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Factorizations for Q-Pascal Matrices of Two Variables

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Factorizations for Q-Pascal Matrices of Two Variables Spec. Matrices 2015; 3:207–213 Research Article Open Access Thomas Ernst Factorizations for q-Pascal matrices of two variables DOI 10.1515/spma-2015-0020 Received July 15, 2015; accepted September 14, 2015 Abstract: In this second article on q-Pascal matrices, we show how the previous factorizations by the sum- mation matrices and the so-called q-unit matrices extend in a natural way to produce q-analogues of Pascal matrices of two variables by Z. Zhang and M. Liu as follows 0 ! 1n−1 i Φ x y xi−j yi+j n,q( , ) = @ j A . q i,j=0 We also find two different matrix products for 0 ! 1n−1 i j Ψ x y i j + xi−j yi+j n,q( , ; , ) = @ j A . q i,j=0 Keywords: q-Pascal matrix; q-unit matrix; q-matrix multiplication 1 Introduction Once upon a time, Brawer and Pirovino [2, p.15 (1), p. 16(3)] found factorizations of the Pascal matrix and its inverse by the summation and difference matrices. In another article [7] we treated q-Pascal matrices and the corresponding factorizations. It turns out that an analoguous reasoning can be used to find q-analogues of the two variable factorizations by Zhang and Liu [13]. The purpose of this paper is thus to continue the q-analysis- matrix theme from our earlier papers [3]-[4] and [6]. To this aim, we define two new kinds of q-Pascal matrices, the lower triangular Φn,q matrix and the Ψn,q, both of two variables. To be able to write down addition and subtraction formulas for the most important q-special functions, i.e. the q-exponential function and the q- trigonometric functions, we need the q-additions. These addition formulas were first published in different notation by Jackson [9] and Exton [8]. In one formula of the present paper we use this q-addition. This paper is organized as follows: In section 2 we give the definitions for q-calculus and definitions and a simple theorem for the matrices. In section 3 we give the factorization and inverse of the extended q-Pascal matrix Φn,q(x, y). Finally, in section 4, we give the factorizations for the generalized symmetric q-Pascal matrix Ψn,q(x, y). 2 Definitions For a full description of all definitions, see the recent book [5]. Thomas Ernst: Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06 Uppsala, Sweden, E-mail: [email protected] © 2015 Thomas Ernst, licensee De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. 208 Ë Thomas Ernst Definition 1. The power function is defined by qa ≡ ealog(q). We always assume that 0 < q < 1. Let δ > 0 be an arbitrary small number. We will use the following branch of the logarithm: −π + δ < Im (log q) ≤ π + δ. This defines a simply connected space in the complex plane. The variables a, b, c, ... 2 C denote certain parameters. The variables i, j, k, l, m, n will denote natural numbers except for certain cases where it will be clear from the context that i will denote the imaginary unit. The q-analogues of a complex number a and the factorial function are defined by: 1 − qa fagq ≡ , q 2 Cnf1g, (1) 1 − q n Y fngq! ≡ fkgq , f0gq! ≡ 1, q 2 C. (2) k=1 Gauss’ q-binomial coefficients are given by ! n fngq! ≡ . (3) k fkgq!fn − kgq! q Definition 2. Let α and β be elements of a ring. The NWA q-addition is given by [1], [5], [6], [10], [11] : n ! X n α ⊕ β n ≡ αk βn−k n ... ( q ) k , = 0, 1, 2, (4) k=0 q Definition 3. If f(x) 2 C[x], the polynomials with complex coefficients, the function ϵ : C[x] 7! C[x] is defined by ϵf(x) ≡ f (qx). (5) We now leave q-calculus and turn our attention to the matrix definitions. In order to be able to write down certain q-matrix multiplication formulas, the following definition will be convenient. Definition 4. Let A and B be two n × n matrices, with matrix index aij and bij, respectively. Then we define n−1 X f (m,i,j) ABf ,q(i, j) ≡ aim bmj q . (6) m=0 Whenever we use a q-matrix multiplication, we specify the corresponding function f(m, i, j). Remark 1. This q-matrix multiplication will be used in formulas (38) and (39). The following matrices, which are used for intermediary calculations, have a relatively simple structure. In section 3 we will encounter similar q-dependent matrices, which enable a multitude of similar formulas. Definition 5. The matrices In,Sn(x),Wn(x, y) ,Dn(x) and Un(x, y) [13, p. 171] are defined by In ≡ diag(1, 1, ... , 1), (7) ( xi−j , if j ≤ i, Sn(x)(i, j) ≡ (8) 0, if j > i, ( xi−j yi+j , if j ≤ i, Wn(x, y; i, j) ≡ (9) 0, if j > i, Dn(x; i, i) ≡ 1, i = 0, ... , n − 1, Dn(x; i + 1, i) ≡ −x, for i = 0, ... , n − 2, (10) Dn(x; i, j) ≡ 0, if j > i or j < i − 1. q-Pascal matrices of two variables Ë 209 −2i −2i−1 Un(x, y; i, i) ≡ y , i = 0, ... , n − 1; Un(x, y; i + 1, i) ≡ −xy , (11) for i = 0, ... , n − 2; Un(x, y; i, j) ≡ 0, if j > i or j < i − 1. The matrices Sn and Dn are used in definition (19). Definition 6. The extended q-Pascal matrix Φn,q(x, y) is given by ! i Φ x y i j ≡ xi−j yi+j n,q( , ; , ) j . (12) q Theorem 2.1. A q-analogue of [13, p. 170]. x1 Φn,q(x1, y1)Φn,q(x2, y2) = Φn,q( ⊕q x2y1, y1y2), y2 ≠ 0. (13) y2 Proof. n ! ! X−1 i k Φ x y Φ x y i j xi−k yi+k xk−j yk+j n,q( 1, 1) n,q( 2, 2)( , ) = 1 1 k 2 2 j = k=0 q q (14) ! n−1 ! ! i i+j X i − j x1 i−k k−j i i+j x1 i−j (y1y2) ( ) (x2y1) = (y1y2) ( ⊕q x2y1) . j k − j y2 j y2 q k=0 q q Definition 7. The matrices Pn,q(x), Pn,k,q(x),Pk,q * (x) and Pn,k,q * (x) are defined by ! i x i j ≡ xi−j i j ... n Pn,q( ; , ) j , , = 0, , − 1, (15) q " T # In−k 0 Pn,k,q(x) ≡ , (16) 0 Pk,q(x) ! i x i j ≡ qx i−j i j ... k Pk,q * ( ; , ) j ( ) , , = 0, , − 1, (17) q " T # In−k 0 Pn,k,q * (x) ≡ . (18) 0 Pk,q * (x) The summation matrix Gn,k(x) and the difference matrix Fn,k(x) are defined by " T # In−k 0 Gn,k(x) ≡ , k = 3, ... , n, 0 Sk(x) (19) " T # In−k 0 Fn,k(x) ≡ , k = 3, ... , n, Fn,n(x) ≡ Dn(x), n > 2. 0 Dk(x) Let the two matrices Ik,q(x) and Ek,q(x) be given by i+1 Ik,q(x; i, i) ≡ 1, i = 0, ... , k − 1, Ik,q(x; i + 1, i) ≡ x(q − 1), i ≤ k − 2, Ik,q(x; i, j) ≡ 0 for other i, j. (20) i−j Ek,q(x; i, j) ≡ hj + 1; qii−j x , i ≥ j, Ek,q(x; i, j) ≡ 0 for other i, j. 210 Ë Thomas Ernst " T # In−k 0 In,k,q(x) ≡ , In,n,q(x) ≡ In . (21) 0 Ik,q(x) " T # In−k 0 En,k,q(x) ≡ , En,n,q(x) ≡ In . (22) 0 Ek,q(x) We call In,k,q(x) the q-unit matrix function. We will use a slightly q-deformed version of the D- and F-matrices: i+1 Dk,q * (x; i, i) ≡ 1, i = 0, ... , k − 1, Dk,q * (x; i + 1, i) ≡ −xq , i ≤ k − 2, (23) Dk,q * (x; i, j) ≡ 0, if j > i or j < i − 1. " T # In−k 0 Fn,k,q * (x) ≡ . (24) 0 Dk,q*(x) i j ( − +1 +j(i−j) i−j q( 2 ) x , if j ≤ i, Gk,q * (x; i, j) ≡ , 0, if j > i, (25) " T # In−k 0 Gn,k,q * (x) ≡ . 0 Gk,q * (x) The inverse of Pk,q * (x) is given by ! i j −1 i i−j − +1 x i j x q( 2 ) i j ... k (Pk,q * ( )) ( , ) = j (− ) , , = 0, , − 1. (26) q 3 Factorization of the extended q-Pascal matrix Φn,q(x, y) We start this section with a couple of lemmata to be able to make quick proofs of the factorization theorems. Lemma 3.1. Four inverse relations. −1 −1 Wn(x, y) = Un(x, y) ; Fn,k(x) = Gn,k(x) , k = 3, ... , n. (27) −1 −1 In,k,q(x) = En,k,q(x); Fn,k,q * (x) = Gn,k,q * (x). (28) The following three lemmata enable a step by step proof of (33). Lemma 3.2. [7] A q-analogue of [12, p.53 (1)]. If n ≥ 3, the q-Pascal matrix Pn,q(x) can be factorized by the summation matrices and by the q-unit matrices as 3 Y (︀ )︀ Pn,q(x) = In,k,q(x)Gn,k(x) Gn,2,q * (x), (29) k=n where the product is taken in decreasing order of k.
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