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Sun, Z.-H. A Note on Wilson’s Theorem and Wolstenholme’s Theorem.

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• TITLE: Some Implications of the Gessel Identity

  AUTHORS: Claire Levaillant

  KEYWORDS: Convolutions Involving Bernoulli Numbers, Truncated Convolutions Involving Bernoulli Numbers, Congruences, Binomial and Multinomial Convolutions of Divided Bernoulli Numbers, Multiple Harmonic Sums, Generalized Harmonic Numbers, Miki Identity, Gessel Identity, Sums of Powers of Integers Weighted by Powers of the Fermat Quotients, Generalization of Kummer’s Congruences, Generalizations of Friedmann-Tamarkine, Lehmer, Ernvall-Metsänkyla’s Congruences, p-Adic Numbers, Weighted Sums of Powers of Integers

  JOURNAL NAME: Applied Mathematics, Vol.14 No.9, September 15, 2023

  ABSTRACT: We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (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 (2021), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsänkyla residues to residues of some full convolutions of the same kind. We also establish some congruences concerning other related weighted sums of powers of integers when these sums are weighted by some analogs of the Teichmüller characters.