ON ao-GENERALIZED FIBONACCI SEQUENCES Walter Motta Departamento de Matematica, CETEC-UFU, Campus Santa Monica, 38400-902 Uberlandia, MG, Brazil e-mail:
[email protected]
Mustapha Rachidi Department de Mathematiques, Faculte des Sciences, Universite Mohammed V, B.P. 1014, Rabat, Morocco e-mail:
[email protected]
Osamu Saeki Department of Mathematics, Faculty of Science, Hiroshima University, Higsasta-Hkoshima 739-8526, Japan e-mail:
[email protected]
1. INTRODUCTION Let aQ,au...,ar_x be arbitrary complex numbers with ar_l^0 (l 0. This completes the proof. • Now we define an oo-generalized Fibonacci sequence as follows. For a sequence {y0, y_u _y_ 2 ,...}eJ 5 we define the sequence {yx, y2, y3,...} by °°
yn = /(y n _i,y„- 2 ,JV 3 >•••) = Z ^ - i ^ - /
(P = *>2>3>•••)•
This is well defined by Lemma 2.2. The sequence {j 7 } /eZ is called an co-generalized Fibonacci sequence associated with the weight sequence {tfj^o- Note that if there exists an integer r > 1 such that at = 0 for all i>r, then the sequence {^}^0 satisfies the condition (2.1) and the above definition coincides with that of weighted r-generalized Fibonacci sequences. Thus oo-generalized sequences generalize weighted r-generalized Fibonacci sequences with r finite. Lemma 23: (1) Suppose that each at is a nonnegative real number and that there exists an S with 0 1). (2.3.1) Then there exists a unique q e~R such that q > S~l, {q~(l+V)}Zo 224
G
X>m^ f(ffl?
( 2
f ^ #~3> • ••) = 1[AUG.
ON oo-GENERALIZED FIBONACCI SEQUENCES
(2) Suppose that there exists an S with 0 S~\ {#~(,+1)},°Lo e X, and f(q~\ q~2, q~3,...) = l. Proof: (1) For JC > R~\ set •••), Si = / ( f t , go, g-i, •••), etc. Lemma 3.1: For all n > 1, we have oo
f
n
J^&^o + E X&^+y-i yz=l
Furthermore, the series on the right-hand side converges absolutely; i.e., the following series converges: 1999]
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ON oo-GENERALIZED FIBONACCI SEQUENCES
\sM + S
Y\sn-j^+J-i\p-il
/=i V;=i J Proof: Note that gt = a0. Then the equality for n = 1 together with the absolute convergence is easily checked. Now assume that, for n, n-l,n-2,...,l, the right-hand side of the equality converges absolutely and that the equality is valid. Then we have oo
i=0
oo f n—i
\
|
oo
=Z Z
Z
H)^, - V
On the other hand, we have w+l
®»+l = U{(^ 'l> -> *r) : ft, ..., O ^ ©IH-1-/}1=1
Then it follows that
*n+i =
Z
H)rV"V
(ij,..., Ir ) €0 „ + 1
This completes the proof. D Lemma 3J:
If Z^ = il#J < 1, then the series T^=0K converges absolutely and is equal to
Proof: First, note that the series l^L^-Vfz1 converges absolutely for \z\r), the above lemma shows that the sequence is asymptotically simple with dominant root q and dominant multiplicity 1 in the terminology of [2]. Remark 3.5: Note that it is easy to construct sequences which satisfy condition (2.1) and which admit a real number S with 0 < S < R satisfying (2.3.1) or (2.3.2), and (3.4.1). For example, take an arbitrary holomorphic function hx{z) defined in a neighborhood of zero. Then the sequence appearing as the coefficients of the power series expansion of the holomorphic function A (z) = hx{z) + a at z = 0 satisfies the above conditions for all a e C with sufficiently large modulus \a\. Remark 3.6: Suppose that each at is a nonnegative real number and that there exists an S with 0a,p and that q2-(a+/3 + a + b)q + (ba + aj3 + aj3) = 0. Furthermore, we see that #0 = 1, g1=a+b, g2 = (a + h) +(aa + gj3), and «+i gf (a+fi + a + b)gn -(ba + ap + afJ)gn_x for n > 2. Therefore, we have gn - Aqn + Brn (n >. 1) for some real numbers A and 5, where r is the solution of the equation r 2 - ( a + / ? + a + d)r + (ba + aj3 + a(3) = 0 with r ^ q. Since \r\< q, we see that hmn_>O0gn/qn exists and is equal to A. The value of A can be calculated by using gl and g2. After tedious but elementary computations, we see that A = (l + aa/(q-a)2 +b/31'(1-P)2)'1 = (l + Z ^ i O " 1 - N o t e t h a t t h e v a I u e ^L\K can be greater than 1. For example, for (a, fi, a, b) = (1,1/2,1,1), the sum is smaller than 1 while, for (a, /?, a, b) = (3,1,1,1), it is greater than 1. Example 4.4: Consider the sequence {at }* 0 with at = ll (i +1)!. Note that, for this sequence, we have h(x) - (ex -1) / x and e(x) - ex - 1 . Hence, the radius of convergence R is equal to oo. In 1999]
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ON oo-GENERALIZED FIBONACCI SEQUENCES
this case, we can easily check that q = (log 2)"1. Hence, we have e'(q~l)~2< 2(log 2)"1 = 2g, which implies that the condition in Lemma 3.3 is satisfied by Remark 3.6. Thus, by an easy calculation, we see that the sequence { g „ C behaves like (log2)"
