A “Generalized Trace Formula” for Bell numbers

The nth Bell number Bn is the number of ways to partition a set of n elements into nonempty subsets. We generalize the “trace formula” of Barsky and Benzaghou [1], which asserts that for an odd prime p and an appropriate constant τp, the relation Bn=-Tr(ϑn-1-τp)Bτp holds in Fp, where ϑ is a root of g˜(x)=xp-x-1 and Tr:Fp[ϑ]⟶Fp is the trace form. We deduce some new interesting congruences for the Bell numbers, generalizing miscellaneous well-known results including those of Radoux [4].