Internat. J. Math. #kh. Sci.
Vol. 6 No. 3 (1983) 545-550
545
FIBONACCI POLYNOMIALS OF ORDER K,
MULTINOMIAL EXPANSIONS AND PROBABILITY
ANDREA$ N. PHILIPPOU
Department of Mathematics
Unlveristy of Patras
Patras, Greece
COSTA$ GEORGHIOU
School of Engineering
Unlverlsty of Patras
Patras, Greece
GEORGE N.. PHILIPPOU
Department of General Studies
Higher Technical Institute
Nicosia, Cyprus
(Received October 31, 1982)
ABSTRACT.
The Fibonacci polynomials of order k are introduced and two expansions of
them are obtained, in terms of the multlnomial and binomial coefficients, respectively.
A relation between them and probability is also established.
izes results of [2]
KEY ORDS AND PHRASES.
The present work general-
[4] and [5].
Fibonacc polynom
of ode k,
eo, mano
d
binomial coeff/e/ents, probability.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
i.
IOA40.
INTRODUCTION.
In the sequel, k is a fixed integer greater than or equal to 2, x is a positive and
finite real number, and n is a nonnegative integer unless otherwise specified.
introduced the Fibonacci polynomials of order M
of their properties.
First we observe that
to be denoted by
f(k)(x)
n
[3], [4], and
(x)(n>_-(r-2)) of [i] for k=r and n>-O.
prove a theorem, which provides two expansions of
ial and binomial coefficients, respectively.
results, give another expansion of
f(k)(x),
n
and study some
are generalized polynomials, appro-
priate extensions for the Fibonacci and Pell numbers of order k
to the r-bonacci polynomials R
f(k)
(x)
n
Motivated
f(k)(x)
n
identlca/
Then we state and
(n>l) in terms of the multimom-
Hoggatt and Bicknell [i], amoung other
in terms of the
A.N. PHILIPPOU, C. GEORGHIOU and G. N. PHILIPP J
546
The latter, however, are less widely
justified k-nomial triangle.
elements of the left
known and used than the multinomial and binomial coefficients, and on this account our
As a corollary to our theorem, we derive several
expansions may be considered better.
results of [2]-[4] and [5].
f(k)(x)
n
We also obtain a relation between
(n->l) and
probability.
2.
THE FIBONACCI POLYNOMIALS OF ORDER K AND MULTINOMIAL COEFFICIENTS.
In this section, we introduce the Fibonacci polynomials of order k and derive two
expansions of them im terms of the multinomial and binomial coefficients, respectively.
The proof is along the lines of [2] and [4].
DEFINITION.
fk)(x)--O, fk)(x)=l,
Fibonacci polynomials of order k if
f(k)(x) <
I
n__
If
{f(k)
(x) }n--0
n
The sequence of polynomials
frjt(x)=O
n
7
i=l
and
if 2 < n < k
(2.1)
k
Z
i=l
xk-i f(k)
(x)
n-i
k
if n
+
i
Hoggatt and Bick [i] call Rn
for -(r-2)<n<-i
n
k-i f(k)(x)
n-i
is said to be the sequel of
(x)--f((x)
r)
(n>-(r-2))
r-bonacci polynomials.
Denoting by F
n(x)
f
(k)
n
(k)
and P
n
respectively, the Fibonacci polynomials [5], the
Fibonacci numbers of order k [3], and the Pell numbers of order k [4], it follows from
(2.1) that
f(2)
(x)
n
Fn(X)
f(k)
(!)
n
f
(k) and F (k) (2)
n
n
P
(k)
n
(2.2)
We now proceed to show the following lemma.
LEMMA.
Let
f
(k)
(x) } 0 be the sequence of Fibonacci polynomials of order k, and
n=
denote its generating function hy
Zk(S ;x)
PROOF.
for 3Nn<_k+l
gk(s;x).
Then, for
s(l)
We see from the definition that f
and xf
(k)
n
(x)
sl <x/(l+xk),
s
k)(x)=xk-I xf(k)
(x)
(X) _f(k) (x) -xkf(k)
n-i
n
n-i
k
(x)
f(k)
(x)=x f(k)...(k)
n-i
n-i Lx)-mn_l_k
for n>k+2
Therefore
(i + x’-)f
f(k__(x)
n
(i
"k)(x)
3 < n < k
n-1
I
k )f(k)(x)
n-i
+
547
EXPANSIONS AND PROBABILITY
MULTIN(4IAL
x
f(k)
(x)
n-l-k
n > k
[(i + xk)]n-2 xk-I
1
_Ix( I + xk)f(k)
n-l(X) ---x f(k)
n-l-k(X)’
+
1
+ 2
2 <n<k+l
(2.3)
n > k
+ 2.
It may be seen, by means of (2.3) and induction on n, that
f(k)
(x)
n
[(l+xk) ]n-2 xk-i
<
(s;x)
which implies the convergence of
for
n>2
Is <x/(l+xk).
(2.4)
Next, by means of (2.3), we
observe that
snf(k)(x)
n
E
(s;x)
n=0
k=l
s
+ xk)]n-2xk-I + I snf n(k) (x),
sn[(l
n-k+2
n2
+
(2.5)
7.
and
F.
n-k+2
1
(k)
Z "_n. (k)l(X)’- r. _n:Zn--..
1(x)
-(l+xk) n-k+2
Zn
x
n-k+2
snf(k)(x)
n
s__(l=xk)
s
k
[(z+
Z
n-0
snfn(k) (x)-s-
k+l
)- ix
1(’;)-Sx
sn[--ix(l+xk
n= 2
2
k=l
z s"[
]n-2xk-i
-sl
snfn<k) (x)
k+l
(Z+x)] n-2 Xk-i
(2.6)
The last two relations give
s
s(s;x) , + Z(z
+
k
x
s
k) k(S;X)
s2
so that
s
gk(s;x)
(l---)
z(z+k_s
k)
x
8
z_-+-’" 2
We will employ the above leema to establish the following expansions of
TnO.
Lt
(a)
{re)
(x) }(R)
n=0
n
.(k)
In+l(x)
nl
be the Fibonacci polynomials of order k.
Z
nl+"
xk(nl
’nk \nl’" "’+nk)
"’nk
+"
+nk)-n
ne0
f(k)(x)
n
Then
(>i).
548
A.N. PHILIPPOU, C. GEORGHIOU and G. N. PHILIPPOU
where the summation is over all noo-negative integers nl,..,nk such that
(b)
E
n/l (x)
ki
x
(-I)
nl+2n2+../knk-n;
(l+xk)
i-O
x
i-O
nl,
where, as ual, Ix] denotes the greatest interger in x.
First we show (a).
PROOF.
_Let n
Let
I,I
k)
<
.o that
lxk[’-/()+. +()kll<l
X
be non-negative integers as specified below Then, using the lemma and the
i(l<i<k)
k
multinomial theorem, and replacing n by n-
E (i-1)n i, we
get,
i=l
E
n-O
snf(k)..
.+tx
s k
[ + (s)2+. ..+() }-I
k s
{l_x
2+.
s
kn
-o’X
n.,,..
s k
}n
r ( ,no
nl+2n2
+. .+knk
nl+ +nk=n
hr.
s
n-O
n
I,...
,nk
nl+2n2+-
(nl+"’+nk Ink(nl+’’’+nk)-n’
(2.8)
nl’" ’nk
+knk=n
from which (a) follows.
We now proceed to establish (b).
Let 0 < s < x/(l +
xk),
so that
iX"S(-l+xk-s
k-
< i.
Then, using the lemma and the binomial theorem, replacing n by n-kl, and setting
B
(k)
n
(x)
.l+xk)n [n/Ik+l)] (-i) ki
i
(----
we ge [:
[1- s_( l+xk-s k)
Z s n.(k)
n/l(X)= (1- s_)
x
x
n-O
(i-
s
)
E
is_ (1+xk-s k)
n-O x
(n ) xki
k)
(l+x
(k+l)i
n>O,
(2.9)
MULTINOHIAL EXPANSIONS AND PROBABILITY
x
(i-
I=0
s
Z s
) n=O
(1-s_)
x
n=l
since
Bo(.k) (x)
i:O
[n/(k+l)]
n
(l+xk)n-(k+l)i X-(n-ki)
nkl
(_l)i
i=O
E s n B (k) (x), by (2.9)
n
n= 0
7.
i+
549
sn[B(k)
(x)
n
i from (2.9).
1
B(k)
n-l(X) ],
(2.10)
"
The last two relations show part (b) of the theorem.
We have the following obvious corollary to the theorem, by means of relation (2.2).
COROLLARY 2.1.
Let F (x),
n
f(k)
n
and
p(k)n
denote the Fibonacci polynomials, the
Fibonacci numbers of order k and the Pell numbers of order k, respectively.
(a)
Fn+ l(x)
(b) (1)
(b) (if)
n;i xn-21
7.
i=0
.(k)
rn+I
n0;
n>-0;
%
nk9
n
I
n
i+2n2+. +knk-n
(k)
I
rn+
2
n
In/(k+l)
(_l)i
?L
i=O
(n-l)/
-2n- 1
(k+l)
i=O
(c) (1)
p(k)
n=l
n
n
1+2n2+
k n
(c) (It)
.I+2
rn+ I (---)
l.l+2k
+kn
k-n
(n-l-ki 2-(k+l)i
(-l)i
(_l)i
i’O
,n
2k(rtl+....h.tk)_n
n->O;
(n;ki)2ki(l+2k)-(k+l)i
n-i [(n-l)/(k+l)]
(-i)
i=O
n>_l;
i
k(nl+...+n>
n/i(+l)
t------)
REMARK.
[n[ki)2-(k+l)i
,nl,
nk9
I
Then,
i
(n-l-ki) 2kl
i
(I+2
k)
(k+l) i
nl.
Part (a) of Corollary 2.1 was proposed by Swamy F51, who appears to be the
first to introduce the Fibonaccl polynomials.
Part (b)(1) was first shown in
while (b) and (c), respectively, were later proved by a different method in [2l and [4].
The following corollary relates the Fibonaccl polynomials of order k to probability.
A.N. PHILIPPOU, C. GEORGHIOU and G. N. PHILIPPOU
550
COROY 2 2.
Let
{f(k)(x))o
be
n
n-0
the sequence of Fibonacci polynomials of order
k, and denote by Nk the number of trials until the occurrence of the kth consecutive
success in independent trials with success probability p (0<p<l).
.(k)
pn+k()n/k Zn+
I (()l/k),
P(Nk.n+k
PROOF.
Then,
n>O.
It follows directly from Theorem 3.1 of [3] and part (a) of the present
theorem.
In partlcular, Corollary 2.2.
reduces to the following results of [2] and [4],
respectively, by means of (2,2).
Let N be as above, and set
k
p-(l+2k) -1
Then
2
P(Nk=n+k)
(1+2
Let N be as above, and set p=I/2.
k
n
...(k)
k)n+k
rn+l’
Then,
1
,.(k)
n In+l
2
P (Nk’n+k)
(2 ii)
n >0
(2 12)
n_>O"
REFERENCES
I.
HOGGATT, VoE. nd BICKNELL, MARORIE.
Generalized Fibonacci Polynomials,
Th___e
Fbonacc_ Quarter!y I__i (1973), 457-465.
2,
PIIILIPPOU, A.N. A Note On Khe Fibonacci Sequence Of Order k and Multinomial Coefficients, Th.e Fib0nacci _Quarte.rly 2.1 (1983), in press.
3.
PHILIPPOU, A.N. and MUWAFI, A.A.
Waiting for the kth Consecutive Success and the
Fibonacci Sequence of Order k, The Fibonacci Quarterly 20 (1982), 28-32.
The Pell Sequence of Order k, Multinomial
Coefficients, and Probability, Submitted for publication (1982).
PHILZPPOU, A.N. and PHILIPPOU, G.N.
5,, SWAMY M.N.S,
Problem B- 74, The Fibonacci Quarterly
3_ (1965), 236.