z��f �z��g�n��� � 0 forn 2 N; n ^ n0.

First we make sure that a sequence, which satisfies a k-th order recurrence relation of type(3) on the sequence of all natural numbers satisfies a k-th order recurrence relation of thesame type on every arithmetic progression mod q for any given q 2 N.

Lemma 1. Let ak 2 q for k � 0; 1; . . . ; kÿ 1 and let q 2 N. Then there exist functionsbl 2 q for l � 0; 1; . . . ; k�qÿ 1� with

�bk�qÿ1��n�zk�qÿ1� � bk�qÿ1�ÿ1�n�zk�qÿ1�ÿ1 � � � � � b0�n��� �zk � akÿ1�n�zkÿ1 � � � � � a0�n��� �cq; k�n�zkq � cq;kÿ1�n�z�kÿ1�q � � � � � cq; 0�n��;

where cq;k 2 q for k � 0; 1; . . . ; k and cq;k �j 0 for at least one k 2 f0; 1; . . . ; kg.P roof. Comparing coefficients in

�bk�qÿ1��n�zk�qÿ1� � bk�qÿ1�ÿ1�n�zk�qÿ1�ÿ1 � � � � � b0�n��� �zk � akÿ1�n�zkÿ1 � � � � � a0�n��

� bk�qÿ1��n�zkq � �bk�qÿ1�ÿ1�n� � bk�qÿ1��n�akÿ1�n� k�qÿ 1���zkqÿ1 � � � �� �b0�n�a1�n� � b1�n�a0�n� 1��z� b0�n�a0�n��! �cq;k�n�zkq � cq; kÿ1�n�z�kÿ1�q � � � � � cq; 0�n��

leads to a system of k�qÿ 1� linear homogeneous equations over q in the k�qÿ 1� � 1unknowns bl 2 q �l � 0; 1; . . . ; k�qÿ 1��. This system possesses a non-trivial solution bl 2 q�l � 0; 1; . . . ; k�qÿ 1��. Because �q�z�;�; �� contains no zero divisors, at least one of theresulting functions cq;k is different from zero. h

Corollary 2. Let the functions cq;k; bl 2 q for k � 0; 1; . . . ; k and l � 0; 1; . . . ; k�qÿ 1� fromthe preceding lemma be well-defined for n ^ n0�q� ^ n0: For a sequence g 2 F, which fulfilsthe assumptions of Theorem 1, Lemma 1 together with Remark 1 implies

cq;k�n�g�n� kq��cq; kÿ1�n�g�n� �kÿ1�q� � � � � � cq; 0�n�g�n� � 0 for n^n0�q�:

203Vol. 74, 2000 Zeros of recurrence sequences

Moreover, we have cq; k �j 0 �j cq; 0, because otherwise g would satisfy recurrences of order% kÿ 1 on every arithmetic progression modulo q. Thus there exists a number N0�q� 2 N,

N0�q� ^ n0�q� with cq;k�n� �j 0 for k � 0; 1; . . . ; kÿ 1 and n ^ N0�q�. Consequently, we have

g�n� kq� � cq;kÿ1�n�cq;k�n� g�n� �kÿ 1�q� � � � � � cq; 0�n�

cq; k�n� g�n� � 0

for n ^ N0�q�.We want to apply the following result of SzemereÂdi (compare [6], p. 244).

Theorem [SzemereÂdi]. For all e > 0 and all k 2 N there exists a number N�k; e� such that ifn > N�k; e� and R 7 �0; n� \ Z with jRj > en then R contains a k term arithmetic progression.

From this Theorem we can obtain

Lemma 2. Let g 2f satisfy an equation (3) of order k 2 N. If there exists an e > 0 with

#fn 2 Nj n < n; g�n� � 0gn

> e�4�

for infinitely many n 2 N, then there exist numbers a; q 2 N with ga; q�n� � 0 for all n 2 N.

P roof. Let e0 :� e=2. According to SzemereÂdi�s theorem, there exists a number N�k; e0�,such that for n > N�k; e0� every set R 7 �n0; n0 � n� \ Z with jRj > e0n contains an arithmetic

progression of length k. AsN�k; e0�

kÿ 1is a finite number, the recurrences

zk� � c�; kÿ1�n�c�; k�n� z�kÿ1�� � � � � � c�;0�n�

c�; k�n�� �

�g�n�� � 0�5�

from Lemma 1 with � %N�k; e0�

kÿ 1form a finite set. Therefore the number

N0 :� max N0������ %

N�k; e0�kÿ 1

� �2 N

exists, and all coefficient functions n 7! c�; k�n�c�; k�n� for k � 0; 1; . . . ; kÿ 1, � � 1; 2; . . . ;

N�k; e0�kÿ 1

� �are well-defined for n ^ N0. According to assumption (4) there exists a number n0 > N0 with

#fn0 % n % n0 �N�k; e0�jg�n� � 0g > e0N�k; e0�:(Observe that

#fn0 % n % n0 �N�k; e0�jg�n� � 0g % e0N�k; e0� for every n0 2 N; n0 ^ N0

would imply

1n

#fn 2 Njn < n; g�n� � 0g %N0

n� nÿN0

N�k; e0�� �

� 1� �

e0N�k; e0�n

%N0

n� nÿN0

N�k; e0� � 1� �

e0N�k; e0�n

� N0

n�1ÿ e0� � 1�N�k; e0�

n

� �e0 % 2e0 � e

for n ^ 2 max fN0=e0;N�k; e0�g contradicting assumption (4).)

204 C. METHFESSEL ARCH. MATH.

Thus the set fn0 % n % n0 �N�k; e0�jg�n� � 0g contains a k term arithmetic progression

with modulus q %N�k; e0�

kÿ 1, say a; a� q; a� �kÿ 1�q. Finally (5) implies ga; q�n� � 0 for

n 2 N. h

Lemma 3. Let g 2f satisfy an equation (3) of order k 2 N, k ^ 2. If there exist a; q 2 N

with ga; q�n� � 0 for n 2 N then for every a0 2 N the sequence ga0;q is recurrent of order kÿ 1.

P roof. Let a; q 2 N. Like in the proof of Lemma 1 we can calculate coefficient functionsck; q;a 2 q for k � 0; 1; . . . ; kÿ 2 and c�q;a 2 q, which are well-defined for n ^ n0�q;a� and atleast one of which is not the zero function, with

ckÿ1;q;a�m�g�m� a� �kÿ 1�q� � ckÿ2; q;a�m�g�m� a� �kÿ 2�q� � . . .

� c0;q;a�m�g�m� a� � c�q;a�m�g�m� � 0�6�

for m 2 N;m ^ n0�q;a�. We may assume that at least one of the ck;q;a is different from zero,because otherwise (6) would imply g�n� � 0 for n 2 N, n ^ n0�q;a�. Putting m :� a� nq anda :� a0 ÿ a together with g�a� nq� � 0 for n 2 N yields the assertion. h

Theorem 1 now follows immediatelly.

3. Proof of Corollary 1 and a remark.

P roof o f the Cor o l l a ry. For every modulus q 2 N there exists a number

k�a; q� :� min fx 2 N��9 f��z� 2 q�z� : deg f��z�

� x; f��z��ga;q�n�� � 0g �a � 0; 1; . . . ; qÿ 1�:Let q 2 N be such that for no a 2 f0; 1; . . . qÿ 1g there exists a d 2 N with k�a� dq; qd� <k�a; q� for all d 2 f0; 1; . . . ; dÿ 1g. Then because of Theorem 1 the sequences ga; q eitherhave a set of zeros of density zero, or ga;q�n� � 0 identically for all sufficiently large n. h

Re ma r k . With the method of proof from 2., but with the following, less deep, theorem ofvan der Waerden/Rabung (compare [4], p. 144) instead of SzemereÂdi�s, we can obtain acorresponding, weaker result.

Theorem [van der Waerden, Rabung]. For any M; k 2 N there is a positive integerNd�M; k� such that any strictly increasing sequence fnigm

i�1 of positive integers withni�1 ÿ ni % M and with nM ÿ n1 ^ Nd�M; k� contains an arithmetic progression of length k.

From this we can obtain

Theorem 2. Let g 2f satisfy a k-th order recurrence (3) with coefficient functions in q forn 2 N; n > n0. Further assume that there does not exist a number q 2 N such that thesubsequences ga; q �a � 0; 1; . . . ; qÿ 1� are recurrent of order kÿ 1. Then for every M 2 N

there exists a number n0 2 N with g�n0 � m� �j 0 for m � 0; 1; . . . ;M.

References

[1] J. W. S. CASSELS, Local fields. London Math. Soc. Stud. Texts 3. Cambridge 1986.[2] P. M. COHN, Skew fields. Theory of general division rings. Encyclopedia Math. Appl. Cambridge

1995.

205Vol. 74, 2000 Zeros of recurrence sequences

[3] OYSTEIN ORE. Theory of non-commutative polynomials. Ann. of Math. II. 34, 480 ± 508 (1933).[4] JOHN R. RABUNG, On applications of van der Waerden�s theorem. Math. Mag. 48, 142 ± 148 (1975).[5] T. N. SHOREY and R. TIJDEMAN, Exponential Diophantine equations. Cambridge Tracts in Math. 87.

Cambridge 1986.[6] E. SZEMEREDI, On sets of integers containing no k elements in arithmetic progesssion. Acta

Arithmetica 27, 199 ± 245 (1975).

Eingegangen am 14. 9. 1998*)

Anschrift der Autorin:

Cordelia MethfesselMathematisches InstitutUniversität FreiburgEckerstr. 1D-79104 Freiburg

206 C. METHFESSEL ARCH. MATH.

*) Eine überarbeitete Fassung ging am 22. 3. 1999 ein.