Some combinatorial matrices and their LU-decomposition

Three combinatorial matrices are considered and their LU-decompositions were found. This is typically done by (creative) guessing, and necessary proofs are more or less routine calculations.


Introduction
Combinatorial matrices often have beautiful LU-decompositions, which leads also to easy determinant evaluations. It has become a habit of this author to try this decomposition whenever he sees a new such matrix.
The present paper contains three independent ones collected over the last one or two years.

A matrix from polynomials with bounded roots
In [11] Kirschenhofer and Thuswaldner evaluated the determinant ≤i,l≤s for t = . Consider the matrix M with entries /(( l) −t ( i− ) ) where s might be a positive integer or in nity. In [11], the transposed matrix was considered, but that is immaterial when it comes to the determinant; we will treat the transposed matrix as well, but the results are slightly uglier. The aim is to provide a completely elementary evaluation of this determinant which relies on the LUdecomposition LU = M, which is obtained by guessing. The additional parameter t helps with guessing and makes the result even more general. We found these results: ; using these formulae, L i,j resp. U j,l can be written in terms of Gamma functions. The proof that indeed j L i,j U j,l = M i,l is within the reach of computer algebra systems (Zeilberger's algorithm). An old version of Maple (without extra packages) provides this summation.
As a bonus, we also state the inverses matrices: the necessary proofs are again automatic.
Consequently the determinant is For t = , this may be simpli ed: the last expression was given in [11]. We used the notation ( n − )!! = · · · · · ( n − ). Now we brie y mention the equivalent results for the transposed matrix:
Note the similarity to Schur's determinant that was used to great success in [9]. This success was based on the two recursions and, with Schur(x) = n≥ an x n , by an = q n an + q +m q n− a n− , leading to Schur's (and thus Lehmer's) determinant plays an instrumental part in proving the celebrated Rogers-Ramanujan identities and generalizations. Lehmer [12] has computed the limit for n → ∞ of the determinant of the matrix M(n). Ekhad and Zeilberger [7] have generalized this result by computing the determinant of the nite matrix M(n). Furthermore, a lively account of how modern computer algebra leads to a solution was given. Most prominently, the celebrated q-Zeilberger algorithm [14] and creative guessing were used.
In this section, the determinant in question is obtained by computing the LU-decomposition LU = M. This is done with a computer, and the exact form of L and U is obtained by guessing. A proof that this is indeed the LU-decomposition is then a routine calculation. From it, the determinant in question is computed by multiplying the diagonal elements of the matrix U. By telescoping, the nal result is then quite attractive, as already stated and proved by Ekhad and Zeilberger [7].

. The LU-decomposition of M
It follows from the basic recursion of the Gaussian q-binomial coe cients [2] that Then we have and all other entries in the U-matrix are zero. Further, and all other entries in the L-matrix are zero. The typical element of the product (LU) i,j , that is ≤k≤n L i,k U k,j is almost always zero; the exceptions are as follows: If i = j, then we get because of the above recursion (1). If i = j − , then we get and if i = j + , then we get This proves that indeed LU = M. Therefore for the determinant of the Lehmer matrix M we obtain the expression Taking the limit n → ∞, leads to the old result by Lehmer for the determinant of the in nite matrix:

Matrices for Fibonacci polynomials
Cigler [5] introduced several matrices that have Fibonacci polynomials as determinants; we will only treat two of them as showcases. The Fibonacci polynomials are our answers will come out in terms of the related polynomials where we write X = x for simplicity. It is easy to check that for instance by comparing coe cients. The rst matrix is ≤i,j<n and we will determine its LU-decomposition M = LU. We obtained For a proof, we do this computation: The determinant is then U , U , . . . U n− ,n− , and by telescoping For completeness, we also factor the transposed matrix as LU = M t : For completeness, we mention another recent paper about matrices and Fibonacci polynomials: [1].