Factorizations for q-Pascal matrices of two variables

© 2015 Thomas Ernst, licensee De Gruyter Open.

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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

Φn,q(x, y) =

⎛

⎝ (︃

i j )︃

q

x^i−jy^i+j

⎞

⎠

n−1

i,j=0

.

We also find two different matrix products for

Ψn,q(x, y; i, j) =

⎛

⎝ (︃i+ j

j )︃

q

x^i−jy^i+j

⎞

⎠

n−1

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,qmatrix 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:

thomas@math.uu.se

Definition 1. The power function is defined by q ^≡e . 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:

{a}q≡ 1 − q^a

1 − q , q^∈C^∖{1^}, (1)

{n}q!^≡

n

∏︁

k=1

{k}q, ^{0^}q!^≡1, q^∈C. (2) Gauss’ q-binomial coefficients are given by

(︃n k )︃

q

≡ {n}q!

{k}q!^{n− k^}q!. (3)

Definition 2. Let α and β be elements of a ring. The NWA q-addition is given by [1], [5], [6], [10], [11] :

(α^⊕qβ)^n≡

n

∑︁

k=0

(︃n k )︃

q

α^kβ^n−k, n = 0, 1, 2, . . . (4)

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 aijand bij, respectively. Then we define

AB_f,q(i, j)^≡

n−1

∑︁

m=0

a_imb_mjq^f(m,i,j). (6)

Whenever we use a q-matrix multiplication, we specify the corresponding function f (m, i, j).

Remark1. 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)

Sⁿ(x)(i, j)^≡

{︃x^i−j, if j ≤ i,

0, if j > i, (8)

Wn(x, y; i, j)^≡

{︃x^i−jy^i+j, if j ≤ i,

0, if j > i, (9)

Dn(x; i, i)^≡1, i = 0, . . . , n − 1, Dn(x; i + 1, i)^≡−x, for i = 0, . . . , n − 2,

Dn(x; i, j)^≡0, if j > i or j < i − 1. (10)

Un(x, y; i, i)^≡y⁻²ⁱ, i = 0, . . . , n − 1; Un(x, y; i + 1, i)^≡−xy⁻²ⁱ⁻¹,

for i = 0, . . . , n − 2; Un(x, y; i, j)^≡0, if j > i or j < i − 1. (11) The matrices Sⁿand Dⁿare used in definition (19).

Definition 6. The extended q-Pascal matrix Φⁿ,q(x, y) is given by

Φn,q(x, y; i, j)^≡ (︃i

j )︃

q

x^i−jy^i+j. (12)

Theorem 2.1. A q-analogue of [13, p. 170].

Φn,q(x1, y1)Φn,q(x2, y2) = Φn,q(x₁

y₂^⊕qx₂y₁, y1y₂), y2≠ 0. (13) Proof.

Φn,q(x1, y1)Φn,q(x2, y2)(i, j) =

n−1

∑︁

k=0

x₁^i−kyⁱ₁^+k (︃

i k

)︃

q

x₂^k−jy^k+j₂ (︃

k j )︃

q

= (︃

i j )︃

q

(y1y₂)^i+j

n−1

∑︁

k=0

(︃

i− j k− j

)︃

q

(x₁

y₂)^i−k(x2y₁)^k−j= (︃

i j )︃

q

(y1y₂)^i+j(x₁

y₂^⊕qx₂y₁)^i−j.

(14)

Definition 7. The matrices Pn,q(x), Pn,k,q(x), Pk,q* (x) and Pn,k,q* (x) are defined by

Pⁿ,q(x; i, j)^≡ (︃i

j )︃

q

x^i−j, i, j = 0, . . . , n − 1, (15)

Pn,k,q(x)^≡

[︃ In−k 0^T 0 Pk,q(x)

]︃

, (16)

Pk,q* (x; i, j)^≡ (︃

i j )︃

q

(qx)^i−j, i, j = 0, . . . , k − 1, (17)

Pn,k,q* (x)^≡

[︃ In−k 0^T 0 Pk,q* (x)

]︃

. (18)

The summation matrix Gn,k(x) and the difference matrix Fn,k(x) are defined by

Gn,k(x)^≡

[︃ In−k 0^T 0 Sk(x)

]︃

, k = 3, . . . , n,

Fn,k(x)^≡

[︃ I_n−k 0^T 0 Dk(x)

]︃

, k = 3, . . . , n, Fⁿ,n(x)^≡Dⁿ(x), n > 2.

(19)

Let the two matrices Ik,q(x) and Ek,q(x) be given by

Ik,q(x; i, i)^≡1, i = 0, . . . , k − 1, Ik,q(x; i + 1, i)^≡x(qⁱ⁺¹− 1), i ≤ k − 2, Ik,q(x; i, j)^≡0 for other i, j.

Ek,q(x; i, j)^{≡ ⟨}j+ 1; q^⟩i−jx^i−j, i ≥ j, Ek,q(x; i, j)^≡0 for other i, j.

(20)

I_n,k,q(x)^≡

[︃ In−k 0^T 0 Ik,q(x)

]︃

, In,n,q(x)^≡In. (21)

En,k,q(x)^≡

[︃ I_n−k 0^T 0 Ek,q(x)

]︃

, Eⁿ,n,q(x)^≡Iⁿ. (22)

We call I_n,k,q(x) the q-unit matrix function. We will use a slightly q-deformed version of the D- and F-matrices:

Dk,q* (x; i, i)^≡1, i = 0, . . . , k − 1, Dk,q* (x; i + 1, i)^≡−xqⁱ⁺¹, i ≤ k − 2,

Dk,q* (x; i, j)^≡0, if j > i or j < i − 1. (23)

Fn,k,q* (x)^≡

[︃ In−k 0^T 0 Dk,q*(x)

]︃

. (24)

Gk,q* (x; i, j)^≡

{︃q(^i−j+12 )^+j(i−j)x^i−j, if j ≤ i,

0, if j > i,, Gn,k,q* (x)^≡

[︃ In−k 0^T 0 Gk,q* (x)

]︃

.

(25)

The inverse of Pk,q* (x) is given by

(Pk,q* (x))⁻¹(i, j) = (︃

i j )︃

q

(−x)^i−jq(^i−j+12 ), i, j = 0, . . . , k − 1. (26)

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.

Wn(x, y) = Un(x, y)⁻¹; F_n,k(x) = G_n,k(x)⁻¹, k = 3, . . . , n. (27)

In,k,q(x)⁻¹= En,k,q(x); Fn,k,q* (x)⁻¹= 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

Pn,q(x) =

3

∏︁

k=n

(︀In,k,q(x)Gn,k(x))︀ Gn,2,q* (x), (29)

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)

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

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

Wn(x, y)Pn,n−1,q* (x

y) = Φn,q(x, y), n ≥ 1. (31)

Proof. For n = 1, Pn,n−1,q* (x) = Iⁿand Wn(x, y) = Φn,q(x, y). Let n > 1. The matrix element (Wn(x, y)Pn,n−1,q* (x

y)(i, j) =

i

∑︁

k=j

x^i−jy^i+j (︃

k− 1 j− 1 )︃

q

q^k−j= x^i−jy^i+j (︃

i j )︃

q

= Φn,q(x, y)(i, j), j ≥ 1. (32)

For j = 0, Wn(x, y)Pn,n−1,q* (^xy; i, 0) = (xy)ⁱ= 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

Φn,q(x, y) = Wn(x, y)

∏︁3 k=n−1

(︂

I_n,k,q(x y)G_n,k(x

y) )︂

Gn,2,q* (x

y), (33)

where the product is taken in decreasing order of k.

Proof. Use (31). At each step the q-unit matrix maps Pn,k,q* (^xy) to Pn,k,q(^xy) by (30) and to Gn,k(^xy) by (29).

Example1. The case n = 4:

⎛

⎜

⎜

⎜

⎝

1 0 0 0

xy y² 0 0

x²y² {2^}qxy³ y⁴ 0 x³y³ {3^}qx²y⁴ {3^}qxy⁵ y⁶

⎞

⎟

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎝

1 0 0 0

xy y² 0 0

x²y² xy³ y⁴ 0 x³y³ x²y⁴ xy⁵ y⁶

⎞

⎟

⎟

⎟

⎠

⎛

⎜

⎜

⎜

⎝

1 0 0 0

0 1 0 0

0 ^x_y(q − 1) 1 0

0 0 ^x_y(q²− 1) 1

⎞

⎟

⎟

⎟

⎠

⎛

⎜

⎜

⎜

⎝

1 0 0 0

0 1 0 0

0 ^x_y 1 0

0 (^x_y)^{2 x}_y 1

⎞

⎟

⎟

⎟

⎠

⎛

⎜

⎜

⎜

⎝

1 0 0 0

0 1 0 0

0 0 1 0

0 0 ^x_yq 1

⎞

⎟

⎟

⎟

⎠ .

(34)

Theorem 3.6. A q-analogue of [13, p.174]. The inverse of the extended q-Pascal matrix is given by Φ_n,q(x, y)⁻¹= Fn,2,q*(︂ x

y )︂ n

∏︁

k=3

(︂

Fn,k

(︂ x y )︂

En,k,q

(︂ x y

)︂)︂

Un(x, y). (35)

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 Qnin [12, p.55]. The matrix Rn,q(x) is defined by Rⁿ,q(x; i, j)^≡

(︃i j )︃

q

x^i+j, i, j = 0, . . . , n − 1. (36)

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

Ψ_n,q(x, y; i, j)^≡ (︃

i+ j j

)︃

q

x^i−jy^i+j, i, j = 0, . . . , n − 1. (37)

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.

Ψn,q(x, y) = Rn,q(xy)Φn,q(y,1

x)^T, (38)

Ψn,q(x, y) = Φⁿ,q(x, y)Pⁿ,q(y

x)^T. (39)

Proof. By the first q-Vandermonde theorem,

Rn,q(xy)Φn,q(y,1

x)^T(i, j) =

n−1

∑︁

k=0

(︃

i k

)︃

q

(xy)^i+ky^j−kx^−j−k (︃

j k

)︃

q

q^k2 = (︃

i+ j j

)︃

q

x^i−jy^i+j. (40)

Φn,q(x, y)Pn,q(y

x)^T(i, j) =

n−1

∑︁

k=0

(︃

i k

)︃

q

x^i−jy^i+j (︃

j k )︃

q

q^k2 = (︃

i+ j j

)︃

q

x^i−jy^i+j. (41)

We have found two different matrix products for the same function.

Example2. The case n = 4:

⎛

⎜

⎜

⎜

⎜

⎝

1 ^yx

y² x²

y³ x³

xy {2^}qy² {3^}qy³

x {4^}qy⁴ x²

x²y² {3^}qxy³ (︀₄

2

)︀

qy⁴ (︀₅

3

)︀

q y⁵

x

x³y³ {4^}qx²y⁴ (︀₅

2

)︀

qxy⁵ (︀₆

3

)︀

qy⁶

⎞

⎟

⎟

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎝

1 0 0 0

xy x²y² 0 0

x²y² {2^}qx³y³ x⁴y⁴ 0 x³y³ {3^}qx⁴y⁴ {3^}qx⁵y⁵ x⁶y⁶

⎞

⎟

⎟

⎟

⎠

⎛

⎜

⎜

⎜

⎝ 1 ^yx

y² x²

y³ x³

0 _x¹2 {2^}q y

x³ {3^}qy² x⁴

0 0 _x¹4 {3^}q y x⁵

0 0 0 _x¹6

⎞

⎟

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎝

1 0 0 0

xy y² 0 0

x²y² {2^}qxy³ y⁴ 0 x³y³ {3^}qx²y⁴ {3^}qxy⁵ y⁶

⎞

⎟

⎟

⎟

⎠

⎛

⎜

⎜

⎜

⎝

1 ^y_x ^y_x²2

y³ x³

0 1 ^{2^}qy

x {3^}qy² x²

0 0 1 ^{3^}qyx

0 0 1

⎞

⎟

⎟

⎟

⎠

. (42)

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

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