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On a congruence of Emma Lehmer related to Euler numbers

John B. Cosgrave, Karl Dilcher (2013)

Acta Arithmetica

A congruence of Emma Lehmer (1938) for Euler numbers E p - 3 modulo p in terms of a certain sum of reciprocals of squares of integers was recently extended to prime power moduli by T. Cai et al. We generalize this further to arbitrary composite moduli n and characterize those n for which the sum in question vanishes modulo n (or modulo n/3 when 3|n). Primes for which E p - 3 0 ( m o d p ) play an important role, and we present some numerical results.

On Bernoulli identities and applications.

Minking Eie, King F. Lai (1998)

Revista Matemática Iberoamericana

Bernoulli numbers appear as special values of zeta functions at integers and identities relating the Bernoulli numbers follow as a consequence of properties of the corresponding zeta functions. The most famous example is that of the special values of the Riemann zeta function and the Bernoulli identities due to Euler. In this paper we introduce a general principle for producing Bernoulli identities and apply it to zeta functions considered by Shintani, Zagier and Eie. Our results include some of...

On certain generalized q-Appell polynomial expansions

Thomas Ernst (2015)

Annales UMCS, Mathematica

We study q-analogues of three Appell polynomials, the H-polynomials, the Apostol-Bernoulli and Apostol-Euler polynomials, whereby two new q-difference operators and the NOVA q-addition play key roles. The definitions of the new polynomials are by the generating function; like in our book, two forms, NWA and JHC are always given together with tables, symmetry relations and recurrence formulas. It is shown that the complementary argument theorems can be extended to the new polynomials as well as to...

On mth order Bernoulli polynomials of degree m that are Eisenstein

Arnold Adelberg, Michael Filaseta (2002)

Colloquium Mathematicae

This paper deals with the irreducibility of the mth order Bernoulli polynomials of degree m. As m tends to infinity, Eisenstein's criterion is shown to imply irreducibility for asymptotically > 1/5 of these polynomials.

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