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 summation 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

⎛(︃ )︃ ⎞n−1 i Φ x y xi−j yi+j n,q( , ) = ⎝ j ⎠ . q i,j=0

We also find two different matrix products for

⎛(︃ )︃ ⎞n−1 i j Ψ x y i j + xi−j yi+j n,q( , ; , ) = ⎝ j ⎠ . 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]

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, ... ∈ 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 {a}q ≡ , q ∈ C∖{1}, (1) 1 − q

n ∏︁ {n}q! ≡ {k}q , {0}q! ≡ 1, q ∈ C. (2) k=1 Gauss’ q-binomial coefficients are given by (︃ )︃ n {n}q! ≡ . (3) k {k}q!{n − k}q! 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 (︃ )︃ ∑︁ n α ⊕ β n ≡ αk βn−k n ... ( q ) k , = 0, 1, 2, (4) k=0 q

Definition 3. If f(x) ∈ C[x], the polynomials with complex coefficients, the function ϵ : C[x] ↦→ 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 ∑︁ 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 (︃ )︃ (︃ )︃ ∑︁−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 ∑︁ 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) ≡ ⟨j + 1; q⟩i−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 ∏︁ (︀ )︀ 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.

Proof. Use the same technique as in Brawer & Pirovino [2], but use the q-unit matrices and the q-Vandermonde theorem.

Lemma 3.3. [7]

In,n−1,q(x)Pn,n−1,q(x) = Pn,n−1,q * (x), n ≥ 1. (30) q-Pascal matrices of two variables Ë 211

Proof. Use the q-Pascal triangle in the end.

Lemma 3.4. A q-analogue of [13, p.173].

x x y Φ x y n Wn( , )Pn,n−1,q * ( y ) = n,q( , ), ≥ 1. (31)

Proof. For n = 1, Pn,n−1,q * (x) = In and Wn(x, y) = Φn,q(x, y). Let n > 1. The matrix element i (︃ )︃ (︃ )︃ x ∑︁ i−j i+j k − 1 k−j i−j i+j i (Wn(x, y)Pn n q * ( )(i, j) = x y q = x y = Φn,q(x, y)(i, j), j ≥ 1. (32) , −1, y j − 1 j k=j q q x i For j = 0, Wn(x, y)Pn,n−1,q * ( y ; i, 0) = (xy) = Φn,q(x; i, 0).

Theorem 3.5. A q-analogue of [13, p.174]. If n ≥ 3, the extended q-Pascal matrix Φn,q(x, y) can be factorized by the W matrix, the summation matrices and by the q-unit matrices as

∏︁3 (︂ x x )︂ x Φ x y x y n,q( , ) = Wn( , ) In,k,q( y )Gn,k( y ) Gn,2,q * ( y ), (33) k=n−1 where the product is taken in decreasing order of k.

x x x Proof. Use (31). At each step the q-unit matrix maps Pn,k,q * ( y ) to Pn,k,q( y ) by (30) and to Gn,k( y ) by (29). Example 1. The case n = 4: ⎛ ⎞ ⎛ ⎞ 1 0 0 0 1 0 0 0 ⎜ 2 ⎟ ⎜ 2 ⎟ ⎜ xy y 0 0 ⎟ ⎜ xy y 0 0 ⎟ ⎜ 2 2 3 4 ⎟ = ⎜ 2 2 3 4 ⎟ ⎝ x y {2}q xy y 0 ⎠ ⎝ x y xy y 0 ⎠ 3 3 2 4 5 6 3 3 2 4 5 6 x y {3}q x y {3}q xy y x y x y xy y ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ (34) 1 0 0 0 1 0 0 0 1 0 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ x ⎟ ⎜ x ⎟ ⎜ ⎟ . ⎝ 0 y (q − 1) 1 0 ⎠ ⎝ 0 y 1 0 ⎠ ⎝ 0 0 1 0 ⎠ x 2 x 2 x x 0 0 y (q − 1) 1 0 ( y ) y 1 0 0 y q 1

Theorem 3.6. A q-analogue of [13, p.174]. The inverse of the extended q-Pascal matrix is given by n (︂ x )︂ ∏︁ (︂ (︂ x )︂ (︂ x )︂)︂ Φ x y −1 x y n,q( , ) = Fn,2,q * y Fn,k y En,k,q y Un( , ). (35) k=3

Proof. Use formulas (27), (28) and (29).

4 Higher q-Pascal matrices

In this section we treat matrices of a slightly different form.

Definition 8. A q-analogue of the matrix Qn in [12, p.55]. The matrix Rn,q(x) is defined by (︃ )︃ i x i j ≡ xi+j i j ... n Rn,q( ; , ) j , , = 0, , − 1. (36) q

A q-analogue of the matrix Ψn[x, y] in [13, p.175]. The generalized symmetric q-Pascal matrix Ψn,q(x, y) is defined by (︃ )︃ i j Ψ x y i j ≡ + xi−j yi+j i j ... n n,q( , ; , ) j , , = 0, , − 1. (37) q 212 Ë Thomas Ernst

Theorem 4.1. A q-analogue of [13, p.175]. In the following two formulas we use the q-matrix multiplication with f(m, i, j) = m2. Ψ x y xy Φ y 1 T n,q( , ) = Rn,q( ) n,q( , x ) , (38)

y Ψ x y Φ x y T n,q( , ) = n,q( , )Pn,q( x ) . (39)

Proof. By the first q-Vandermonde theorem,

n−1 (︃ )︃ (︃ )︃ (︃ )︃ ∑︁ i j 2 i j xy Φ y 1 T i j xy i+k yj−k x−j−k qk + xi−j yi+j Rn,q( ) n,q( , x ) ( , ) = k ( ) k = j . (40) k=0 q q q

n−1 (︃ )︃ (︃ )︃ (︃ )︃ y ∑︁ i j 2 i j Φ x y T i j xi−j yi+j qk + xi−j yi+j n,q( , )Pn,q( x ) ( , ) = k k = j . (41) k=0 q q q

We have found two different matrix products for the same function. Example 2. The case n = 4:

⎛ y y2 y3 ⎞ 1 x x2 x3 3 4 ⎜ { } 2 { } y { } y ⎟ ⎜ xy 2 q y 3 q x 4 q x2 ⎟ ⎜ 5 ⎟ = ⎜ x2y2 { } xy3 (︀4)︀ y4 (︀5)︀ y ⎟ ⎝ 3 q 2 q 3 q x ⎠ x3y3 { } x2y4 (︀5)︀ xy5 (︀6)︀ y6 4 q 2 q 3 q ⎛ ⎞ ⎛ y y2 y3 ⎞ 1 0 0 0 1 x x2 x3 2 ⎜ xy x2y2 ⎟ ⎜ 1 { } y { } y ⎟ ⎜ 0 0 ⎟ ⎜ 0 x2 2 q x3 3 q x4 ⎟ 2 2 3 3 4 4 y = ⎜ x y { }q x y x y ⎟ ⎜ 1 { } ⎟ ⎝ 2 0 ⎠ ⎝ 0 0 x4 3 q x5 ⎠ 3 3 4 4 5 5 6 6 x y { }q x y { }q x y x y 1 3 3 0 0 0 x6 ⎛ ⎞ ⎛ y y2 y3 ⎞ 1 0 0 0 1 x x2 x3 2 ⎜ xy y2 ⎟ ⎜ { } y { } y ⎟ ⎜ 0 0 ⎟ ⎜ 0 1 2 q x 3 q x2 ⎟ 2 2 3 4 y . (42) ⎜ x y { }q xy y ⎟ ⎜ ⎟ ⎝ 2 0 ⎠ ⎝ 0 0 1 {3}q x ⎠ 3 3 2 4 5 6 x y {3}q x y {3}q xy y 0 0 1

We note that we needed a q-matrix multiplication for the two last formulas and thus no inverse was available.

References

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