BINOMIAL IDENTITIES INVOLVING THE GENERALIZED FIBONACCI TYPE POLYNOMIALS EMRAH KILIC AND NURETTIN IRMAK Abstract. We present some binomial identities for sums of the bivariate Fibonacci polynomials and for weighted sums of the usual Fibonacci polynomials with indices in arithmetic progression.
1. Introduction The Fibonacci polynomials Fn (x) and Lucas polynomials Ln (x) are satis…ed the recursion un+1 (x) = xun (x) + un 1 (x) with F0 (x) = 0, F1 (x) = 1; and, L0 (x) = 2, L1 (x) = x; for n > 0: The bivariate Fibonacci fFn (x; y)g and Lucas fLn (x; y)g polynomial sequences are satis…ed the recursion Un+1 (x; y) = xUn (x; y) + yUn
1
(x; y)
with F0 (x; y) = 0; F1 (x; y) = 1 and L0 (x; y) = 2; L1 (x; y) = x; for n > 0: The sequences fFn (x; y)g and fLn (x; y)g can be expressed as n
n
Fn (x; y) =
and Ln (x; y) =
n
+
n
;
where and are roots of 2 x y = 0: When x = y = 1; Ln (1; 1) = ln (nth Lucas number) and Fn (1; 1) = fn (nth Fibonacci number). General background material on Fibonacci type polynomials can be found in [2, 4]. In [3], the authors obtained various results for these polynomials sequences. Thus we recall that for any positive integer r; we have Fr(n+1) (x; y) = Lr (x; y) Frn (x; y) r
z r Fr(n
1)
(x; y)
(1.1)
r
where Lr (x; y) = r + r , z r = ( ) = ( y) . In this paper we use Fn and Ln instead of Fn (x; y) and Ln (x; y) ; respectively. 1 P Let H (t; x; y) = Fr(n+1) tn : Clearly, n=0
H (t; x; y) =
1 X
Fr(n+1) tn =
n=0
1
Fr : Lr t + z r t2
1991 Mathematics Subject Classi…cation. 11B37, 11B39. Key words and phrases. Fibonacci Polynomials, Binomial Identities. 1
(1.2)
2
EM RAH KILIC AND NURETTIN IRM AK
and further 1 Lr t + z r t2
1
1 X
=
(Lr
m=0 1 X m X
=
m=0 n=0 1 X 2m X
=
m=0 n=m
m
z r t) tm m n ( z r ) Lm r n m n
1 bn=2c X X n
=
n=0 m=0
By (1.2-1.3), clearly Fr(n+1) = Fr
bn=2c
X
n
m=0
m m
m m m
Lnr
L2m r Lnr
2m rm
y
n
2m
n m+n
t
n m n
( zr )
t
m
( z r ) tn :
m(r+1)
( 1)
(1.3)
:
(1.4)
The following sums can be found in [10]: k Y
X
a1 +a2 +
Fal +1 (x) =
+ak =n l=1
bn=2c
X
n+k
1
m
m
m=0
n + k 1 2m n x k 1
2m
:
In [11], the author generalized the sums above as for k; m > 0 and n > 0; X
a1 +a2 +
mn
fm(a1 +1) : : : fm(ak+1 +1) = ( i)
+ak+1 =n
k+1 fm (k) U 2k k! n+k
im lm : 2
(k) Un
(x) denotes the kth derivation of the Chebyshev polynomials of second where kind Un (x) with respect to x: Also in [9], the authors gave a generalization of (??) as k Y
X
a1 +a2 +
Fal +1 (x; y) =
+ak =n l=1
bn=2c
X
m=0
n+k
1 m
m
n + k 1 2m n x k 1
2m m
y :
In similar meaning, some authors also derived various identities including Fibonacci and Lucas numbers (see [5, 8, 12]). In this paper, we consider both the Fibonacci polynomials and the bivariate generalized Fibonacci polynomials with indices in arithmetic progress. We generalize the results of [9]. 2. Sums of Bivariate Generalized Fibonacci Polynomials Theorem 1. For k; n; r > 0,
a1 +
= Frk
X
k Y
Fr(al +1)
+ak =n l=1
bn=2c
X
m=0
n+k
1 m
where al > 0; for l = 1; 2; : : : ; k:
m
n + k 1 2m (n Lr k 1
2m) rm
y
m(r+1)
( 1)
BINOM IAL IDENTITIES
3
Proof. Denote the l1 th derivation of H (t; x; y) according to the bivariate Lucas l1 +l2 H(t;x;y) polynomials Lr ; and, the l2 th derivation of H (t; x; y) according to ( z) by @@Ll1 @( : z)l2 By (1.2) and induction, we write @ 2l+1 H (t; x; y) @Ll+1 r @
l zr )
(
= =
@ 2l+1 @Ll+1 r @
Fr Lr t + z r t2
l zr )
1 ( 3l+1 Fr (2l + 1)!t 2l+2
Lr t + z r t2 )
(1
= Fr = Fr
1 X
n=0 1 X
(l+1;l)
Fr(n+1) tn (l+1;l)
Fr(n+2l+2) tn+2l+1
n=l
= Fr
1 X
(l+1;l)
Fr(j+3l+2) tj+3l+1 :
j=0
And @ 2l H (t; x; y) @Llr @
l zr )
(
= =
@ 2l l zr )
1 ( Fr (2l + 1)!t3l
@Llr @
Fr Lr t + z r t2
2l+1
Lr t + z r t2 ) 1 X (l;l) = Fr Fr(n+1) tn (1
= Fr
n=0 1 X
(l;l)
Fr(n+2l+1) tn+2l
n=l
= Fr
1 X
(l;l)
Fr(j+3l+1) tj+3l :
j=0
Let 1 X
n=0
k Y
X
tn
a1 +a2 +
Fr(al +1) =
+ak =n l=1
1 X
Fr(n+1) tn
n=0
!k
= I:
(2.1)
For k = 2l + 2; we obtain I
= =
Fr2l+1 @ 2l+1 H (t; x; y) 1+3l (l+1) l (2l + 1)!t @Lr @ ( zr ) 1 (l+1;l) Fr2l+1 P Fr(n+3l+2) tn : (2l+1)!
(2.2)
n=0
From (2.1) and (2.2), we get 1 X
X
n=0 a1 +a2 +
k Y
+ak =n l=1
1 Fr2l+1 X (l+1;l) Fr(al +1) t = F tn : (2l + 1)! n=0 r(n+3l+2) n
(2.3)
4
EM RAH KILIC AND NURETTIN IRM AK
By (1.4), we write (l+1;l)
Fr(n+3l+2) = Fr
= Fr
@
2l+1
(l+1) @Lr @
(
b(n+2l)=2c
X
= Fr
X
m=0
n + 3l + 1 m
m
2m)
Ln+3l+1 r
2m
1
m ( zr ) A
m l
( zr )
bn=2c
X (n + 2l + 1 m)!Lr(n 2m) ( z r )m m! (n 2m)! m=0
bn=2c
X (n + k
m=0
So
@
b(n+3l+1)=2c
(n + 3l + 1 m)! L(n+2l (m l)! (n + 2l 2m)! r
m=l
= Fr
l zr )
0
(n 2m)
m
m)!Lr ( zr ) : m! (n 2m)!
(l+1;l)
Fr2l+1 Fr(n+3l+2) (2l + 1)! =
bn=2c Frk X
k!
= Frk
(n + k
m=0
bn=2c
X
(n 2m)
n+k
1
m
n + k 1 2m (n Lr k 1
m
m=0
m
( zr ) m)!Lr m! (n 2m)!
2m) rm
z
m
( 1) : (2.4)
For k = 2l + 1; from (2.3-2.4), we have the conclusion. For k = 2l + 1, the proof is similarly obtained. When r = x = y = 1 in Theorem 1; we obtain the result of [9]. 3. Weighted Sums for the Fibonacci polynomials We derive some identities involving the terms of sequence fFrn (x)g : Theorem 2. For k; n > 0; X
a1 +a2 +
= Frk
k Y
(al + 1) Fr(al +1) (x)
+ak =n l=1
minfbn=2c;k 1g b(n 2m)=2c
X
m=0
n
2m j
X j=0
j
n
n
2m + 2k 1 n 2m j
2m + 2k 2 2k 1
j
where al > 0; for l = 1; 2; : : : ; k: P1 Proof. Let G (t; x) = n=0 (n + 1) Fr(n+1) tn : Thus G (t; x) =
Fr 1
(1
k 1 m
n 2m 2j
(Lr (x))
:
r
( 1) t2 r
2:
Lr (x) t + ( 1) t2 )
(3.1)
BINOM IAL IDENTITIES
5
By induction on (3.1), we obtain r+1
Fr 1 + ( 1) t2 (l + 1)!tl @ l G (t; x) = : l+2 r @Llr (1 Lr (x) t + ( 1) t2 )
(3.2)
Since r+1 2
1 + ( 1)
k X1
k 1
=
t
k
1 m
m=0
m(r+1) 2m
( 1)
t
and from (3.1-3.2), we get 1 X
a1 +a2 +
n=0
k Y
X
tn
r+1 2
Frk 1 + ( 1)
=
2k
r
Frk
r+1 2
1
1 + ( 1) (2k
(2k (n
k 1
t
1)!
1)! n=0
1 X
(2k 2)
(n + 2k
1) Fr(n+2k
1)
(x) tn
n=0
1 minfbn=2c;k X X
Frk 1
=
k
t
Lr (x) t + ( 1) t2 )
(1 =
(al + 1) Fal +1 (x)
+ak =n l=1
1g
m(r+1) k 1 m
( 1)
m=0
(2k 2) 2m+2k 1)
2m + 2k
1) Fr(n
(x) tn :
(3.3)
From (3.3) and (2.4), we derive X
a1 +a2 +
=
k Y
(al + 1) Fr(al +1) (x)
+ak =n l=1
Frk 1 (2k 1)!
minfbn=2c;k 1g
X
m(r+1)
( 1)
(2k 2) 2m+2k 1)
m(r+1)
( 1)
= Fr
minfbn=2c;k 1g b m=0
n
2m j
2m + 2k
1 (n
2m + 2k (2k 1)!
2m c 2 X n
1)
1)
n
2m + 2k 1 n 2m j
2m + 2k 2 2k 1
j
k 1 m
n 2m 2j
(Lr (x))
(2k 2) 2m+2k 1)
Fr(n
n
j=0
j
k m
m=0
X
(n
(x)
minfbn=2c;k 1g
X
1 m
m=0
Fr(n = Frk
k
(x)
6
EM RAH KILIC AND NURETTIN IRM AK
Theorem 3. For k; n; r > 0, k Y
X
a1 +a2 +
=
Frk
(al + 2) Fr(al +1) (x)
+ak =n l=1
minfk;ng b(n j)=2c b(n j 2i)=2c
X j=0
n
j
X
1 X
m
n
j
2i + 2k 2k 1
(n + 2) Fr(n+1) (x) tn =
n=0
Since (2
k
Lr (x) t) =
j n
2
Proof. By taking y = 1 in (1.2), we write ! 1 X @ n+2 @ Fr(n+1) (x) t = @t @t n=0
Then
n
i+j
( 1)
m=0
i=0
2i m
X
j
m
2 (i k) 1 k 2 2i m n 2i m
(Lr (x))
Fr t2 r Lr (x) t + ( 1) t2
1
Fr (2
Lr (x) t)
j k j
m(r+1)
( 1)
:
:
2:
r
Lr (x) t + ( 1) t2 )
(1
k X k j ( Lr (x) t) 2k j j=0
j
and
1
X 1 = 2 1+t i=0
i
t2
;
using the same method given in the previous theorem gives us 1 X
tn a1 +
n=0
=
=
X
(2k
(n + 2k
j=0
(2k 2) j 2i+2k 1)
Frk (2 (1
(2k 2)
1) Fr(n+2k
1)
k
Lr (x) t) r
2k
Lr (x) t + ( 1) t2 ) (x) tn
n=0
j)=2c 1 minfk;ng X X b(nX
1)! n=0
Fr(n
(al + 2) Fr(al +1) (x) =
+ak =n l=1 1 k X
Frk (2 Lr (x) t) (1 + t2 ) (2k 1)! Frk
k Y
i+j
( 1)
(n
j
2 (i + k)
1)
i=0
(x) tn :
Considering the proof of previous theorem, we have the conclusion. Identities for the bivariate Fibonacci polynomials Fn (x; y) generate identities for specially multiplicative functions at prime powers fpn 1 with x = fp and y = fp21 , and vice versa, see [1, 6, 7]. References [1] P. Haukkanen, A note on a curious polynomial identity. Nieuw Arch.Wisk. (4.2) 13 (1995), 181-186. [2] A. F. Horadam, A synthesis of certain polynomial sequences. Applications of Fibonacci numbers, Vol. 6, 215-229, Kluwer, Dordrecht, 1996. [3] E. K¬l¬ç and P. Stanica, Factorizations and representations of binary polynomial recurrences with matrix methods, accepted in Rocky Mount. J. Math. (to appear) [4] T. Koshy, Fibonacci and Lucas numbers with applications. Wiley-Interscience, New York, 2001. [5] D. Liu, C. Li and C. Yang, Some identities involving square of Fibonacci numbers and Lucas numbers, Chin. Quart. J. Math. 19(1) (2004), 67-68.
BINOM IAL IDENTITIES
7
[6] P. J. McCarthy, R. Sivaramakrishnan, Generalized Fibonacci sequences via arithmetical functions. Fibonacci Quart. 28.4 (1990), 363-370. [7] J. McLaughlin, N. J. Wyshinski, Further combinatorial identities deriving from the nth power of a 2 2 matrix. Discrete Appl. Math. 154(8) (2006), 1301-1308. [8] Y. Sun, Numerical triangle and several classical sequences, Fibonacci Quart. (to appear). [9] Q. Yan, Y. Sun and T. Wang, Identities involving the Fibonacci polynomials, Ars Combin. 93 (2009), 87-96. [10] Y. Yuan and W. Zhang, Some identities involving the Fibonacci polynomials, Fibonacci Quart. 40 (2002), 314-318. [11] W. Zhang, Some identities involving the Fibonacci and Lucas numbers, Fibonacci Quart. 42 (2004), 149-154. [12] F. Zhao and T. Wang, Some identities involving the generalized Fibonacci and Lucas number, Ars Combin. 72 (2004), 311-318. TOBB University of Economics and Technology, Mathematics Department 06560 Ankara Turkey E-mail address :
[email protected] Nigde University, Mathematics Department 51241 Nigde Turkey E-mail address :
[email protected]
