A new class of polynomials with applications (Q1813222)

The author extends some ideas of his paper 'An application of the umbral calculus', J. Math. Anal. Appl., to appear. In the Euclidean plane with origin 0 we choose axes \(0X\) and \(0Y\) at an angle \(\pi/3.\) Let \(\b{i}\) and \(\b{j}\) be the unit vectors along these axes. A point \((i,j)\), where \(i\) and \(j\) are non-negative integers, is associated with a vector \(i\b{i}+j\b{j}\). The set of such points is denoted by \(S\). The author considers linear connected graphs joining the origin to a point \((i,j)\) so that the vertices of the graph belong to \(S\) and the edges are parallel to \(\b{i}\), \(\b{j}\) or \(\b{i}-\b{j}\). Let \(N(i,j,L)\) denote the number of such graphs with length \(L\). An explicit formula for \(N(i,j,L)\) is obtained by means of umbral calculus using a suitable Sheffer sequence of polynomials.