Multinomial matrices

The paper contains two theorems about multinomial matrices. In the first one it is shown that Mn,q,γ = (mα,β) with mα,β = αβ11·α βqq, α, β, γ ε{lunate} Zq+, 0≤γi ≤αi, γi≤βi, and Σqi=1 αi = Σqi=1 βi = n is nonsingular. In the second one we give explicit expressions for the eigenvalues of D = (dα,β) with dα,β = (nβ)αβ1 1·αβqq, α, β ε{lunate} Zq+, and Σqi = 1 αi = Σqi=1 βi = n. The Bernstein operators from approximation theory are generalized and used to obtain the results of the second theorem