LDU decomposition of an extension matrix of the Pascal matrix

Let Ωm,n(α,β,γ)Ωm,n(α,β,γ) denote a set of all elements of weighted lattice paths with weight (α,β,γ)(α,β,γ) in the xy  -plane from (0,0)(0,0) to (m,n)(m,n) such that a vertical step V=(0,1)V=(0,1), a horizontal step H=(1,0)H=(1,0), and a diagonal step D=(1,1)D=(1,1) are endowed with weights α,βα,β, and γγ respectively and let ω(Ωm,n(α,β,γ))ω(Ωm,n(α,β,γ)) denote the weight of Ωm,n(α,β,γ)Ωm,n(α,β,γ) defined byω(Ωm,n(α,β,γ))=∑p∈Ωm,n(α,β,γ)ega(p)where ω(p)ω(p) is the product of the weights of all its steps in pp. A matrix A=[aij]A=[aij] is called a lattice path matrix with weight (α,β,γ)(α,β,γ) if aij=ω(Ωi,j(α,β,γ))aij=ω(Ωi,j(α,β,γ)) for a triple α,βα,β, and γγ of real numbers . In this paper, we present LDU decomposition of lattice path matrices with weight (α,β,γ)(α,β,γ) and related properties for every triple α,βα,β, and γγ of real numbers, and a necessary and sufficient condition in which the symmetric lattice path matrices are positive definite. We also investigate the relationship between the lattice path matrices and generalized Pascal matrices.