FIBONACCI POLYNOMIALS OF ORDER K, MULTINOMIAL EXPANSIONS AND PROBABILITY

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. The present work general- izes results of

[2] [4]

and

[5].

KEY ORDS AND PHRASES. Fibonacc polynom of ^{ode k,} eo, mano ^d

binomial coeff/e/ents, probability.

1980

MATHEMATICS SUBJECT CLASSIFICATION CODE. IOA40.

i. 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.

Motivated

f(k) (x)

and study

some

introduced the Fibonacci polynomials of order

M

to be denoted by n

of their properties. First we observe that

f(k)(x)

^are generalized polynomials, appro- priate extensions for the ^Fibonacci and Pell numbers of order k

[3], [4],

and identlca/

to the r-bonacci polynomials R (x)(n>_-(r-2)) of [i] for k=r and

n>-O.

Then we state and n

prove a theorem, ^which provides two expansions of

f(k)(x)

(n>l) in terms of the multimom- n

ial and binomial coefficients, respectively.

Hoggatt

and Bicknell

[i],

amoung other results, give another expansion of

f(k)(x),

^{in terms of the}

n

546 A.N. PHILIPPOU, C. GEORGHIOU and G. N. PHILIPP J

elements of the left justified k-nomial triangle. The latter, however, are less widely known and used than the multinomial and binomial coefficients, and on this account our expansions may be considered better. As a corollary to our theorem, we derive several results of

[2]-[4]

and

[5].

We also obtain a relation between

f(k)(x)

(n->l) and

n 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. The sequence of polynomials

{f(k)

n (x)

}

n--0 ^{is said to} be the sequel of Fibonacci polynomials of order k if

fk)(x)--O, fk)(x)=l,

^and

n⁷

k-i ^f(k)(x)

_n-i ^{if 2 < n < k}

f(k)(x)

_{n__}

<

^i=l

I

^i=lk

^{Z xk-i f(k)}

^{n-i(x) if n k}

⁺

^{i (2.1)}

If

frjt (x)=O

^for -(r-2)<n<-i Hoggatt and Bick [i] call R

(x)--f ((x)

^r) (n>-(r-2))

n n

r-bonacci polynomials.

(k) (k)

Denoting by F

n(x)

^fn ^{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(k) ^(k)

^{(k) (k)}

f(2)

n _(x)

Fn(X)

n ^{(!) f}n ^{and F}n ^{(2) P}n ^(2.2)

We now proceed to show the following lemma.

LEMMA.

Let f(k) (x)

}

n=0 be the sequence of Fibonacci polynomials of order k, and denote ^its generating function

hy gk(s;x).

^{Then, for}

sl <x/(l+xk),

s(l)

s

Zk(S

^;x)

k-I

xf(k) ^{_f(k) kf(k)}

PROOF. We see from the definition that f

k)(x)=x

(X)

(x) (x)

n n-i -x

n-i for 3Nn<_k+l and xf(k)

(x)

f(k)

_(x)=x^k

f(k)...(k) ^(x)

for n>k+2 Therefore

n n-i n-i

Lx)-mn_l_k

MULTIN(4IAL EXPANSIONS AND PROBABILITY 547

(i

+

x’-)f

"k)(x)

3 < n ^< k

+

1

f(k__(x)

^n-1

n ⁽ⁱ

+ k )f(k)(x)

^I

^f(k) (x)

^{n >} k

+

2

n-i x n-l-k

[(i ⁺

^x

^k)]n-2

^{xk-I 2} <n<k+l

_Ix( ^{I + xk)f(k)} n-l(X) ---x

¹

^{f(k) n-l-k(X)’}

^{n > k}

⁺

^2.

(2.3)

It

may be seen, by means of (2.3) and induction on n, that

f(k)

n

(x)

^<

[(l+x ^{k) ]n-2}

^{xk-i n>2}

which implies the convergence of

(s;x)

^for

Is <x/(l+xk).

observe that

(s;x) ^E snf(k)(x)

n=0 n

(2.4) Next, by means of (2.3), we

k=l

s

+

^7.

sn[(l ⁺

^x

^{k)]n-2xk-I +}

n2

I

snf

^(k)

n-k+2 ⁿ

(x),

(2.5)

and

F.

snf(k)(x) -(l+x ^k)

n-k+2 ⁿ

_n.

(k)

1 _n:(k)

Z " Zn ^l(X)’-

x

^r. Zn--.. ^1(x)

n-k+2 n-k+2

snf ^(k)

s__(l=x ^{k) Z}

^(x)-s-

n-0 n

n=

2

sn[--ix(l+xk ^]n-2xk-i -sl

^k+l

^snf

^n<k)(x)

s k ik+l 2 k=l

[(z+ ^)- x 1(’;)-Sx ^{z s"[ (Z+x)]}

^{n-2 k-iX}

^(2.6)

The last two relations give

s k

s(s;x) , + Z(z ⁺

^x

k) ^s

2

s

k(S;X)

so that

s

(l---)

8

gk(s;x)

z(z+k_s k) z_-+-’"

²

x

We will employ the above leema to establish the following expansions of

f(k)(x) (>i).

n

TnO.

Lt

{re)

(x)

}(R)

n n=0 be the Fibonacci polynomials of order k. Then

nl+" "’+nk) ^{xk(nl +" +nk)-n}

^ne0

(a)

^.(k)

Z

In+ ^l(x) _{nl ’nk} \nl’" "’nk

548 A.N.

PHILIPPOU,

C. GEORGHIOU and G. N. PHILIPPOU

where the summation is over all noo-negative integers

nl,..,n

k such that

nl+2n2+../knk-n;

(b)

n/l ^{(x) E (-I)}

^x

ki

(l+xk)

i-O

x i-O

nl, where, as ual,

Ix]

denotes the greatest interger in x.

PROOF. First we show (a). Let

I,I

^<

k)

.o that

lxk[’-/()+.

X

+()kll<l

_Let n

i(l<i<k)

be non-negative integers as specified below Then,using the lemma and the k

multinomial theorem, and replacing n by n-

E

(i-1)n_i, we get, i=l

E snf(k)..

{l_x^{k s}

+ s)2+.

^{s k}

}-I

.+tx [ ( ..+()

n-O

s

2+.

^{s k}

^}n

kn

r ( ^,no

-o’X

^n.,,..

nl+ +nk=n

nl+2n2

+.

.+kn k

hr.

_s

(nl+"’+nk Ink(nl+’’’+nk)-n’

n-O n

I,... ,nk nl’" ’nk

nl+2n2 ^{+- +knk=n}

(2.8)

from which (a) follows.

S(-l+xk

^k-

We now proceed to establish (b). Let 0 < s <

x/(l + xk),

so that

iX"

^{-s < i.}

Then, using the lemma and the binomial theorem, replacing n by n-kl, and setting

.l+xk)n [n/Ik+l)]

ⁱ

(n )

^(l+x

^k)

n (-i)

ki ^xki

^(k+l)i

B(k)

(x)

(---- ^{n>O, (2.9)}

we ge[:

Z

s

^n.(k) n/l(X)=

^(1-

s_)

^[1-

s_( l+xk-s k)

n-O

^{x x}

s

is_

(i-

) ^E (1+xk-s k)

n-O ^x

MULTINOHIAL EXPANSIONS AND PROBABILITY 549

x I=0 i:O s sn

[n/(k+l)]

(i-

)

^Z

(_l)i nkl (l+xk)n-(k+l)i

X^-(n-ki)

n=O i=O

(1-s_) E

sn B(k)

(x), by (2.9) x n=0 ⁿ

since

Bo(.k)

i+ 7.

sn[B(k)

(x) ¹

B(k)

n

" ^{n-l(X) ],}

n=l

(2.10)

(x) i from (2.9). 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), f(k)

^and

p(k)n

^{denote the Fibonacci} polynomials, ^the

n n

Fibonacci numbers of order k and the Pell numbers of order k, respectively. Then, (a)

Fn+ ^l(x)

^7.

n;i ^xn-21

i=0 (b)(1) .(k)

rn+

^I

_%

nI

nk9

n

i+2n2 ^+. +knk-n

In/(k+l)

(k) 2ⁿ

(b)(if)

rn+

^I

n0;

n>-0;

?L ^(_l)i [n[ki)2-(k+l)i

i=O

(c)(1)

p(k)

n=l

/ (k+l)

-2n-

¹

(n-l) ^(-l)i _(n-l-ki ^2-(k+l)i

^n>_l;

i=O ⁱ

k(nl+...+n> 2k(rtl+....h.tk)_n

n->O;

,nl,

^,n

nI

nk9

n

1+2n2+

^+kn

^k-n

n/i(+l)

.I+2^{k n} (c)(It)

rn+

I

(---)

i’O

(_l)i (n;ki)2ki(l+2k)-(k+l)i

l.l+2k

^n-i

[(n-l)/(k+l)]

(n-l-ki)

t---)

^(-i)i i

2kl

(I+2

k)

^(k+l)i

i=O nl.

REMARK. 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.

550 A.N.

PHILIPPOU,

C. GEORGHIOU and G. N. PHILIPPOU

COROY 2 2. Let

{f(k)(x))o

n n-0 be the sequence of Fibonacci polynomials of order

k,

and denote by N

k the number of trials until the occurrence of the kth consecutive success in independent trials with success probability p (0<p<l). Then,

P(Nk.n+k pn+k()n/k Zn+

^.(k)I

(()l/k),

n>O.

PROOF. 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, bymeans of (2,2).

Let

N

k be as above, and set

p-(l+2k)

^-1 Then 2n

...(k)

>0 (2 ii)

P(Nk=n+k)

(1+2

k)n+k ^rn+l’

ⁿ

Let N

k be as

above,

and set

p=I/2.

Then,

1 ,.(k) n_>O" (2 12)

P

(Nk’n+k)

2

n In+l

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 Co-

efficients, 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.

PHILZPPOU, A.N.

and

PHILIPPOU,

G.N. ^{The Pell} Sequence of Order k, Multinomial Coefficients, and Probability, Submitted for publication (1982).

5,, SWAMY ^M.N.S,

^{Problem B-}

^74,

^The Fibonacci Quarterly

3_

(1965), 236.

Mathematical Problems in Engineering

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