A Formula for Explicit Solutions of Certain Linear Recursions on Polynomial Sequences
Heinrich Niederhausen · 1985
for all n 1. Such a recursion describes the number of arrangements of n dumbbells on a 2 (n 1+x) array of compartments (McQuistan and Lichtman, [2, 1970]). Umbral Calculus can be helpful in solving such a recursion, if we nd a suitable, known polynomial sequence to connect to (pn). The problem becomes usually more transparent if we introduce the associated linear operators. In our example, de ne P by Ppn = pn 1 for all n 0 (pm = 0 for m < 0). Furthermore, we make use of the shiftoperator