Download Full text

LAMBERT SERIES AND THE SUMMATION OF RECIPROCALS
IN CERTAIN F I B O N A C C I - L U C A S - T Y P E SEQUENCES
R.
Andre-Jeannin
Ecole d'Ingenieurs de Sfax, Tunisie
(Submitted July 1988)
1.
Introduction
Consider the sequence of real numbers defined by the recurrence relation
(1.1)
Wn = pWn.x
+
Wn-2,
where p is a strictly positive real number.
est us here are:
Special cases of (Wn) which inter-
an - 3 n
Un =
_
(Fibonacci-type sequence),
(1.3)
Vn = an + 3 n
(Lucas-type sequence),
where
+ /p2 + 4
a = p
(1.2)
and
2
(1.4)
/p2 + 4
2
p
*="—
It is clear that
(1.5)
a3 = -1, a > 1, -~1 < 3 < 0.
On the other hand, the Lambert series is defined by
(1.6)
L(x) = ±
— ^ - ,
|x| < 1.
It has been known for a long time (see Horadam [1] for complete references)
that
7T- = (a - B M M B 2 ) -
E
n= 1
L(^)],
U
2n
*= 1 l 2 n - l
The purpose of this paper is to establish the following result.
Theorem
(1.7)
1:
E
= 2(a - 3)[M3 2 ) - 2L(B4) + 2L(38)] + 3;
jr-^j
2.
Lemma
(2.D
1:
E
, ^"2w+1
^2= 0 1 -
1990]
P r e l i m i n a r y Lemma
X
Z n + i
= L
^) -
L
(^2);
223
LAMBERT SERIES AND THE SUMMATION OF RECIPROCALS IN CERTAIN FIBONACCI-LUCAS-TYPE SEQUENCES
(2.2)
X
£
n = L{x) n= l 1 + Xn
2L(x2);
(2.3)
£ ~ ^ — T T T = L ^ ~ 3 L (^ 2 ) + 2L(x^).
n=0 1 + X 2 n+ 1
(2.1) is obviously true, whereas (2.2) follows from the identity
2x In
n
n
1-x
1 +x
I -
x2n'
and (2.3) follows from
n=0 1 + X2n+l
«= i 1 + x n
« - l 1 + OJ2?2 "
3.
Proof of Theorem 1
Lemma 2:
(3-D
2 i ; 4 r =1 + E7r4-;
(3-2)
2 E - ^ - i
n=ia
Proof:
Un
a
«= l a Fn
up
n=
+
1 vn.Un + 1
E - 7^ -
n=
l ^z n + l
F i r s t , we have
aUn + 1 + y„ = _ L - [ a ( a » + l - (-1)» + 1 - L - ) + a" - (-1)" ^
p.W+1 ,
1
= — — (an+2
+ an) = -
1 \
(a + -
= an + l .
Thus,
1
a»*/w
+" a^Un
UnUn + }
+1
By adding
ig this term by term, we
we find (3.1) since U\ follows the
the same pattern if we observe
c
that
1.
The proof
of (3.2)
uVn + i + Vn = (a - G)aw + 1 .
Thus,
1
n
u Vn
1
+
n+1
a
Vn+l
_ a - g
" VnVn + l'
Now, adding t h i s term by term, we find
( 3 . 2 ) s i n c e ^i = p .
Lemma 3:
(3.3)
£ - J — = (a - £ ) [ L ( 3 2 ) - 2L(3 4 ) + 2Z,(3 8 )];
n = 1 a 6/n
Proof;
1
^
_ ^ _= ^
n
a - B^ia [/n
1
2
„~ia *
-"(-1)*
„~1 1 ~ 3 ^
Using ( 1 . 6 ) with
hand, we have:
224
x = 3
4
=
+
^
32n
„~i 1 -
(-l)n32w
n^0 1 + 3 ^
+ 2
and ( 2 . 3 ) with x = 6 2 , we find
(3.3).
On t h e o t h e r
[Aug.
LAMBERT SERIES AND THE SUMMATION OF RECIPROCALS IN CERTAIN FIBONACCI-LUCAS-TYPE SEQUENCES
y
-
=y
=v
-
=v
-
i*+n+2
+v —
ihn+Z'
„t-i anVn n^Y a2n + {-l)n
„^i 1 + ( - l ) n 3 2 n
n=i 1 + 3 ^
n=o 1
4
Using (2.2) with x = 3 and (2.1) with x = 3 2 , we find (3.4). This concludes
the proof of Lemma 3. Now the proof of the theorem follows immediately from
Lemmas 2 and 3.
4.
4.1
Special Cases
Fibonacci-Lucas Sequences
Let p = 1 in (1.1) to obtain
a
+ Wn-2>
Wn = K-l
1 - /5
l + i/5
»
=
£/n = F n is the Fibonacci sequence and Vn = Ln is the Lucas sequence.
(1.7) and (1.8) take the following form:
1
£
n= 1 ^n^
2/5 L
n+ \
1
0
2
- 2L
3 -
rc= 1 -^n^n + 1
4.2
.3 - /5\
/5
2L
11 - 3 / 5 \
+ 2L
47 - 21/5
/47 - 21/5
2
1 - /5
1 - /5
2/5
v5
Pell a n d P e l l - L u c a s
Equations
Sequences
Let p = 2 i n ( 1 . 1 ) t o o b t a i n
Wn = 2Wn_l + Wn_2,
a = 1 + /2,
3 = 1-
£/n = P n is the Pell sequence, Vn = § n is
(1.7) and (1.8) take the form:
/2..
the I. Pell-Lucas
sequence.
Equations
= 4/2[L(3 - 2/2) - 2L(17 - 12/2) + 2L(577 - 408/2)] + 1 - / 2 ;
£
n = 1 Pyi^n + l
~
1
1
rr~
53 7 ^ 3n + l= -7=[i(3
- 2/2) - 2L(577 - 408/2) ] +
il
5.
1-/2
4/2
Generalization
The following theorem generalizes the above result. It is given without
proof, since the methods required exactly parallel those of Section 3. We
assume that K is an odd integer.
Theorem
2:
1
^
n=iUknUk{n
+ l)
2(a - 3 ) ,
[L($2k)
Uk
1
»E
^i^„^(n+i)
- 2 L ( 3 4 k ) + 2 L ( 3 8 k ) ] + ~z\
,
Uk
8k
-[£(B2k) - 2L(f3
)] +
N
(a - B ) V ^
'
" -
"
Co - B)J/fc^
For the proof, the reader will need the following lemmas.
Lemma
2':
2n£= i a
2 y
1990]
1
kn
[/fen
akt/k
1
1
'« Vk
T/
fen
+ ^rz=E1
,
^n^(n+l)'
N
"
n=l
1
Vkn Vk(n
+ l)
225
LAMBERT SERIES AND THE SUMMATION OF RECIPROCALS IN CERTAIN FIBONACCI-LUCAS-TYPE SEQUENCES
Lemma
3':
Z
nn
n^i~aknkUkn
t
n=
= (^ ~ 3)[£(32/c) - 2L(^)
-J^T— = L{^k)
i aKnV-,
kn
+ 2L(B8^)];
- 2L(38fe).
Reference
1.
A. F. Horadam. "Elliptic Functions and Lambert Series in the Summation of
Reciprocals in Certain Recurrence-Generated Sequences." Fibonacci
Quarterly 26.2 (1988):98-114.
Applications of Fibonacci Numbers
Volume 3
New Publication
Proceedings of 'The Third International Conference
on Fibonacci Numbers and Their Applications*
Pisa, Italy, July 25-29, 1988.
edited by G.E. Bergum, A.N. Fhilippoii and A.F. Horadam
This volume contains a selection of papers presented at the Third International Conference on
Fibonacci Numbers and Their Applications. The topics covered include number patterns,
linear recurrences and the application of the Fibonacci Numbers to probability, statistics, differential equations, cryptography, computer science and elementary number theory. Many of
the papers included contain suggestions for other avenues of research.
For those interested in applications of number theory, statistics and probability, and
numerical analysis in science and engineering.
1989, 392 pp. ISBN 0—7923—0523—X
Hardbound Dfl. 195.00/ £65.00/US $99.00
A.M.S. members are eligible for a 25% discount on this volume providing they order directly
from the publisher. However, the bill must be prepaid by credit card, registered money order
or check. A letter must also be enclosed saying "I am a member of the American
Mathematical Society and am ordering the book for personal use."
14
KLUWER
ACADEMIC
PUBLISHERS
P.O. Box 322, 3300 AH Dordrecht, The Netherlands
P.O. Box 358, Accord Station, Hingham, MA 02018-0358, U.S.A.
226
[Aug.