A Recursive Formula for Sums of Values of Degenerate Falling Factorials

Abstract

The classical Faulhaber's formula expresses the sum of a fixed positive integer powers of the first n positive integers in terms of Bernoulli polynomial. As a degenerate version of this, we may consider sums of values of degenerate falling factorials, which reduce to aforementioned sum as lambda tends to 0. The aim of this note is to derive a recursive formula for sums of values of degenerate falling factorials by using probabilistic method. In this manner, we obtain a new recursive formula for such sums, which involves the (signed) Stirling numbers of the first kind.