For a given sequence {an}n=0∞, the Hankel matrix is defined by

One can obtain a combinatorial matrix having interesting properties from a Hankel matrix. For example, the Hilbert matrix H_n = [h_ij ] is defined by hij=1i+j-1 (for more details, see [2, 4]) and the Filbert matrix ℱ_n = [f_ij ] is defined by fij=1Fi+j-1 (see [1, 6]). Clearly they are of the forms

where F_n is nth Fibonacci number.

Throughout this paper, we will use the q-Pochhammer symbol

and the Gaussian q-binomial coefficients

It is clearly that

where (nk) is the usual binomial coefficient.

The Cauchy binomial theorem is given by

and Rothe’s formula (see [2]) is given by

Prodinger [3] consider the reciprocal super Catalan matrix M with the entries mij=i!(i+j)!j!(2i)!(2j)! and obtain explicit formulae for its LU-decomposition, the LU-decomposition of its inverse, and some related matrices. For all results, q-analogues are also presented.

We rewrite the matrix M in terms of three binomial coefficients, two of them is in the reciprocal form, as shown

By inspiring the matrix M, we consider two kinds variants of it by keeping only one binomial coefficient in reciprocal form in the first one and two binomial coefficients in reciprocal forms in the second one. Also we will add two additional parameters to each one. Now we define these matrices: The first one is the matrix A = [a_ij] of order n with the entries

and the second one is the n × n matrix B with the entries

where m and t are nonnegative integers and all indices of these matrices start at (0, 0).

We write the matrices and ℬ which are the q-analagues of the matrices A and B, respectively. We give explicit expressions for LU-decompositions of these matrices and their inverses.

By help of a computer, LU-decompositions of these matrices were firstly obtained and then we have achieved the formulas by certain skills especially guessing skill. Using q-Zeilberger algorithm [5] and elementary matrix operations, the proofs are given as combinatorial identities. We will discuss a few of them here rather than all of them.

The matrix = [â_ij ] has the entries a^ij=[2i+mi]q[2j+tj]q-1[i+ji]q for 0 ≤ i, j ≤ n − 1. Now we will give expressions for LU-decompositions L₁U₁ = A and L₂U₂ = A⁻¹, for L1-1,U1-1 and for L2-1,U2-1, by the following theorem.

Theorem 2.1

For m,t ≥ 0,

and

In this section, the matrix ℬ = [b̂_ij] is defined with entries

for 0 ≤ i, j ≤ n−1. Now we will give expressions for LU-decompositions L₃U₃ = ℬ and L₄U₄ = ℬ⁻¹, for L3-1,U3-1 and for L4-1,U4-1 without proof by the following theorem

Theorem 3.1

For m, t ≥ 0 and 0 ≤ i, j ≤ n − 1,

and

In this section, we have the following results without proof by using the results of Theorem 2.1 with the fact given in (1.1). For 0 ≤ i, j < n − 1,

In this section, we have the following results without proof by using the results of Theorem 2.2 with the fact given in (1.1). For 0 ≤ i, j < n − 1,