A q-analogue for bi s nomial coefficients and generalized Fibonacci sequences

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Combinatorics/Number theory

A q-analogue for bi ^s nomial coefficients and generalized Fibonacci sequences

Un q-analogue pour les coefficients bi

^s

nomiaux et les suites de Fibonacci généralisées

Hacène Belbachir

^a

, Athmane Benmezai

^b,c

aUSTHB, Faculty of Mathematics, RECITS Laboratory, DG-RSDT, BP 32, El Alia 16111, Bab Ezzouar, Algiers, Algeria

bUniversity of Dely Brahim, Fac. of Eco. & Manag. Sc., RECITS Laboratory, DG-RSDT, Rue Ahmed Ouaked, Dely Brahim, Algiers, Algeria cUniversity of Oran, Faculty of Sciences, BP 1524, ELM Naouer, 31000, Oran, Algeria

a r t i c l e i n f o a b s t r a c t

Article history:

Received 22 October 2013

Accepted after revision 21 January 2014 Available online 4 February 2014 Presented by the Editorial Board

A new q-analogue of bi^snomial coefficients is proposed according to the generalized q-Fibonacci sequence suggested by Cigler’s approach.

©2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

r é s u m é

Nous proposons une nouvelle variante de q-analogue pour les coefficients binomiaux généralisés appelés coefficients bi^snomiaux. Elle est basée sur les suites q-Fibonacci proposées par Cigler.

©2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

1. Introduction

Fors^{1, bis}nomial coefficients denoted by_n

k

sare considered as extensions of binomial coefficients_n

k

and are obtained by the multinomial expansion:

1

+

^x

+

^x2

+ · · · +

^xs

n

=

k⁰

n k

s

x^k

,

(1)

where_n

k

1=_n

k

is the classical binomial coefficient, and fork>nsork<0,_n

k

s=^0.

They satisfy the following recursion:

n k

s

=

n

j=⁰

n j

j k

−

^j

s−¹

.

(2)

For an appropriate introduction of this numbers, see Andrews and Baxter[1], Belbachir etal.[6], Bollinger[7]and Smith and Hogatt[9].

E-mail addresses:[email protected],[email protected](H. Belbachir),[email protected](A. Benmezai).

1631-073X/$ – see front matter ©2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.crma.2014.01.009

Using the classical binomial and multinomial coefficients (see[2,6]), we obtain the following formulae:

n k

s

=

j₁+^j2+···+^js=^k

n j₁

j₁ j₂

· · ·

j_s−1

j_s

,

(3)

n k

s

=

j₁+^2j2+···+^sjs=^k

n j1

,

j2

, . . . ,

js

,

(4)

n k

s

=

k/(

s+1) j=⁰

( −

¹

)

^j

n

j

n

−

¹

+

^k

−

^j

(

s

+

¹

)

n

−

¹

.

(5)

The following recursion is also satisfied:

n k

s

=

s

j=0

n

−

¹ k

−

^j

s

.

The coefficients q _k

2

n k

are considered as a q-analogue of binomial coefficients. They appear as the coefficients of the binomial expansion ofx, ysuch that yx=^qxy:

(

x

+

^y

)

ⁿ

=

n

k=0

n k

x^n−ky^k

,

or as the coefficients of the following product:

(

1

+

^z

)(

1

+

^qz

)

1

+

^q2z

· · ·

1

+

^qn−1z

=

n

k=0

n k

q^k2z^k

.

(6)

2. Aq-analogue of bi^snomial coefficients

Fora=^eis+12π , the multinomial expansion gives:

_s

j=0

x^j

n

=

_s

r=1

x

−

^ar

n

=

s

r=¹

_n

j=0

n j

x^j

−

^ar

n−j

=

ns

k=0

j1+j2+···+js=k

n j₁

n j₂

· · ·

n

j_s

(−

¹

)

^ns−ka^{sr=1r(n−jr)}

x^k

.

By identification, we obtain a new expression of the bi^snomial coefficients:

Theorem 2.1.The following identity holds:

n k

s

=

j1+j2+···+js=k

n j₁

n j₂

· · ·

n

j_s

(−

¹

)

^ka^{−sr=1r jr}

.

(7)

Now, we are able to propose a definition of theq-analogue of the bi^snomial coefficients.

Definition 2.1.We define aq-analogue of bi^snomial coefficients by:

n k

s

=

j1+j2+···+js=k

n j1

1

n j2

1

· · ·

ⁿ_j

s

1

(−

¹

)

^ka^{−rs=1r jr}

.

(8)

wheren j

1=^q2jn j

.

Remark 2.2.Fors=1, we obtain the classicalq-binomial coefficientsq _k

2

n k

. We use the notationn k

1to indicate the term q^k2n

k

.

According to relation (7), these coefficients seem to be natural as the classical q-analogues of bi^snomial coefficients.

Moreover, the product_n−1

j=⁰(1+^qjz+ · · · +(q^jz)^s)appears as a naturalq-analogue of multinomial expansion; we will show below thatn

k

sis the coefficient of thek-th term of the product.

Theorem 2.3.We have:

n

−1 j=⁰

1

+

^qjz

+ · · · +

q^jz

s

=

ns

k=⁰ n k

s

z^k

.

(9)

Proof. Knowing thata=^{eis2+π1, we have}(1+^z+^z2+ · · · +^zs)=s

j=¹(z−^aj). Hence, n

−¹

j=0

1

+

q^jz

+ · · · +

q^jz

s

=

s

r=¹

z

−

a^r

qz

−

a^r

· · ·

qⁿ⁻¹z

−

a^r

=

s

r=¹

_n

k=0

n j

q^2jz^j

−

^ar

n−j

=

ns

k=0

j1+···+js=k

n j₁

· · ·

ⁿ_j

s

q^rs=1

_jr

2

(−

1

)

^ns−ka^{sr=1r(n−jr)}

z^k

=

ns

k=⁰ n k

s

z^k

.

The last line is deduced from the previous one, by:a^{r=1s rn}=(−¹)^sn. 2

Remark 2.4.Our definition for theq-analogue of bi^snomial coefficients and the definition suggested by Andrews and Baxter are two different approaches (see for example the cases=2). Each one has its advantages.

Theq-binomial coefficientsn k

1satisfy two recursions:

n k

1

=

^{qk n}

−

¹ k

1

+

^{qk−1 n}

−

¹ k

−

1

1

,

(10)

n k

1

=

ⁿ

−

¹ k

1

+

qⁿ⁻¹ n

−

¹ k

−

¹

1

.

(11)

Using the two relations:

n

−¹ j=0

1

+

q^jz

+ · · · +

q^jz

s

=

1

+

z

+ · · · +

z^s

ⁿ

⁻²

j=0

1

+

q^j+1z

+ · · · +

q^j+1z

s

,

n

−¹ j=0

1

+

^qjz

+ · · · +

q^jz

s

=

1

+

^qn−1z

+ · · · +

qⁿ⁻¹z

s

ⁿ

⁻²

j=0

1

+

^qjz

+ · · · +

q^jz

s

,

we establish the following theorem, which generalizes the recursions(10)and(11)to theq-bi^snomial coefficients.

Theorem 2.5.The q-analogue of bi^snomial coefficients satisfy the two following recursions:

n k

s

=

s

j=⁰

q^k−j n

−

¹ k

−

j

s

,

(12)

n k

s

=

s

j=⁰

q^(n−1)j n

−

¹ k

−

j

s

.

(13)

These two recursions areq-analogue of the generalized Pascal formula satisfied by the bi^snomial coefficients:

n k

s

=

s

j=0

n

−

¹ k

−

j

s

.

3. Generalized Fibonacci sequences andq-analogue

Belbachir and Bencherif[3]consider the sequences the sequences(Φn^(s))n−^sdefined by:

Φ

₀^(s)

, Φ

₋^(s₁⁾

, . . . , Φ

₋^(s_s⁾

= (

1

,

0

,

0

, . . . ,

0

), Φ

_n^(s)

= Φ

_n^(s₋⁾₁

+ Φ

_n^(s₋⁾₂

+ · · · + Φ

_n^(s₋⁾_s−1

(

n

¹

),

as the generalized Fibonacci sequences, and in[6], Belbachir etal.establish the following:

Φ

_n^(s)

=

sm

−^r

k=0

n

−

k k

s

,

wheremis given by the extended Euclidean algorithm for division: 0^rsandm(s+¹)−^r=^{n. As the}q-analogue of the generalized Fibonacci sequences(Φ_n^(s))n−^s, we introduce the following.

Definition 3.1.Fors1, we define theq-analogue of the generalized Fibonacci polynomials by:

F⁽_n^s₊⁾₁

(

z

) =

sm

−r

k=⁰

q^k n

−

^k k

s

z^k

,

where 0^rsandm(s+¹)−^r=^n.

Theorem 3.1.The generalized q-Fibonacci polynomials satisfy the two following recursions:

F⁽_n^s₊⁾₁

(

z

) =

s

j=⁰

q^n−jz

j

F_n^(s₋⁾_j

z

q^j

,

(14)

F⁽_n^s₊⁾₁

z

q

=

s

j=0

z^jF_n^(s₋⁾_j

(

z

),

(15)

with the initials:

F⁽₁^s)

(

z

),

F⁽₀^s)

(

z

),

F⁽₋^s)₁

(

z

), . . . ,

F⁽₋^s)_s+1

(

z

)

= (

1

,

0

,

0

, . . . ,

0

).

Proof. Forn<0, the termF⁽_n^s₊⁾₁(z)is computed over an empty summation index which gives F⁽_n^s₊⁾₁(z)=0; otherwise, we have:

s

j=⁰

q^n−jz

j

F⁽_n^s₋⁾_j

z

q^j

=

s

j=⁰

q^n−jz

j k0

q^k n

−

¹

−

^j

−

^k k

s

z q^j

k

=

s

j=0 k0

q^{(n−j−k)j+k} n

−

¹

−

^j

−

^k k

s

z^k+j

=

s

j=0 k0

q^{(n−k)j+k−j} n

−

¹

−

^k k

−

^j

s

z^k

=

k0

q^k

_s

j=⁰

q^(n−k−1)j n

−

¹

−

^k k

−

^j

p

z^k

.

According to(13), we obtain:

s

j=⁰

(

q^n−jz

)

^jF⁽_n^s₋⁾_j

(

^z q^j

) =

k⁰

q^k n

−

k k

s

z^k

=

^F_n^(s₊⁾₁

(

z

).

Also, we have:

s

j=⁰

z^jF⁽_n^s₋⁾_j

(

z

) =

s

j=⁰ z^j

k0

q^k n

−

¹

−

^j

−

^k k

s

z^k

=

s

j=⁰ k0

q^k n

−

¹

−

^j

−

^k k

s

z^k+j

=

k⁰

_s

j=⁰

q^k−j n

−

¹

−

^k k

−

j

s

z^k

.

According to(12), we obtain:

s

j=0

z^jF⁽_n^s₋⁾_j

(

z

) =

k⁰ n

−

^k

k

s

z^k

=

F⁽_n^s₊⁾₁

(

^z q

). 2

Fors=1, we recover theq-Fibonacci polynomials suggested by J. Cigler[8], see also[4,5]

F⁽_n¹₊⁾₁

(

z

) =

Fn+¹

(

z

) =

k0

q^k n

−

^k k

1

z^k

,

which satisfy the two following recursions:

F_n+1

(

z

) =

^Fn

(

z

) +

^qn−1zFn−¹

z q

,

(16)

Fn+¹

z q

=

^Fn

(

z

) +

^zFn−¹

(

z

).

(17)

The equalities (16) and (17) are respectively a specialization (s=¹) of the recursions (14) and (15) satisfied by the q-analogue of the generalized Fibonacci polynomials.

Acknowledgement

The authors would like to thank the anonymous referee for careful reading and suggestions that improved the clarity of this manuscript.

References

[1]G.E. Andrews, J. Baxter, Lattice gas generalization of the hard hexagon model IIIq-trinomials coefficients, J. Stat. Phys. 47 (1987) 297–330.

[2]H. Belbachir, Determining the mode for convolution power of discrete uniform distribution, Probab. Eng. Inf. Sci. 25 (4) (2011) 469–475.

[3]H. Belbachir, F. Bencherif, Linear recurent sequences and powers of a square matrix, Integers 6 (2006) A12.

[4]H. Belbachir, A. Benmezai, Expansion of Fibonacci and Lucas polynomials: An answer to Prodinger’s question, J. Integer Seq. 15 (2012), Article 12.7.6., 5 pp.

[5]H. Belbachir, A. Benmezai, An alternative approach to Cigler’sq-Lucas polynomials, J. Appl. Math. Comput. 226 (2014) 691–698.

[6]H. Belbachir, S. Bouroubi, A. Khelladi, Connection between ordinary multinomials, Fibonacci numbers, Bell polynomials and discrete uniform distribution, Ann. Math. Inform. 35 (2008) 21–30.

[7]R.C. Bollinger, A note on PascalT-triangles, Multinomial coefficients and Pascal pyramids, Fibonacci Q. 24 (1986) 140–144.

[8]J. Cigler, A new class ofq-Fibonacci polynomials, Electron. J. Comb. 10 (2003), Article R19.

[9]C. Smith, V.E. Hogatt, Generating functions of central values of generalized Pascal triangles, Fibonacci Q. 17 (1979) 58–67.