LDU decomposition of an extension matrix of the Pascal matrix

Abstract

Let Ω m , n ( α , β , γ ) denote a set of all elements of weighted lattice paths with weight ( α , β , γ ) in the xy-plane from ( 0 , 0 ) to ( m , n ) such that a vertical step V = ( 0 , 1 ) , a horizontal step H = ( 1 , 0 ) , and a diagonal step D = ( 1 , 1 ) are endowed with weights α , β , and γ respectively and let ω ( Ω m , n ( α , β , γ ) ) denote the weight of Ω m , n ( α , β , γ ) defined by ω ( Ω m , n ( α , β , γ ) ) = ∑ p ∈ Ω m , n ( α , β , γ ) ega ( p ) where ω ( p ) is the product of the weights of all its steps in p . A matrix A = [ a ij ] is called a lattice path matrix with weight ( α , β , γ ) if a ij = ω ( Ω i , j ( α , β , γ ) ) for a triple α , β , and γ of real numbers . In this paper, we present LDU decomposition of lattice path matrices with weight ( α , β , γ ) and related properties for every triple α , β , and γ 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.