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 λ 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.