Some implications of the Gessel identity

TL;DR

This paper extends classical congruences involving sums of powers of integers weighted by Fermat quotients to higher powers, utilizing the Gessel identity to connect residues of Bernoulli number convolutions with established number theoretic residues.

Contribution

It generalizes existing congruences to the third power of Fermat quotients and links Bernoulli number convolutions with these residues using the Gessel identity.

Findings

Extended congruences to third power of Fermat quotients.

Connected Bernoulli number convolutions with Fermat quotient residues.

Established new congruences involving weighted sums and Teichmuller characters.

Abstract

We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), Ernvall-Metsankyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2001), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsankyla residues to some full convolutions of the same kind. We also establish some congruences concerning other related weighed sums of powers of integers when these sums are weighted by some analogs of the Teichmuller characters.