On an extension of a recurrent relation from combinatorics

Abstract

The following recurrent relation/partial difference equation wn,k = awn−1,k−1 + bwn−1,k, where k,n,a,b ∈ ℕ, appears in a problem in combinatorics. Here we show that an extension of the recurrent relation is solvable on, the, so called, combinatorial domain C={(n,k)∈ ℕ20:0≤k≤n}∖{(0,0)}, when its coefficients and the boundary values wj,0, wj,j, j ∈ ℕ, are complex numbers, by presenting a representation of the general solution to the recurrent relation on the domain in terms of the boundary values. As a special case we obtain a solution to the problem in combinatorics in an elegant way. From the general solution along with an application of the linear first-order difference equation is also obtained the solution of the recurrent relation in the case wj,j=c ∈ C, j ∈ ℕ.