Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 264842, 18 pages doi:10.1155/2012/264842 Research Article Some Properties of the p, q-Fibonacci and p, q-Lucas Polynomials GwangYeon Lee1 and Mustafa Asci2 1 2 Department of Mathematics, Hanseo University, Seosan, Chungnam 356-706, Republic of Korea Department of Mathematics, Science and Arts Faculty, Pamukkale University, Denizli, Turkey Correspondence should be addressed to GwangYeon Lee, gylee@hanseo.ac.kr Received 23 May 2012; Revised 10 July 2012; Accepted 11 July 2012 Academic Editor: Shan Zhao Copyright q 2012 G. Lee and M. Asci. 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. 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. 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 fn x xfn−1 x fn−2 x, 1.1 where f1 x 1, f2 x x, and n ≥ 3. Notice that fn 1 Fn , the nth Fibonacci number. The Fibonacci polynomials studied by Jacobsthal were defined by Jn x Jn−1 x xJn−2 x, 1.2 where J1 x 1 J2 x and n ≥ 3. The Pell polynomials pn x are defined by pn x 2xpn−1 x pn−2 x, 1.3 2 Journal of Applied Mathematics where p0 x 0, p1 x 1 and n ≥ 2. The Lucas polynomials Ln x, originally studied in 1970 by Bicknell, are defined by Ln x xLn−1 x Ln−2 x, 1.4 where L0 x 2, L1 x x, and n ≥ 2. Horadam 2 introduced the polynomial sequence {wn x} defined recursively by wn x pxwn−1 x qxwn−2 x, n ≥ 2, 1.5 where w0 x c0 , w1 x c1 xd , pxc2 xd , qx c3 xd 1.6 in which c0 , c1 , c2 , c3 are constants and d 0 or 1. Special cases of the wx with given initial conditions are given in Table 1. For a fixed n, Brawer and Pirovino 3 defined the n × n lower triangular Pascal as matrix Pn pi,j i,j1,2,...,n Pi,j ⎧ ⎪ ⎪ ⎨ i−1 j −1 ⎪ ⎪ ⎩0 if i ≥ j, 1.7 otherwise. 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: Fn fij Fi−j1 if i − j 1 ≥ 0, 0 if i − j 1 < 0. 1.8 Also in 10 factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices are studied. The inverse of this matrix is given as follows; ⎡ Fn−1 1 ⎢−1 ⎢ ⎢−1 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ .. ⎣ . 0 0 ··· 0 ··· 0 ··· . −1 −1 1 . . .. .. .. .. . . . . · · · 0 −1 −1 0 0 1 0 −1 1 ⎤ 0 0⎥ ⎥ 0⎥ ⎥ .. ⎥ ⎥. .⎥ ⎥ ⎥ 0⎦ 1 1.9 Journal of Applied Mathematics 3 Table 1: Special cases of the wx with given initial conditions are given. px x 2x 1 3x 2x qx 1 1 2x −2 −1 w0 x 0, w1 x 1 Fibonacci Polynomial Pell Polynomial Jacobsthal Poly Fermat Polynomial Cheby. Poly. 2nd kind w0 x 2, w1 x x Lucas Polynomial Pell-Lucas Polynomial Jacobsthal-Lucas Poly. Fermat-Lucas Polynomial Cheby. Poly. first kind The Riordan group was introduced by Shapiro et al. in 6 as follows. n Let R rij i,j≥0 be an infinite matrix with complex entries. Let ci t ∞ n≥0 rn,i t be the generating function of the ith column of R. We call R a Riordan matrix if ci t gtfti , where gt 1 g1 t g2 t2 g3 t3 · · · , ft t f2 t2 f3 t3 · · · . 1.10 In this case we write R gt, ft and we denote by R the set of Riordan matrices. Then the set R is a group under matrix multiplication ∗, with the following properties: R1 gt, ft ∗ ht, lt gthft, lft, R2 I 1, t is the identity element, R3 the inverse of R is given by R−1 1/gft, ft, where ft is the compositional inverse of ft, that is, fft fft t, R4 if a0 , a1 , a2 , . . .T is a column vector with the generating function At, then multiplying R gt, ft on the right by this column vector yields a column vector with the generating function Bt gtAft. 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 hx-Fibonacci polynomials, where hx is a polynomial with real coefficients. The hx-Fibonacci polynomials {Fh,n x}∞ n0 are defined by the recurrence relation Fh,n1 x hxFh,n x Fh,n−1 x, n ≥ 1, with initial conditions Fh,0 x 0, Fh,1 x 1. In this paper, we introduce a generalization of the hx-Fibonacci polynomials. 2.1 4 Journal of Applied Mathematics Definition 2.1. Let px and qx be polynomials with real coefficients. The p, q-Fibonacci are defined by the recurrence relation polynomials {Fp,q,n x}∞ n0 Fp,q,n1 x pxFp,q,n x qxFp,q,n−1 x, n≥1 2.2 with the initial conditions Fp,q,0 x 0 and Fp,q,1 x 1. For later use Fp,q,2 x px, Fp,q,3 x p2 x qx, Fp,q,4 x p3 x 2pxqx, Fp,q,5 x p4 x 3p2 xqx q2 x · · · . as the following definition. Now, we introduce p, q-Lucas polynomials {Lp,q,n x}∞ n0 Definition 2.2. The p, q-Lucas polynomials {Lp,q,n x}∞ are defined by the recurrence n0 relation Lp,q,n x Fp,q,n1 x qxFp,q,n−1 x. 2.3 Also for later use Lp,q,0 x 2, Lp,q,1 x px, Lp,q,2 x p2 x 2qx, Lp,q,3 x p x 3pxqx, Lp,q,4 x p4 x 4p2 xqx 2q2 x · · · . In 12, the authors defined hx-Lucas polynomials as follows: 3 Lh,n1 x hxLh,n x Lh,n−1 x, n ≥ 1, 2.4 with initial conditions Lh,0 x 2, Lh,1 x hx. However, we defined p, q-Lucas polynomials in the Definition 2.2 which is different from hx-Lucas polynomials. From the Definition 2.2, for px 1 and qx 1, we obtain the usual Lucas numbers. And, for px hx and qx 1, we obtain the hx-Lucas polynomials. For the special cases of px and qx, we can get the polynomials given in Table 1. The generating function gF t of the sequence {Fp,q,n x} is defined by gF t ∞ Fp,q,n xtn . 2.5 n0 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 gF t t . 1 − pxt − qxt2 2.6 Journal of Applied Mathematics 5 Proof. Let gF t be the generating function of the p, q-Fibonacci polynomial sequence Fp,q,n x, then gF t ∞ Fp,q,n xtn n0 Fp,q,1 xt Fp,q,2 xt2 ∞ Fp,q,n xtn n3 t t2 px ∞ pxFp,q,n−1 x qxFp,q,n−2 x tn n3 t t2 px ∞ pxFp,q,n−1 xtn n3 t t2 px t ∞ qxFp,q,n−2 xtn n3 ∞ pxFp,q,n−1 xtn−1 t2 n3 t t2 px tpx ∞ 2.7 qxFp,q,n−2 xtn−2 n3 ∞ ∞ Fp,q,n xtn t2 qx Fp,q,n xtn n2 t t2 px tpx n1 ∞ ∞ Fp,q,n xtn − t t2 qx Fp,q,n xtn n1 n1 t t2 px tpx gF t − t t2 qxgF t. By taking gF t parenthesis we get gF t t . 1 − pxt − qxt2 2.8 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 gL t 2 − pxt . 1 − pxt − qxt2 2.9 The Binet formula is also very important in Fibonacci numbers theory. Now we can get the Binet formula of p, q-Fibonacci polynomials. Let αx and βx be the roots of the characteristic equation t2 − pxt − qx 0 2.10 6 Journal of Applied Mathematics of the recurrence relation 2.2. Then αx px p2 x 4qx 2 βx , px − p2 x 4qx 2 . 2.11 Note that αxβx px and αxβx −qx. Now we can give the Binet formula for the p, q-Fibonacci and p, q-Lucas polynomials. Theorem 2.5. For n ≥ 0 Fp,q,n x αn x − βn x , αx − βx 2.12 Lp,q,n x α x β x. n n Proof. The theorem can be proved by mathematical induction on n. Lemma 2.6. For n ≥ 1, tn Fp,q,n xt qxFp,q,n−1 x. 2.13 Proof. From the characteristic equation of the p, q-Fibonacci polynomials we have t2 pxt qx Fp,q,2 xt qxFp,q,1 x. 2.14 By induction on n we get tn1 tn t Fp,q,n xt qxFp,q,n−1 x t Fp,q,n xt2 qxtFp,q,n−1 x Fp,q,n x pxt qx qxtFp,q,n−1 x Fp,q,n xpx qxFp,q,n−1 x t qxFp,q,n x 2.15 Fp,q,n1 xt qxFp,q,n x. Thus we have tn Fp,q,n xt qxFp,q,n−1 x. 2.16 Journal of Applied Mathematics 7 Theorem 2.7. Let Lp,q,n x Fp,q,n1 x qxFp,q,n−1 x. Then for n ≥ 3, Lp,q,n x pxLp,q,n−1 x qxLp,q,n−2 x. 2.17 Proof. If n 3 then Lp,q,3 p3 x 3pxqx pxLp,q,2 x qxLp,q,1 x. 2.18 By induction on n we have pxLp,q,n x qxLp,q,n−1 x px Fp,q,n1 x qxFp,q,n−1 x qx Fp,q,n x qxFp,q,n−2 x pxFp,q,n1 x pxqxFp,q,n−1 x qxFp,q,n x q2 xFp,q,n−2 x pxFp,q,n1 x qxFp,q,n x qx pxFp,q,n−1 x qxFp,q,n−2 x 2.19 Fp,q,n2 x qxFp,q,n x Lp,q,n1 x. 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 1 1 Q . 1 0 2.20 Actually, in 12, Nalli and Pentti defined the matrix Qh x as follows Qh x hx 1 . 1 0 2.21 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 Qp,q x px qx . 1 0 2.22 8 Journal of Applied Mathematics Theorem 2.9. Let n ≥ 1. Then n Qp,q x Fp,q,n1 x qxFp,q,n x . Fp,q,n x qxFp,q,n−1 x 2.23 Proof. We can prove the theorem by induction on n. The result holds for n 1. Suppose that it holds for n m m ≥ 1. Then m1 m Qp,q x Qp,q x · Qp,q x Fp,q,m1 x Fp,q,m x F x p,q,m2 Fp,q,m1 x qxFp,q,m x qxFp,q,m−1 x qxFp,q,m1 x qxFp,q,m x px qx 1 0 2.24 which completes the proof. Corollary 2.10. Let m, n ≥ 0. Then Fp,q,mn1 x Fp,q,m1 xFp,q,n1 x qxFp,q,m xFp,q,n x. 2.25 If an integer a / 0 divides an integer b, we denote a | b. Corollary 2.11. For k ≥ 1, Fp,q,n x | Fp,q,kn x. 2.26 n Corollary 2.12. The roots of characteristic equation of Qp,q x are αn x and βn x. Corollary 2.13. For n ≥ 1 Fp,q,n x n−1/2 n0 n − i n−2i p xqi x. i 2.27 The following identities of which originated from Koshy 1998 1 are a generalization of Koshy’s results. Theorem 2.14. For k ≥ 1, one has ⎧ j k ⎪ ⎪ ⎪ Fp,q,nkj xp x Fp,q,j x − Fp,q,nkkj x Fp,q,k−j x −qx , ⎪ n ⎨ pk x − Lp,q,k x 1 Fp,q,kij x k k ⎪ Fp,q,nkj xp xFp,q,j x−Fp,q,nkkj xFp,q,j−k x −qx ⎪ i0 ⎪ ⎪ , ⎩ pk x − Lp,q,k x 1 j < k, otherwise. 2.28 Journal of Applied Mathematics 9 Proof. We know that αx − βx p2 x 4qx, 2.29 Lp,q,n x αn x βn x. Since Fp,q,n x αn x − βn x/ p2 x 4qx, from Theorem 2.5, we have n Fp,q,kij x i0 n αkij x − βkij x i0 p2 x 4qx 1 n n α x αki x − βj x βki x j p2 x 4qx i0 i0 1 βnkk x − 1 αnkk x − 1 αj x k − βj x k α x − 1 β x − 1 p2 x 4qx 1 p2 x 4qx 2.30 αnkkj x − αj x βnkkj x − βj x − αk x − 1 βk x − 1 . Set 1/ p2 x 4qx Ax and Cx Then we have n Fp,q,kij x k j Fp,q,k−j x −qx k Fp,q,j−k x −qx if j < k, 2.31 otherwise. Ax − αk x βk x 1 αxβx × αnkkj xβk x − αj xβk x − αnkkj x αj x − βnkkj xαk x i0 αk xβj x βn x Ax pk x − Lp,q,k x 1 1 1 1 Fp,q,j x − Fp,q,nkkj x × Fp,q,nkj pk x Ax Ax Ax αk xβj x − αj xβk x Fp,q,nkj xpk x Fp,q,j x − Fp,q,nkkj x Cx pk x − Lp,q,k x 1 . 2.32 Thus the proof is completed. 10 Journal of Applied Mathematics Theorem 2.15. For n ≥ 0 one has n n k0 k pk xqn−k xFp,q,k x Fp,q,2n x. 2.33 Proof. By Theorem 2.5 we have n n αk x − βk x n k n k p xqn−k xFp,q,k x p xqn−k x k k αx − βx k0 k0 1 αx − βx 1 αx − βx n n k n−k k k p xq x α x − β x k k0 n n k0 − k n n k0 k pxαx qn−k x k k pxβx q n−k x n n 1 pxαx qx pxβx qx . αx − βx 2.34 Since αx and βx are the solutions of the equation t2 − pxt − qx 0, α2 x pxαx qx, β2 x pxβx qx. 2.35 Thus we have 2 n 2 n n α x − β x n k n−k p xq xFp,q,k x k αx − βx k0 2.36 Fp,q,2n x. The proof is completed. Theorem 2.16. For n ≥ 0, one has n n k n−k n k n p x −qx −2qx Fp,q,2n−k x. Fp,q,k x k k k0 k0 2.37 Journal of Applied Mathematics 11 Proof. By Theorem 2.5 we have n n k0 n n−k n−k αk x − βk x n k Fp,q,k x p x −qx p x −qx k k αx − βx k0 k 1 αx − βx 1 αx − βx n n k0 k n n k0 − k n−k pxαx −qx k n n k0 n−k k α x − βk x p x −qx k k n−k k pxβx −qx n n 1 pxαx − qx pxβx − qx . αx − βx 2.38 Since αx and βx are the solutions of the equation t2 − pxt − qx 0, pxαx − qx α2 x − 2qx, pxβx − qx β2 x − 2qx. 2.39 Thus we have n n 2 α x − 2qx − β2 x − 2qx n−k Fp,q,k x pk x −qx k αx − βx n n k0 n n k0 k k −2qx Fp,q,2n−k x. 2.40 The proof is completed. Corollary 2.17. For n ≥ 0 one has n n n−k p x−1k Lp,q,k x Lp,q,n x. k k0 Proof. Since px − αx −qx/αx and px − βx −qx/βx, we have n n k0 k p n−k n n n−k p x−1k αk x βk x x−1 Lp,q,k x k k0 k n n n−k p x−1k −αxk k k0 2.41 12 Journal of Applied Mathematics n k n n−k p x−1k −βx k k0 n n px − αx px − βx qx n qx n − − αx βx n αn x βn x −qx n αxβx Lp,q,n x. 2.42 Corollary 2.18. For n ≥ 0, Fp,q,2n x Fp,q,n xLp,q,n x. 2.43 m Fp,q,nm x − −qx Fp,q,n−m x Fp,q,m xLp,q,n x. 2.44 Corollary 2.19. For n ≥ m, Proof. From the Binet formula of the p, q-Fibonacci and Lucas polynomials we have m Fp,q,nm x − −qx Fp,q,n−m x m αn−m x − βn−m x αnm x − βnm x − −qx αx − βx αx − βx m αnm x − βnm x − −qx αn−m x − βn−m x . αx − βx Since αxβx −qx then m Fp,q,nm x − −qx Fp,q,n−m x m n−m αnm x − βnm x − αxβx α x − βn−m x αx − βx αnm x − βnm x − αn xβm x − αm xβn x αx − βx 2.45 Journal of Applied Mathematics m α x − βm x − αn x βn x αx − βx 13 αm x − βm x n α x βn x αx − βx Fp,q,m xLp,q,n x. 2.46 Corollary 2.20. For n ≥ 0 one has n n k0 k pk xqn−k xLp,q,k x Lp,q,2n x. 2.47 Theorem 2.21. For n ≥ 1, one has 2 Fp,q,n−1 xFp,q,n1 x − Fp,q,n x −1n qn−1 x. 2.48 Proof. We will prove the theorem by mathematical induction on n. Since 2 Fp,q,0 xFp,q,1 x − Fp,q,1 x 0px − 12 0 − 1 −11 q0 x , 2.49 the given statement is true when n 1. Now, we assume that it is true for an arbitrary positive integer k, that is, 2 Fp,q,k−1 xFp,q,k1 x − Fp,q,k x −1k qk−1 x. 2.50 Then 2 Fp,q,k xFp,q,k2 x − Fp,q,k1 x 1 Fp,q,k1 x − qxFp,q,k−1 x px 2 × pxFp,q,k1 x qxFp,q,k x − Fp,q,k1 x 1 2 qxFp,q,k1 xFp,q,k x pxFp,q,k1 x px −q2 xFp,q,k xFp,q,k−1 x − pxqxFp,q,k−1 xFp,q,k1 x 2 − Fp,q,k1 x 14 Journal of Applied Mathematics qx q2 x 2 Fp,q,k1 xFp,q,k xFp,q,k1 Fp,q,k xFp,q,k−1 x x− px px 2 − qxFp,q,k−1 xFp,q,k1 x − Fp,q,k1 x qx q2 x Fp,q,k1 xFp,q,k x − Fp,q,k xFp,q,k−1 x px px − qxFp,q,k−1 xFp,q,k1 x. 2.51 By assumption we have 2 Fp,q,k xFp,q,k2 x − Fp,q,k1 x qx Fp,q,k1 xFp,q,k x px − q2 x 2 Fp,q,k xFp,q,k−1 x − qx −1k qk x Fp,q,k x px qx q2 x Fp,q,k1 xFp,q,k x − Fp,q,k xFp,q,k−1 x px px 2 − qxFp,q,k x − −1k qk x qx Fp,q,k1 xFp,q,k x − qxFp,q,k x px qx Fp,q,k−1 x Fp,q,k x −1k1 qk x × px qx Fp,q,k1 xFp,q,k x px − qxFp,q,k x 1 Fp,q,k1 x −1k1 qk x px −1k1 qk x. 2.52 Thus the formula works for n k 1. So by mathematical induction, the statement is true for every integer n ≥ 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 Fx Fp,q,i,j x 3.1 Journal of Applied Mathematics 15 is defined as follows: ⎡ 1 0 0 0 ⎤ ⎢ . ⎥ ⎢ px 1 0 0 .. ⎥ ⎢ ⎥ ⎢ .. ⎥ Fx ⎢ ⎥ 2 ⎢ p x qx px 1 0 . ⎥ ⎢ 3 ⎥ ⎣p x 2pxqx p2 x qx px 1 ⎦ ··· ··· gFx t, fFx t . 3.2 The matrix Fx is an element of the set of Riordan matrices. Since the first column of Fx is 1, px, p2 x qx, p3 x 2pxqx, p4 x 3p2 xqx q2 x, . . . T , 3.3 n 2 then it is obvious that gFx t ∞ n0 Fp,q,n xt 1/1 − pxt − qxt . In the matrix Fx each entry has a rule with the upper two rows, that is, Fp,q,n1,j x pxFp,q,n,j x qxFp,q,n−1,j x. 3.4 Fx gFx t, fFx t 1 , t ; 1 − pxt − qxt2 3.5 Then fF t t, that is, hence Fx is in R. Similarly we can define the p, q-Lucas polynomials matrix. The infinite p, q-Lucas polynomials matrix Lx Lp,q,i,j x 3.6 Lx gLx t, fLx t 2 − pxt , t . 1 − pxt − qxt2 3.7 can be written as In this section we give two factorizations of Pascal Matrix involving the p, qFibonacci polynomial matrix. For these factorizations we need to define two matrices. Firstly we define an infinite matrix Mx mij x as follows: mij x i−1 i−2 i−3 − px − qx . j −1 j −1 j −1 3.8 16 Journal of Applied Mathematics We have the infinite matrix Mx as follows: ⎡ ⎤ 0 ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ Mx ⎢1 − px − qx 2 − px 1 0 ..⎥. ⎢ ⎥ ⎣1 − px − qx 3 − 2px − qx 3 − px 1 ⎦ ··· 1 1 − px 0 1 0 0 3.9 Now we can give the first factorization of the infinite Pascal matrix via the infinite p, q-Fibonacci polynomial matrix and the infinite matrix Mx defined in 3.8 by the following theorem. Theorem 3.1. Let Mx be the infinite matrix as in 3.8 and Fx be the infinite p, q-Fibonacci polynomial matrix; then, P x Fx ∗ Mx, 3.10 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: P t 1 , , 1−t 1−t Fx 1 , t . 1 − pxt − qxt2 3.11 Now we can find the Riordan representation of the infinite matrix Mx gMx t, fMx t 3.12 as follows: ⎡ ⎤ 0 ⎢ 0 ⎥ ⎢ ⎥ ⎢ ..⎥ ⎥ Mx ⎢ ⎢1 − px − qx 2 − px 1 0 .⎥. ⎢ ⎥ ⎣1 − px − qx 3 − 2px − qx 3 − px 1 ⎦ ··· 1 1 − px 0 1 0 0 3.13 From the first column of the matrix Mx we obtain gMx t 1 − pxt − qxt2 . 1−t 3.14 Journal of Applied Mathematics 17 From the rule of the matrix Mx, fMx t t/1 − t. Thus Mx 1 − pxt − qxt2 t , 1−t 1−t 3.15 . Finally by the Riordan representations of the matrices Fx and Mx we complete the proof. Now we define the n × n matrix Rx rij x as follows: rij x i−1 i−1 i−1 − px − qx . j −1 j j 1 3.16 We have the infinite matrix Rx as follows. ⎤ 0 ⎢ 0 ⎥ ⎥ ⎢ ⎢ ..⎥ ⎥ Rx ⎢ ⎢ 1 − 2px − qx 2 − px 1 0 .⎥. ⎥ ⎢ ⎣1 − 3px − 3qx 3 − 3px − qx 3 − px 1 ⎦ ··· ⎡ 1 1 − px 0 1 0 0 3.17 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 Rx be the matrix as in 3.16. Then P Rx ∗ Fx. 3.18 We can find the inverses of the matrices by using the Riordan representations of the matrices easily. Corollary 3.3. One has F−1 x 1 − pxt − qxt2 , t −1 M x t 1t , 1 2 − px t − 1 − px − qx t2 1 t 1 − pxt − qxt2 −1 ,t . L 2 − pxt 3.19 Acknowledgment The authors would like to thank the referees for helpful comments and pointing out some typographical errors. 18 Journal of Applied Mathematics References 1 T. Koshy, Fibonacci and Lucas Numbers with Applications, Toronto, New York, NY, USA, 2001. 2 A. F. 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