Some Properties of the (p,q) -Fibonacci and (p,q) -Lucas Polynomials

Abstract

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called (p,q) -Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving (p,q) -Fibonacci polynomials.

(ProQuest: ... denotes non-US-ASCII text omitted.)

GwangYeon Lee 1 and Mustafa Asci 2

Recommended by Shan Zhao

1, Department of Mathematics, Hanseo University, Seosan, Chungnam 356-706, Republic of Korea
2, Department of Mathematics, Science and Arts Faculty, Pamukkale University, Denizli, Turkey

Received 23 May 2012; Revised 10 July 2012; Accepted 11 July 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Large classes of polynomials can be defined by Fibonacci-like recurrence relation and yield Fibonacci numbers [1]. Such polynomials, called the Fibonacci polynomials, were studied in 1883 by the Belgian Mathematician Eugene Charles Catalan and the German Mathematician E. Jacobsthal. The polynomials _fn (x) studied by Catalan are defined by the recurrence relation [figure omitted; refer to PDF] where _f1 (x)=1, _f2 (x)=x , and n...5;3 . Notice that _fn (1)=_Fn , the n th Fibonacci number. The Fibonacci polynomials studied by Jacobsthal were defined by [figure omitted; refer to PDF] where _J1 (x)=1=_J2 (x) and n...5;3 . The Pell polynomials _pn (x) are defined by [figure omitted; refer to PDF] where _p0 (x)=0, _p1 (x)=1 and n...5;2 . The Lucas polynomials _Ln (x) , originally studied in 1970 by Bicknell, are defined by [figure omitted; refer to PDF] where _L0 (x)=2, _L1 (x)=x , and n...5;2 .

Horadam [2] introduced the polynomial sequence {_wn (x)} defined recursively by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] in which _c0 ,_c1 ,_c2 ,_c3 are constants and d=0 or 1 . Special cases of the w(x) with given initial conditions are given in Table 1:

Table 1: Special cases of the w(x) with given initial conditions are given.

p(x )	q ( x )	w0 ( x ) = 0, w1 ( x ) = 1	w0 (x)=2 , w1 (x)=x
x	1	Fibonacci Polynomial	Lucas Polynomial
2x	1	Pell Polynomial	Pell-Lucas Polynomial
1	2 x	Jacobsthal Poly	Jacobsthal-Lucas Poly.
3x	-2	Fermat Polynomial	Fermat-Lucas Polynomial
2x	-1	Cheby. Poly. 2nd kind	Cheby. Poly. first kind

For a fixed n , Brawer and Pirovino [3] defined the n×n lower triangular Pascal as matrix _Pn =_{[pi,j ]i,j=1,2,...,n} [figure omitted; refer to PDF]

The Pascal matrices have many applications in probability, numerical analysis, surface reconstruction, and combinatorics. In [4] the relationships between the Pascal matrix and the Vandermonde, Frobenius, Stirling matrices are studied. Also in [4] other applications in stability properties of numerical methods for solving ordinary differential equations are shown. In [5-8] the binomial coefficients, fibonomial coefficients, Pascal matrix, and its generalizations are studied. The authors in [9] factorized the Pascal matrix involving the Fibonacci matrix.

Lee et al. [10] defined the n×n Fibonacci matrix as follows: [figure omitted; refer to PDF] Also in [10] factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices are studied. The inverse of this matrix is given as follows; [figure omitted; refer to PDF]

The Riordan group was introduced by Shapiro et al. in [6] as follows.

Let R=_{[rij ]i,j...5;0} be an infinite matrix with complex entries. Let _ci (t)=^{∑n...5;0∞}_rn,i^tn be the generating function of the i th column of R . We call R a Riordan matrix if _ci (t)=g(t)^[f(t)]i , where [figure omitted; refer to PDF] In this case we write R=(g(t),f(t)) and we denote by ... the set of Riordan matrices. Then the set ... is a group under matrix multiplication * , with the following properties:

(R1) (g(t),f(t))*(h(t),l(t))=(g(t)h(f(t)),l(f(t))) ,

(R2) I=(1,t) is the identity element,

(R3) the inverse of R is given by ^R-1 =(1/g(f¯(t)),f¯(t)) , where f¯(t) is the compositional inverse of f(t) , that is, f(f¯(t))=f¯(f(t))=t ,

(R4) if ^{(_a0 ,_a1 ,_a2 ,...)T} is a column vector with the generating function A(t) , then multiplying R=(g(t),f(t)) on the right by this column vector yields a column vector with the generating function B(t)=g(t)A(f(t)) .

This group has many applications. Three of them are given in [6] such as Euler's problem of the King walk, binomial and inverse identities, and Bessel-Neumann expansion.

Riordan arrays are also useful for solving the combinatorial sums by the help of generating functions. For example, in [11], Cheon, Kim, and Shapiro have many results including a generalized Lucas polynomial sequences from Riordan array and combinatorial interpretations for a pair of generalized Lucas polynomial sequences.

2. The (p,q) -Fibonacci and (p,q) -Lucas Polynomials with Some Properties

In [12], the authors introduced the h(x) -Fibonacci polynomials, where h(x) is a polynomial with real coefficients. The h(x) -Fibonacci polynomials ^{{_Fh,n (x)}n=0∞} are defined by the recurrence relation [figure omitted; refer to PDF] with initial conditions _Fh,0 (x)=0 , _Fh,1 (x)=1 .

In this paper, we introduce a generalization of the h(x) -Fibonacci polynomials.

Definition 2.1.

Let p(x) and q(x) be polynomials with real coefficients. The (p,q) -Fibonacci polynomials ^{{_Fp,q,n (x)}n=0∞} are defined by the recurrence relation [figure omitted; refer to PDF] with the initial conditions _Fp,q,0 (x)=0 and _Fp,q,1 (x)=1 .

For later use _Fp,q,2 (x)=p(x) , _Fp,q,3 (x)=^p2 (x)+q(x) , _Fp,q,4 (x)=^p3 (x)+2p(x)q(x) , _Fp,q,5 (x)=^p4 (x)+3^p2 (x)q(x)+^q2 (x)... .

Now, we introduce (p,q) -Lucas polynomials ^{{_Lp,q,n (x)}n=0∞} as the following definition.

Definition 2.2.

The (p,q) -Lucas polynomials ^{{_Lp,q,n (x)}n=0∞} are defined by the recurrence relation [figure omitted; refer to PDF]

Also for later use _Lp,q,0 (x)=2 , _Lp,q,1 (x)=p(x) , _Lp,q,2 (x)=^p2 (x)+2q(x) , _Lp,q,3 (x)=^p3 (x)+3p(x)q(x) , _Lp,q,4 (x)=^p4 (x)+4^p2 (x)q(x)+2^q2 (x)... .

In [12], the authors defined h(x) -Lucas polynomials as follows: [figure omitted; refer to PDF] with initial conditions _Lh,0 (x)=2 , _Lh,1 (x)=h(x) . However, we defined (p,q) -Lucas polynomials in the Definition 2.2 which is different from h(x) -Lucas polynomials. From the Definition 2.2, for p(x)=1 and q(x)=1 , we obtain the usual Lucas numbers. And, for p(x)=h(x) and q(x)=1 , we obtain the h(x) -Lucas polynomials.

For the special cases of p(x) and q(x) , we can get the polynomials given in Table 1.

The generating function _gF (t) of the sequence {_Fp,q,n (x)} is defined by [figure omitted; refer to PDF] We know that the generating function _gF (t) is a convergence formal series.

Theorem 2.3.

Let _gF (t) be the generating function of the (p,q) -Fibonacci polynomial sequence _Fp,q,n (x) . Then [figure omitted; refer to PDF]

Proof.

Let _gF (t) be the generating function of the (p,q) -Fibonacci polynomial sequence _Fp,q,n (x) , then [figure omitted; refer to PDF] By taking _gF (t) parenthesis we get [figure omitted; refer to PDF] The proof is completed.

Corollary 2.4.

Let _gL (t) be the generating function of the (p,q) -Lucas polynomial sequence _Lp,q,n (x) . Then [figure omitted; refer to PDF]

The Binet formula is also very important in Fibonacci numbers theory. Now we can get the Binet formula of (p,q) -Fibonacci polynomials. Let α(x) and β(x) be the roots of the characteristic equation [figure omitted; refer to PDF] of the recurrence relation (2.2). Then [figure omitted; refer to PDF]

Note that α(x)+β(x)=p(x) and α(x)β(x)=-q(x) . Now we can give the Binet formula for the (p,q) -Fibonacci and (p,q) -Lucas polynomials.

Theorem 2.5.

For n...5;0 [figure omitted; refer to PDF]

Proof.

The theorem can be proved by mathematical induction on n .

Lemma 2.6.

For n...5;1 , [figure omitted; refer to PDF]

Proof.

From the characteristic equation of the (p,q) -Fibonacci polynomials we have [figure omitted; refer to PDF] By induction on n we get [figure omitted; refer to PDF] Thus we have [figure omitted; refer to PDF]

Theorem 2.7.

Let _Lp,q,n (x)=_Fp,q,n+1 (x)+q(x)_Fp,q,n-1 (x) . Then for n...5;3 , [figure omitted; refer to PDF]

Proof.

If n=3 then [figure omitted; refer to PDF] By induction on n we have [figure omitted; refer to PDF]

In [12], the author introduced the matrix _Qh (x) that plays the role of the Q -matrix. The Q -matrix is associated with the Fibonacci numbers and is defined as [figure omitted; refer to PDF] Actually, in [12], Nalli and Pentti defined the matrix _Qh (x) as follows [figure omitted; refer to PDF]

We now introduce the matrix _Qp,q (x) which is a generalization of the _Qh (x) .

Definition 2.8.

Let _Qp,q (x) denote the 2×2 matrix defined as [figure omitted; refer to PDF]

Theorem 2.9.

Let n...5;1 . Then [figure omitted; refer to PDF]

Proof.

We can prove the theorem by induction on n . The result holds for n=1 . Suppose that it holds for n=m (m...5;1) . Then [figure omitted; refer to PDF] which completes the proof.

Corollary 2.10.

Let m,n...5;0 . Then [figure omitted; refer to PDF]

If an integer a...0;0 divides an integer b, we denote a|"b .

Corollary 2.11.

For k...5;1 , [figure omitted; refer to PDF]

Corollary 2.12.

The roots of characteristic equation of ^Qp,qn (x) are ^αn (x) and ^βn (x) .

Corollary 2.13.

For n...5;1 [figure omitted; refer to PDF]

The following identities of which originated from Koshy (1998) [1] are a generalization of Koshy's results.

Theorem 2.14.

For k...5;1 , one has [figure omitted; refer to PDF]

Proof.

We know that [figure omitted; refer to PDF] Since _Fp,q,n (x)=(^αn (x)-^βn (x))/^p2 (x)+4q(x) , from Theorem 2.5, we have [figure omitted; refer to PDF] Set 1/^p2 (x)+4q(x)=A(x) and [figure omitted; refer to PDF] Then we have [figure omitted; refer to PDF] Thus the proof is completed.

Theorem 2.15.

For n...5;0 one has [figure omitted; refer to PDF]

Proof.

By Theorem 2.5 we have [figure omitted; refer to PDF]

Since α(x) and β(x) are the solutions of the equation ^t2 -p(x)t-q(x)=0 , [figure omitted; refer to PDF] Thus we have [figure omitted; refer to PDF] The proof is completed.

Theorem 2.16.

For n...5;0 , one has [figure omitted; refer to PDF]

Proof.

By Theorem 2.5 we have [figure omitted; refer to PDF] Since α(x) and β(x) are the solutions of the equation ^t2 -p(x)t-q(x)=0 , [figure omitted; refer to PDF] Thus we have [figure omitted; refer to PDF] The proof is completed.

Corollary 2.17.

For n...5;0 one has [figure omitted; refer to PDF]

Proof.

Since p(x)-α(x)=-q(x)/α(x) and p(x)-β(x)=-q(x)/β(x) , we have [figure omitted; refer to PDF]

Corollary 2.18.

For n...5;0 , [figure omitted; refer to PDF]

Corollary 2.19.

For n...5;m , [figure omitted; refer to PDF]

Proof.

From the Binet formula of the (p,q) -Fibonacci and Lucas polynomials we have [figure omitted; refer to PDF] Since α(x)β(x)=-q(x) then [figure omitted; refer to PDF]

Corollary 2.20.

For n...5;0 one has [figure omitted; refer to PDF]

Theorem 2.21.

For n...5;1 , one has [figure omitted; refer to PDF]

Proof.

We will prove the theorem by mathematical induction on n . Since [figure omitted; refer to PDF] the given statement is true when n=1 .

Now, we assume that it is true for an arbitrary positive integer k , that is, [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] By assumption we have [figure omitted; refer to PDF] Thus the formula works for n=k+1 . So by mathematical induction, the statement is true for every integer n...5;1 .

3. The Infinite (p,q) -Fibonacci and (p,q) -Lucas Polynomial Matrix

In this section we define a new matrix which we call (p,q) -Fibonacci polynomials matrix. The infinite (p,q) -Fibonacci polynomials matrix [figure omitted; refer to PDF] is defined as follows: [figure omitted; refer to PDF] The matrix ...(x) is an element of the set of Riordan matrices. Since the first column of ...(x) is [figure omitted; refer to PDF] then it is obvious that _g...(x) (t)=^∑n=0∞ ..._Fp,q,n (x)^tn =1/(1-p(x)t-q(x)^t2 ) . In the matrix ...(x) each entry has a rule with the upper two rows, that is, [figure omitted; refer to PDF] Then _fF (t)=t , that is, [figure omitted; refer to PDF] hence ...(x) is in ... .

Similarly we can define the (p,q) -Lucas polynomials matrix. The infinite (p,q) -Lucas polynomials matrix [figure omitted; refer to PDF] can be written as [figure omitted; refer to PDF]

In this section we give two factorizations of Pascal Matrix involving the (p,q) -Fibonacci polynomial matrix. For these factorizations we need to define two matrices. Firstly we define an infinite matrix M(x)=(_mij (x)) as follows: [figure omitted; refer to PDF]

We have the infinite matrix M(x) as follows: [figure omitted; refer to PDF]

Now we can give the first factorization of the infinite Pascal matrix via the infinite (p,q) -Fibonacci polynomial matrix and the infinite matrix M(x) defined in (3.8) by the following theorem.

Theorem 3.1.

Let M(x) be the infinite matrix as in (3.8) and ...(x) be the infinite (p,q) -Fibonacci polynomial matrix; then, [figure omitted; refer to PDF] where P is the usual Pascal matrix.

Proof.

From the definitions of the infinite Pascal matrix and the infinite (p,q) -Fibonacci polynomial matrix we have the following Riordan representations: [figure omitted; refer to PDF] Now we can find the Riordan representation of the infinite matrix [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF] From the first column of the matrix M(x) we obtain [figure omitted; refer to PDF] From the rule of the matrix M(x), _fM(x) (t)=t/(1-t) . Thus [figure omitted; refer to PDF] Finally by the Riordan representations of the matrices ...(x) and M(x) we complete the proof.

Now we define the n×n matrix R(x)=(_rij (x)) as follows: [figure omitted; refer to PDF] We have the infinite matrix R(x) as follows. [figure omitted; refer to PDF]

Now we can give second factorization of Pascal matrix via the infinite (p,q) -Fibonacci polynomial matrix by the following corollary:

Corollary 3.2.

Let R(x) be the matrix as in (3.16). Then [figure omitted; refer to PDF]

We can find the inverses of the matrices by using the Riordan representations of the matrices easily.

Corollary 3.3.

One has [figure omitted; refer to PDF]

Acknowledgment

The authors would like to thank the referees for helpful comments and pointing out some typographical errors.

References

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