Some implications of the Gessel identity

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.