PORTUGALIAE MATHEMATICA
Vol. 53 Fasc. 2 – 1996
A NOTE ON THE SEQUENCE (Wn )n≥0 OF A.F. HORADAM
Gheorghe Udrea
1. In 1965, A.F. Horadam [2] considered the sequence (Wn )n≥0 , defined by
the following conditions:
(1.1)
Wn = Wn (a, b; p, q)
= p · Wn−1 − q · Wn−2 ,
W0 = a, W1 = b ,
where n ∈ IN, n ≥ 2, and a, b, p, q ∈ Z.
For this sequence, one has (if p2 − 4q 6= 0)
(1.2)
Wn =
A · αn − B · β n
,
α−β
where A = b − a · β, B = b − a · α, α and β being the roots of the associated
characteristic equation of the sequence (Wn )n≥0 :
λ2 − p · λ + q = 0
(1.3)
(see [5], p. 138).
The sequence (Wn )n≥0 generalizes the Fibonacci sequence (Fn )n≥0 , since
(1.4)
αn − β n
α−β
√ ¶¸
√ ¶
µ
·µ
1− 5 n
1
1+ 5 n
−
,
=√ ·
2
2
5
Fn = Wn (0, 1; 1, −1) =
n ∈ IN .
The sequence (Wn )n≥0 generalizes, also, other important sequences, for instance:
a) the fundamental Lucas sequence (Hn )n≥0 , where
(1.5)
αn − β n
=
α−β
p
p
µ
¶
¶ ¸
·µ
1
p − p2 − 4q n
p + p2 − 4q n
p
=
−
;
·
2
2
p2 − 4q
Hn = Wn (0, 1; p, q) =
Received : March 31, 1995; Revised : July 21, 1995.
144
G. UDREA
b) the Lucas sequence (Ln )n≥0 , where
√ ¶¸
√ ¶
µ
·µ
1− 5 n
1+ 5 n
1
−
Ln = Wn (2, 1; 1, −1) = √
2
2
5
√ ¶n−1 ¸
√ ¶n−1 µ
·µ
1− 5
2
1+ 5
−
;
+√ ·
2
2
5
(1.6)
c) the Pell sequence (Pn )n≥0 , where
(1.7)
√ ·µ
√ ¶¸
√ ¶
µ
αn − β n
2
1− 2 n
1+ 2 n
Pn = Wn (0, 1; 2, −1) =
;
−
=
·
α−β
4
2
2
d) the Tagiuri sequence (Tn )n≥0 , where
(1.8)
( ·µ
√ ¶¸
√ ¶
µ
1− 5 n
1
1+ 5 n
−
Tn = Wn (a, b; 1, −1) = √ · b ·
2
2
5
√ µ
√ µ
√ ¶n
√ ¶n ¸)
·
1+ 5
1− 5
1+ 5
1− 5
,
−
−a·
·
·
2
2
2
2
n ∈ IN .
Also, we have
def.
EW = −A · B = p · a · b − q · a2 − b2 .
(1.9)
2. If we put p = 2x, q = 1, a = 1, b = x or b = 2x, x ∈ C, in (1.1),
we obtain the sequences of Chebyshev polynomials of the first and second kind,
respectively:
(2.1)
Tn (x) = Wn (1, x; 2x, 1) ,
(2.2)
Un (x) = Wn (1, 2x; 2x, 1) ,
n ≥ 0, x ∈ C ,
n ≥ 0, x ∈ C .
Also, we obtain
(2.10 )
ET = x 2 − 1 ,
x∈C,
respectively
(2.20 )
EU = −1 .
Some important properties of the polynomials (2.1) and (2.2) are given by
the formulas
(2.3)
Tn (cos ϕ) = cos nϕ ,
n ∈ IN, ϕ ∈ C ,
A NOTE ON THE SEQUENCE (Wn )n≥0 OF A.F. HORADAM
145
and
(2.4)
sin nϕ
,
sin ϕ
Un−1 (cos ϕ) =
n ∈ IN∗ , ϕ ∈ C, sin ϕ 6= 0 .
From (2.1) and (2.2) it follows that the sequence of Chebyshev polynomials of
the first kind, (Tn )n≥0 , and the sequence of Chebyshev polynomials of the second
kind, (Un )n≥0 , have the same properties as the sequence (Wn )n≥0 (see [3], [4], [5],
[8], [10]).
3. Conversely, we have:
Ã
p
p± p2 −4q √
p
A) α, β =
= q·
√ ±
2
2· q
where cos ϕ =
p
√
2· q ,
sµ
¶
!
2
p
√
√ − 1 = q · (cos ϕ ± i · sin ϕ) ,
2· q
ϕ ∈ C;
αn − β n
sin nϕ
√
= ( q)n−1 ·
=
α−β
sin ϕ
µ
¶
n−1
p
.
· Un−1 (cos ϕ) = q 2 · Un−1
√
2· q
Hn = Wn (0, 1; p, q) =
=q
n−1
2
Hence
(3.1)
Hn = q
n−1
2
· Un−1
µ
¶
p
√ ,
2· q
n ∈ IN∗ .
1
· (A · αn − B · β n ) =
α−β
i
h
1
√
√
=
· A · ( q)n · (cos nϕ + i · sin nϕ) − B · ( q)n · (cos nϕ − i · sin nϕ) =
α−β
√
i
( q)n h
=
· (A − B) · cos nϕ + i · (A + B) · sin nϕ
α−β
B) Wn =
n
= q2 ·
·
·
A−B
A+B
· cos nϕ + i ·
· sin nϕ
α−β
α−β
2·b−a·p
· sin nϕ
= q · a · cos nϕ + √
2 q · sin ϕ
n
2
n
·
= q 2 · a · Tn (cos ϕ) +
n
2
·
= q · a · Tn
µ
p
√
2· q
¶
¸
¸
2·b−a·p
· Un−1 (cos ϕ)
√
2· q
µ
¸
2·b−a·p
p
+
· Un−1
√
√
2· q
2· q
¶¸
.
146
G. UDREA
So that, we obtain
·
n
(3.2) Wn = q 2 · a · Tn
µ
p
√
2· q
¶
+
µ
2·b−a·p
p
· Un−1
√
√
2· q
2· q
¶¸
,
n ∈ IN∗ .
Also, we get
µ
·
n
p
b
Wn = q 2 · √ · Un−1
√
q
2· q
(3.3)
¶
− a · Un−2
µ
p
√
2· q
¶¸
n ∈ IN ,
,
n ≥ 2, since Tm (x) = x · Um−1 (x) − Um−2 (x), (∀) x ∈ C, (∀) m ∈ IN, m ≥ 2.
On the other hand,
√
i
( q)n h
Wn =
· (A − B) · cos nϕ + i · (A + B) · sin nϕ
α−β
n
=
q2
1
2
q · 2 · i · sin ϕ
"
· p
(A −
A−B
· cos nϕ
+ (i · (A + B))2
B)2
#
q
i · (A + B)
·
sin
nϕ
·
+p
(A − B)2 + (i · (A + B))2
(A − B)2 + (i · (A + B))2
√
n
·
¸
A−B
A+B
2· E ·q2
p
√
√
·
· cos nϕ + i ·
· sin nϕ
=
p2 − 4q
2 E
2 E
√
n
i
2 E ·q2 h
=p 2
· cos φ · cos nϕ − sin φ · sin nϕ
p − 4q
√
√
n
n
µ ³
¶
2 E ·q2
φ´
2 E ·q2
· cos(nϕ + φ) = p 2
· Tn cos ϕ +
,
=p 2
n
p − 4q
p − 4q
p
A−B
√
where cos ϕ = 2√
q , ϕ ∈ C, and cos φ = 2 E , φ ∈ C (see [6], [9]).
Consequently,
√
n
2 E ·q2
(3.4)
Wn (a, b; p, q) = p 2
· cos(nϕ + φ) , n ∈ IN ,
p − 4q
where cos ϕ =
p
√
2 q,
ϕ ∈ C, and cos φ =
a·
√
p2 −4q
√
,
2 E
φ ∈ C.
Remarks I. We have:
µ
¶
µ
¶
i
1
1
= in−1 · Un−1 −
=
· Un−1
i
2i
2
µ ¶
µ ¶
i
i
= in−1 · (−1)n−1 · Un−1
= i3(n−1) · Un−1
, n ∈ IN∗ .
2
2
(3.3)
a) Fn = Wn (0, 1; 1, −1) = in ·
147
A NOTE ON THE SEQUENCE (Wn )n≥0 OF A.F. HORADAM
Hence,
(3.5)
b)
Fn+1 = i
3n
· Un
µ ¶
i
,
2
n ∈ IN .
µ
¶
i
=
Ln = Wn (2, 1; 1, −1) = 2 · i · Tn −
2
µ ¶
µ ¶
i
i
n
n
3n
= 2 · i · (−1) · Tn
= 2 · i · Tn
,
2
2
(3.2)
n
n ∈ IN .
So that, we have
(3.6)
Ln = i
3n
c) Pn = Wn (0, 1; 2, −1) = i
· 2 · Tn
n−1
µ ¶
i
,
2
· Un−1
n ∈ IN .
µ ¶
1
i
= in−1 · Un−1 (−i) =
= in−1 · (−1)n−1 · Un−1 (i) = i3(n−1) · Un−1 (i) ,
n ∈ IN∗ .
Hence,
Pn+1 = i3n · Un (i) ,
(3.7)
d)
(3.2)
·
Tn = Wn (a, b; 1, −1) = in · a · Tn
·
µ
¶
µ
n ∈ IN .
1
2i
¶
µ
+
µ
2b − a
1
· Un−1
2i
2i
¶¸
¶¸
2b − a
i
· Un−1 −
2i
2
µ ¶
·
µ ¶¸
i
i
2b − a
n
n
n−1
= i · a · (−1) · Tn
+
· (−1)
· Un−1
2
2i
2
·
¸
2b − a
n
n 1
n
n−1
n−1
= i · a · (−1) · · i · Ln +
· (−1)
·i
· Fn =
2
2i
2b − a
Ln − F n
a
· Fn = a ·
+ b · Fn = a · Fn−1 + b · Fn , n ∈ IN∗ ,
= · Ln +
2
2
2
= in · a · Tn −
i
2
+
since Ln − Fn = 2Fn−1 , (∀) n ∈ IN∗ .
Remarks II. We observe that the identity
(∗)
Wn · Wn+r+s − Wn+r · Wn+s = EW · q n · Hr · Hs
(see [4]), is a consequence of the formula (3.4).
148
G. UDREA
Indeed, we have:
i) Wn · Wn+r+s
√
n
2 E
=p 2
· q 2 · cos(nϕ + φ)
p − 4q
√
³
´
n+r+s
2 E
· q 2 · cos (n + r + s) ϕ + φ
·p 2
p − 4q
=
ii)
Wn+r · Wn+s =
³
´
2n+r+s
4E
2
·
cos(nϕ
+
φ)
·
cos
(n
+
r
+
s)
ϕ
+
φ
;
·
q
p2 − 4q
³
´
³
´
2n+r+s
4E
2
·
q
·
cos
(n+r)
ϕ
+
φ
·
cos
(n+s)
ϕ
+
φ
,
p2 −4q
n, r, s ∈ IN .
From i) and ii) we obtain
Wn · Wn+r+s − Wn+r · Wn+s =
·
³
´
2n+r+s
4E
= 2
· q 2 · cos(nϕ + φ) · cos (n + r + s) ϕ + φ
p − 4q
³
´
³
− cos (n + r) ϕ + φ · cos (n + s) ϕ + φ
=
p2
−
2n+r+s
4E
·q 2 ·
− 4q
³
"
³
´¸
´
cos (2n + r + s) · ϕ + 2φ + cos(r + s) ϕ
´
2
cos (2n + r + s) ϕ + 2φ + cos(r − s) ϕ
2
#
2n+r+s
4E
cos(r + s) ϕ − cos(r − s) ϕ
·q 2 ·
− 4q
2
2n+r+s
4E
= 2
· q 2 · (−1) · sin rϕ · sin sϕ
p − 4q
2n+r+s
4E
sin rϕ sin sϕ
= (− sin2 ϕ) · 2
·q 2 ·
·
p − 4q
sin ϕ sin ϕ
2n+r+s
4E
· q 2 · Ur−1 (cos ϕ) · Us−1 (cos ϕ)
= (cos2 ϕ − 1) · 2
p − 4q
·µ
¸
¶
µ
¶
¶
µ
2n+r+s
p 2
4E
p
p
=
· q 2 · Ur−1 √ · Us−1 √
−1 · 2
√
2 q
p − 4q
2 q
2 q
µ
µ
¶
¶
2n+r+s−2
p
p
2
=E·q
,
· Ur−1 √ · Us−1 √
2 q
2 q
=
p2
149
A NOTE ON THE SEQUENCE (Wn )n≥0 OF A.F. HORADAM
i.e.,
(α)
Wn · Wn+r+s − Wn+r · Wn+s =
=E·q
2n+r+s−2
2
· Ur−1
µ
p
√
2 q
¶
· Us−1
µ
¶
p
√ ,
2 q
n, r, s ∈ IN∗ .
On the other hand, we have
E · q n · Hr · Hs = E · q n · q
=E·q
r−1
2
2n+r+s−2
2
¶
µ
µ
s−1
p
p
√ · q 2 · Us−1 √
2 q
2 q
µ
µ
¶
¶
p
p
,
· Ur−1 √ · Us−1 √
2 q
2 q
· Ur−1
¶
i.e.,
n
(β)
E ·q ·Hr ·Hs = E ·q
2n+r+s−2
2
·Ur−1
µ
¶
¶
µ
p
p
√ ·Us−1 √ ,
2 q
2 q
n, r, s ∈ IN∗ .
From (α) and (β) we obtain the identity (∗), q.e.d..
4. It is well-known that if A =
(4.1)
³a b´
c d
∈ M2 (C), then

n−1
n−1
n
n

 λ1 − λ 2 − δ · λ1 − λ 2
· I2 ,
A
A n = λ1 − λ 2
λ1 − λ 2


λ1 6= λ2 ,
n · λn−1 · A − (n − 1) · δA · λn−2 · I2 , λ1 = λ2 = λ ,
³
´
where δA = det A = ad − bc, I2 = 10 01 , λ1 and λ2 being the roots of the
associated characteristic equation of the matrix A:
λ2 − (a + d) · λ + δA = 0 .
(4.10 )
On the other hand, we have
µ
¶
n−1
λn1 − λn2
a+d
= Wn (0, 1; a + d, δA ) = Hn = δA 2 · Un−1 √
,
λ1 − λ 2
2 δA
δA 6= 0 .
Hence,
n−1
(4.2)
·
=δ
n−1
2
µ
¶
µ
¶
¸
1
a+d
a+d
√
· A − δA2 · Un−2 √
· I2
2 δA
2 δA
µ
¶
µ
¸
·
¶ µ
¶
1
a+d
a+d
a+d
· A−
· I2 + δA2 · Tn √
· I2 ,
· Un−1 √
2
2 δA
2 δA
An = δA 2 · Un−1
if λ1 6= λ2 and δA 6= 0.
150
G. UDREA
For δA = 1 we obtain, from (4.2),
¶
µ
(4.3)
Also, we have, for δA = −1,
(4.4)
¶
µ
a+d
a+d
· A − Un−2
· I2
2
2
¶ µ
¶
¶
µ
µ
a+d
a+d
a+d
· I 2 + Tn
· A−
· I2 .
= Un−1
2
2
2
An = Un−1
n
A =i
3(n−1)
·
· Un−1
µ
a+d
i·
2
¶
· A + i · Un−2
µ
¶
¸
a+d
· i · I2 ,
2
where i2 = −1, n ∈ IN∗ .
From (4.4) we deduce, for a + d = 1,
An = Fn · A + Fn−1 · I2 ,
(4.5)
n ∈ IN∗ ,
where (Fn )n≥0 is the Fibonacci sequence.
Remarks III.
a) For A =
³
1 1
1 0
´
we have δA = −1 and a + d = 1. Hence
n
A = Fn · A + Fn−1 · I2 =
Ã
Fn+1 Fn
Fn
Fn−1
!
,
i.e.,
µ
(4.6)
1 1
1 0
¶n
=
Ã
Fn+1 Fn
Fn
Fn−1
!
,
(∀) x ∈ IN∗ ,
(see [1]).
From (4.6) we deduce
Fn+1 · Fn−1 − Fn2 = det
µ
1 1
1 0
¶n
µ
= det
µ
1 1
1 0
¶¶n
= (−1)n ,
i.e.,
Fn+1 · Fn−1 − Fn2 = (−1)n ,
(4.7)
n ∈ IN∗ .
The identity (4.7) is in fact the Catalan’s Identity for m = n and r = 1 (see
(4.1)).
b) For A =
³
3 1
−1 0
´
we have a + d = 3 and δA = 1.
A NOTE ON THE SEQUENCE (Wn )n≥0 OF A.F. HORADAM
151
Hence,
n
A = Un−1
= Un−1
= (−1)
= (−1)
µ ¶
3
2
µ µ
· A − Un−2
3
− −
2
·
n−1
¶¶
· Un−1
·
n−1
3
2
· I2
· A − Un−2
µ
µ
3
−
2
· Un−1 T2

µ ¶
¶
µ µ
3
− −
2
· A + Un−2
µ ¶¶
i
2
µ ¶
µ
¶¶
3
−
2
· I2
¶
µ
· I2
· A + Un−2 T2
¸
µ ¶¶
i
2
µ ¶
· I2

i
i
U2(n−1)−1

µ 2¶ · A +
µ ¶2 · I2 
= (−1)n−1 · 


i
i
U2−1
U2−1
2
2
µ ¶
¸
µ ¶
·
i
i
n−1 1
= (−1)
· · U2n−1
· A + U2n−3
· I2
i
2
2
µ ¶
·
µ ¶
¸
i
i
= i2n−3 · U2n−1
· A + U2n−3
· I2
2
2
 U2n−1
h
= i2n−3 · i2n−1 · F2n · A + i2n−3 · F2(n−1) · I2
= F2n · A − F2(n−1) · I2 =
µ
i
F2(n+1) F2n
−F2n
−F2(n−1)
¶
¸
,
i.e.,
Ã
(4.8)
3 1
−1 0
!n
=
Ã
F2(n+1) F2n
−F2n
−F2(n−1)
!
,
where n ∈ IN∗ and (Fn )n≥0 is the Fibonacci sequence (see [13]).
From (4.8) we get
1=
Ã
det
Ã
3 1
−1 0
!!n
= det
Ã
= det
Ã
3 1
−1 0
!n
F2(n+1) F2n
−F2n
−F2(n−1)
=
!
2
= F2n
− F2(n−1) · F2(n+1) ,
i.e.,
(4.9)
2
F2n
− F2(n−1) · F2(n+1) = 1 ,
(∀) n ∈ IN∗ .
Also, the identity (4.9) is in fact the Catalan’s Identity for m = 2n and r = 2
(see (4.11)).
152
G. UDREA
Remark IV.
a) Clearly, in (4.8) we utilized the identity
(4.10)
Un−1 (Tk (x)) =
Unk−1 (x)
,
Uk−1 (x)
(∀) n, k ∈ IN∗ , x ∈ C ;
b) The Catalan’s Identity for Fibonacci numbers is
(4.11)
2
,
Fm−r · Fm+r + (−1)m−r · Fr2 = Fm
where m, r ∈ IN, m ≥ 2, m > r (see [7]).
5. In [11] and [12] I established the following identities for the Chebyshev
polynomials (Tn )n≥0 and (Un )n≥0 , respectively:
2
2
Tn · Tn+2r − (x2 − 1) · Ur−1
= Tn+r
,
2
2
Tn+2r · Tn+4r − (x2 − 1) · Ur−1
= Tn+3r
,
(5.1)
2
2
Tn · Tn+4r − (x2 − 1) · U2r−1
= Tn+2r
,
2
2
4 · Tn · Tn+r · Tn+2r · Tn+3r + (x2 − 1)2 · Ur−1
· U2r−1
=
= (Tn · Tn+3r + Tn+r · Tn+2r )2 ,
2
2
4 · Tn+r · Tn+2r · Tn+3r · Tn+4r + (x2 − 1)2 · Ur−1
· U2r−1
=
= (Tn+2r · Tn+3r + Tn+r · Tn+4r )2 ,
2
4
4 · Tn+r · Tn+2r
· Tn+3r + (x2 − 1)2 · Ur−1
=
2
= (Tn+r · Tn+3r + Tn+2r
)2 ,
n, r ∈ IN∗ ;
2
2
Um · Um+2r + Ur−1
= Um+r
,
2
2
Um+2r · Um+4r + Ur−1
= Um+3r
,
(5.2)
2
2
Um · Um+4r + U2r−1
= Um+2r
,
2
2
4 · Um · Um+r · Um+2r · Um+3r + Ur−1
· U2r−1
=
= (Um · Um+3r + Um+r · Um+2r )2 ,
2
2
=
· U2r−1
4 · Um+r · Um+2r · Um+3r · Um+4r + Ur−1
= (Um+2r · Um+3r + Um+r · Um+4r )2 ,
2
4
4 · Um+r · Um+2r
· Um+3r + Ur−1
=
2
= (Um+r · Um+3r + Um+2r
)2 ,
m, r ∈ IN∗ .
A NOTE ON THE SEQUENCE (Wn )n≥0 OF A.F. HORADAM
153
Now, we define Sk = Tk (A) and Rk = Uk (A), A ∈ Mp (C), p, k ∈ IN, p ≥ 2
(Mp (C) is the set of square matrix of order p with complex elements).
From (5.1) and (5.2) we obtain the following theorems:
Theorem (S). For the sequence (Sn )n≥0 the product of any two distinct
elements of the set
n
Sn , Sn+2r , Sn+4r ; 4 · Sn+r · Sn+2r · Sn+3r
o
2(t−1)
2
increased by (−1)t · (A2 − I2 )t · Rh−1
· Rk−1 , where t = 1 or 2 depending on
whether 2 or 4 factors occur in the product and Rh−1 , Rk−1 are suitable elements
of the sequences (Rn )n≥0 , is a “perfect square” (in Mp (C)), (∀) n, r ∈ IN, (∀) A ∈
Mp (C), p ∈ IN, p ≥ 2.
Theorem (R). For the sequence (Rn )n≥0 the product of any two distinct
elements of the set
n
Rn , Rn+2r , Rn+4r ; 4 · Rn+r · Rn+2r · Rn+3r
o
2(t−1)
2
increased by Rh−1
· Rk−1 , where t = 1 or 2 depending on whether 2 or 4
factors occur in the product and Rh−1 , Rk−1 are suitable elements of the sequence
(Rn )n≥0 , is a “perfect square” (in Mp (C)), (∀) n, r ∈ IN, (∀) A ∈ Mp (C), p ∈ IN,
p ≥ 2.
Remarks V.
a) The analogues of the Catalan’s Identity for the sequences (Sn )n≥0 and
(Rn )n≥0 can be considered as the identities
(5.3)
2
Sn−r · Sn+r − (A2 − I2 ) · Rr−1
= Sn2 ,
respectively
2
Rn−r · Rn+r + Rr−1
= Rn2 ,
(5.4)
where m, r ∈ IN∗ , m > r.
³a b´
b) Since for every A =
∈ M2 (C) we have A2 = (a + d) · A − δA · I2 , it
c d
results
(5.5)
An+2 = (a + d) · An+1 − δA · An ,
(∀) n ∈ IN∗ .
154
G. UDREA
fn = An we obtain the sequence (W
fn )n≥0 :
If we denote W
(5.6)

fn+2 = (a + d) · W
fn+1 − δA · W
fn ,
W
f
W0 = I 2 ,
n ∈ IN ,
f1 = A .
W
For this sequence we have (see [4])
i.e.,
2
2
2
EW
e = (a + b) · I2 · A − δA · I2 − A = (a + b) · A − δA · I2 − A = O2 ,
(5.7)
Consequently,
EW
e = O2 =
µ
0 0
0 0
¶
.
Theorem (O). The set
n
fn , W
fn+2r , W
fn+4r ; 4 · W
fn+r · W
fn+2r · W
fn+3r
W
o
,
n, r ∈ IN, is a (trivial) solution, in M2 (C), of the Problem of Diophantos–Fermat
(see [4], [10]).
REFERENCES
[1] Hoggatt, V.E. (and other) – A matrix generation of Fibonacci identities for
F2nk , Fibonacci Assoc., Santa Clara, California, 1980.
[2] Horadam, A.F. – Basic properties of a certain generalized sequence of numbers,
Fibonacci Quart., 3 (1965), 161–176.
[3] Horadam, A.F. – Tschebyscheff and other functions associated with the sequence
{Wn (a, b; p, q)}, Fibonacci Quart., 7 (1969), 14–22.
[4] Horadam, A.F. – Generalization of a result of Morgado, Portugaliae Math., 44
(1987), 313–336.
[5] Horadam, A.F. and Shannon, A.G. – Generalization of identities of Catalan
and other, Portugaliae Math., 44 (1987), 137–148.
[6] Harman, C.T. – Complex Fibonacci numbers, Fibonacci Quart., 19 (1981), 82–86.
[7] Morgado, J. – Some remarks on an identity of Catalan concerning the Fibonacci
numbers, Portugaliae Math., 39 (1980), 341–348.
[8] Morgado, J. – Note on some results of A.H. Horadam and A.G. Shannon concerning a Catalan’s identity on Fibonacci numbers, Portugaliae Math., 44 (1987),
243–252.
[9] Pethe, S. and Horadam, A.F. – Generalized Gaussian Fibonacci numbers, Bull.
Austr. Math. Soc., 33 (1986), 37–48.
A NOTE ON THE SEQUENCE (Wn )n≥0 OF A.F. HORADAM
155
[10] Shannon, A.G. – Fibonacci numbers and Diophantine quadruples: generalizations of results of Morgado and Horadam, Portugaliae Math., 45 (1988), 165–169.
[11] Udrea, Gh. – A problem of Diophantos–Fermat and Chebyshev polynomials of
the second kind, Portugaliae Math., 52 (1995), 301–304.
[12] Udrea, Gh. – A problem of Diophantos–Fermat and Chebyshev polynomials of
the first kind (unpublished).
[13] Werman, N. and Zeilberger, D. – A bijective proof of Cassini’s Fibonacci
identity, Discrete Math., 58(1) (1986), 109.
Gheorghe Udrea,
Str. Unirii–Siret, Bl. 7A, Sc. 1, Ap. 17,
TÂRGU-JIU, Cod 1400, Judeţul GORJ – ROMÂNIA