ON THE GENERATING FUNCTIONS OF A PERIODIC INFINITE ORDER LINEAR RECURRENCE ANTÓNIO BRAVO A AND HENRIQUE M. OLIVEIRA B
Abstract. Let G be the space of generating functions of a periodic in…nite order linear recurrence. In this paper we provide an explicit procedure for computing a basis of G.
1. Introduction and Statement of the Main Result Let N = f0; 1; 2; : : :g be the set of the natural numbers and p a positive integer. In this paper we study a particular class of vectorial homogeneous recurrences of the type
(1)
xn+1 =
+1 X i=0
Ai xn i , for all n 2 N,
where xn 2 Cp and fAn gn2N is an in…nite sequence of p p matrices with complex entries. As in [3], we call this type of in…nite order linear recurrences generalized Fibonacci 1 recurrences, or F ib1 p recurrences for short. A F ibp recurrence is said to be s+ periodic (with s 2 Z ) if An+s = An for all n 2 N: The main result of this work concerns the study of the solutions of a s-periodic F ib1 p recurrence. Before stating the main result of this work, we introduce some notation. Throughout the paper, C [[z]] denotes the commutative ring of formal power series with complex coe¢ cients. Matrices with entries in C and C [[z]] will be m n denoted respectively as elements of Cm n and C [[z]] . As usual, the entry (i; j) of a matrix A will be denoted by A (i; j) : The in…nite-dimensional vector spaces over C M Q p C , U= Cp and V = n2Z
n2N
will play an important role in this discussion. We write u and v to denote the vectors of U and V , with components un 2 Cp and vn 2 Cp , i.e., u = (un )n2N = (u0 ; u1 ; : : :) , with un 2 Cp , un = 0 for almost all n,
and v = (vn )n2N = (: : : ; v
1 ; v0 ; v1 ; : : :) ,
with vn 2 Cp .
Date : November 4th, 2013 – AMS: 15A15, 39A06, 92D25,39A06. Key words and phrases. Generalized Fibonacci recurrences, kneading determinant, in…nite vectorial recurrence, periodic in…nite recurrence, formal power series. 1
2
A. BRAVO & H. M . OLIVEIRA
We write e1 ; : : : ; ep to denote the vectors of the standard basis of Cp . The symbols e , with 2 Z+ , denote the vectors of the standard basis of U : e1 = (e1 ; 0; 0; : : :) , e2 = (e2 ; 0; 0; : : :) , : : : , ep = (ep ; 0; 0; : : :) , ep+1 = (0; e1 ; 0; : : :) , ep+2 = (0; e2 ; 0; : : :) , : : : , e2p = (0; ep ; 0; : : :) , :::
where 0 denotes the zero vector of Cp . A vector v = (vn )n2N 2 V is said to be a solution of the F ib1 p recurrence (1), if the set fn 0 : vn 6= 0g is …nite and vn+1 =
+1 X i=0
Ai vn i , for all n 2 N.
The subspace of V whose vectors are the solutions of the F ib1 p recurrence is denoted by S. Naturally, there exists an isomorphism : U 7! S, where (u) = (vn )n2N ,
denotes the unique vector of S satisfying v
n
= un for all n 2 N.
The vector (u) 2 S is called the solution of the F ib1 p recurrence for the initial condition u 2U . The vector space U is called space of initial conditions. In order to analyze the asymptotic behaviour of a solution (u) = (vn )n2Z 2 S, we de…ne the generating function G (u) as the formal power series with coe¢ cients in Cp X G (u) = vn z n . n 0
p
Alternatively, G (u) can be de…ned as an element of the C-vector space C [[z]] such that G (u) = (G1 (u) ; : : : ; Gp (u)) , with X G (u) = vn( ) z n 2 C [[z]] , n 0
( ) vn
where denotes the -component of vn with respect to the standard base of Cp . Naturally, the map p G : U 7! C [[z]] u ! G (u) , is linear. We call G = G (U ) the space of generating functions of the F ib1 p recurrence. Notice that, from the linearity of the map G, we have (2)
G (u) =
X
c G (e ) ,
1
where (c ) 2Z+ denote the coordinates of u with respect to the standard basis, (e ) 2Z+ , of U . Therefore, the set G (e ) : spans G.
2 Z+ ,
GENERATING FUNCTIONS OF PERIODIC RECURRENCES
3
Now, we can state the main result of this work concerning the important class of periodic F ib1 p recurrences. As we will see, this result combined with the main theorem of [1] provides an explicit procedure for computing a basis of G. Theorem 1.1. If a F ib1 p recurrence is s-periodic, then all generating functions G (u) 2 G are rational function of z. Moreover, the generating functions G (e1 ) ; : : : ; G e(s+1)p span the space G. Hence, dim G (s + 1) p. 2. Proof of Theorem 1.1 The notions of kneading matrix and kneading determinant of a F ib1 p recurrence, introduced in [1] (see also [2]), are the main ingredients of the proof of Theorem 1.1. In order to improve the readability of the proof of Theorem 1.1, we present a brief description of the main ideas of [1]. Let = 1; : : : ; p and be a positive integer. Analogously to the kneading theory, stated by Milnor and Thurston in [4], we introduce the ( ; )-kneading increment of a F ib1 p recurrence as the following formal power series P A ( ; p) z n 2 C [[z]] , if p divides , Pn 0 n+q 1 K( ; )= n otherwise; n 0 An+q ( ; r) z 2 C [[z]] ,
where q and r denote, respectively, the quotient and the remainder of the integer division of by p. Notice that, in the particular case ; = 1; : : : ; p the kneading increment is given by X K( ; )= An ( ; ) z n , n 0
and the p
p-matrix of formal power series 0 1 K (1; 1) K (1; p) B C p .. .. .. K=@ A 2 C [[z]] . . . K (p; 1) K (p; p)
p
,
receives the name of kneading matrix of the F ib1 p recurrence. Finally, we de…ne kneading determinant of the F ib1 p recurrence by = det (I
zK) 2 C [[z]] ,
where I denotes the p p-identity matrix. Similarly, for each = 1; : : : ; p and 2 Z+ we de…ne the extended kneading matrix K ( ) and the extended kneading determinant ( ) of a F ib1 p recurrence by setting 0 1 K (1; 1) K (1; p) K (1; ) B C .. .. .. .. B C (p+1) (p+1) . . . . (3) K ( ) = B , C 2 C [[z]] @ K (p; 1) A K (p; p) K (p; ) ( ; 1) ( ; p) ( ; ) where (i; j) is the usual Kronecker delta function and ( ) = det (I where I denotes the (p + 1)
zK ( )) 2 C [[z]] ,
(p + 1) identity matrix.
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A. BRAVO & H. M . OLIVEIRA
Next, we recall the main result of [1], which establishes a main relationship between the generating functions G (e ) and the kneading determinants and ( ) of a F ib1 recurrence. p
Theorem 2.1. For every = 1; ::; p and every vector e of the standard basis of U , the generating function G (e ) of a F ib1 p recurrence satis…es the following equation in C [[z]]
1
zG (e ) = 1
( ).
We have all the ingredients needed to prove Theorem 1.1. First, observe that for any = 1; : : : ; p and 2 Z+ the kneading increment 1 K ( ; ) of a s-periodic F ibp recurrence is a rational function of z. Indeed, as An+s = An for all n 2 N, one gets
(4)
K( ; )=
( P
n 0
P
An+q
1
( ; p) z n ,
if p divides ,
n
An+q ( ; r) z , otherwise, Ps 1 k z sm , if p divides , 0 k=0 Ak+q 1 ( ; p) z Ps 1 k z sm , otherwise, 0 k=0 Ak+q ( ; r) z
n 0
=
=
=
8 P > < m
P > : m 8 P s 1 > < k=0 Ak+q > : 8 < :
P sm ( ; p) z k , if p divides , m 0z Ps 1 P k sm , otherwise, k=0 Ak+q ( ; r) z m 0z
Ps
1 k=0
Ps
1 k=0
1
Ak+q 1 ( ;p)z k , 1 zs
Ak+q ( ;r)z 1 zs
if p divides ,
k
,
otherwise,
where q and r denote, respectively, the quotient and the remainder of the integer division of by p. So, the kneading determinant , as well any extended kneading determinant ( ) of a s-periodic F ib1 p recurrence are rational functions of z. But, by Theorem 2.1, this means that any generation function G (e ) is rational too. Combining this with (2) we …nally conclude that any generating function G (u) of a s-periodic F ib1 p recurrence is a rational function of z. This is precisely what is stated in the …rst part of Theorem 1.1. Next, we prove the second part of Theorem 1.1. Evidently, if q 2 N and r 2 N denote, respectively, the quotient and the remainder of the division of 2 Z+ by p, then the quotient and the remainder of the division of + sp by p are, respectively, q + s and r. So, since the F ib1 p recurrence
GENERATING FUNCTIONS OF PERIODIC RECURRENCES
is s-periodic, one gets by (4) 8 < K ( ; + sp) = : 8 < = : 8 < = :
Ps
1 k=0
Ps
1 k=0
Ps
1 k=0
Ps
1 k=0
Ps
1 k=0
Ps
1 k=0
Ak+q+s 1 zs
1(
;p)z k
Ak+q+s ( ;r)z k , 1 zs Ak+q 1 ( ;p)z 1 zs Ak+q ( ;r)z 1 zs
if p divides
+ sp,
otherwise,
k
,
if p divides
+ sp,
k
,
Ak+q 1 ( ;p)z 1 zs Ak+q ( ;r)z 1 zs
,
5
otherwise, k
,
if p divides ,
k
,
otherwise,
= K ( ; ),
for all
2 Z+ . Combining this with (3), one obtains
= 1; : : : ; p and
K ( ) = K ( + sp) , for all
= 1; : : : ; p and
> p.
Therefore, ( ) = det (I
zK ( )) = det (I
zK ( + sp)) =
( + sp) ,
and, by Theorem 2.1, we can write G (e ) = G (e This proves that G (e
+sp )
+sp ) ,
for all
= G (e ) for all
= 1; : : : ; p and
> p. Consequently,
2 Z+ = fG (e ) :
G (e ) :
> p.
= 1; : : : ; p (1 + s)g ,
which proves that G (e1 ) ; G (e2 ) ; : : : ; G ep(1+s) span G. Hence, dim G p (1 + s). This is precisely what is stated in the second part of Theorem 1.1. We …nish the paper with two simple examples to illustrate the role played by Theorems 1.1 and 2.1 in the explicit computation of a basis of G. Example 2.2. Consider the 1-periodic F ib1 2 recurrence xn+1 =
+1 X i=0
Ai xn i , for all n 2 N,
with An =
1 3
By (4), the kneading increments are ( 2 is even, 1 z , if K (1; ) = 1 is odd, 1 z , if for all (5)
2 4
, for all n 2 N:
and K (2; ) =
(
4 1 z, 3 1 z,
if
is even,
if
is odd,
2 Z+ . Therefore, K=
1 1 z 3 1 z
2 1 z 4 1 z
,
= det (I
zK) =
4z 2 (1
7z + 1 2
z)
.
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A. BRAVO & H. M . OLIVEIRA
On the other hand, as 8 > > > > > > > < K ( )= > > > > > > > : one has (6)
8 < :
0 @ 0 @
( ) = det (I
( ; )(
2 1 z 4 1 z
2 1 z 4 1 z
( ; 1)
( ; 2)
( ; )
1 1 z 3 1 z
2 1 z 4 1 z
1 1 z 3 1 z
( ; 1)
( ; 2)
( ; )
1
A,
if
is even,
A,
if
is odd,
1
zK ( )) =
z ) 2 ( ;2)z 2 +2z 3 +2z 2 7z+1 , if (1 z)2 4z 3 +7z 2 z )+2 ( ;2)z 2 (2z 1) z 3 +3z 2 7z+1 (1 z)2
( ; )(7z
2
1 1 z 3 1 z
4z
3
is even, , if
is odd.
From Theorem 1.1, the generating functions G (e1 ), G (e2 ), G (e3 ) and G (e4 ) span G. By (5), (6) and Theorem 2.1, the computation of the generating functions is straightforward 5z 2 6z + 1 3z 2 + 3z G (e1 ) = ; 2 , 2 4z 7z + 1 4z 7z + 1 G (e2 ) =
2z 2 + 2z 2z 2 ; 4z 2 7z + 1 4z 2
3z + 1 7z + 1
,
G (e3 ) =
z2 + z 3z 2 + 3z ; 4z 2 7z + 1 4z 2 7z + 1
,
2z 2 + 4z 2z 2 + 2z ; . 4z 2 7z + 1 4z 2 7z + 1 Thus, G (e1 ), G (e2 ), G (e3 ), G (e4 ) are linearly independent. Therefore, dim G = 4. G (e4 ) =
Example 2.3. Now, consider the 1-periodic F ib1 2 recurrence xn+1 =
+1 X i=0
with
Ai xn i , for all n 2 N,
1 1 , for all n 2 N. 1 1 As in the previous example, the generating functions G (e1 ), G (e2 ), G (e3 ), G (e4 ) span G. Now, by Theorem 2.1, one gets An =
G (e1 ) =
2z 3z
z 1 ; 1 3z 1
, G (e2 ) =
z
3z
2z ; 1 3z
1 1
,
and z
;
z
. 3z 1 3z 1 Thus, G (e1 ), G (e2 ), G (e3 ) are linearly independent and dim G = 3. G (e3 ) = G (e4 ) =
Acknowledgements The authors were supported by CAMGSD-LARSys and CEAF through FCT Program POCTI-FEDER.
GENERATING FUNCTIONS OF PERIODIC RECURRENCES
7
References [1] João F. Alves, António Bravo and Henrique M. Oliveira. Kneading determinants of in…nite order linear recurrences. arXiv preprint: http://arxiv.org/abs/1307.3474, 7 2013. [2] João F. Alves, António Bravo and Henrique M. Oliveira. Populations dynamics with in…nite leslie matrices. arXiv preprint: http://arxiv.org/abs/1307.4036, 7 2013. [3] Benaissa Bernoussi, Walter Motta, Mustapha Rachidi and Osamu Saeki. On periodic 1generalized …bonacci sequences. Fibonacci Quarterly, 42(4):361–367, 2004. [4] John Milnor and William Thurston. On iterated maps of the interval, volume 1342 of Lecture Notes in Mathematics, pages 465–563. Springer, Berlin, 1988. a Centro
de Análise Funcional e Aplicações, b Centro de Análise Matemática Geometria e Sistemas Dinâmicos, a;b Mathematics Department, Technical Institute of Lisbon, University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal, Email Adresses: a
[email protected], b
[email protected] (corresponding author)
