A new class of identities involving Cauchy numbers, harmonic numbers and zeta values

Bernard Candelpergher (1) , Marc-Antoine Coppo (1)

1 JAD - Laboratoire Jean Alexandre Dieudonné

Improving an old idea of Hermite, we associate to each natural number $k$ a modified zeta function of order $k$. The evaluation of the values of these functions $F_k$ at positive integers reveals a wide class of identities linking Cauchy numbers, harmonic numbers and zeta values.

Mots clés: Cauchy numbers; Bell polynomials; Harmonic numbers; Laplace-Borel transform; Mellin transform; Zeta values; Ramanujan summation; Hermite's formula.; Hermite's formula

Domaines: Théorie des nombres [math.NT]; Combinatoire [math.CO]

Bernard Candelpergher, Marc-Antoine Coppo. A new class of identities involving Cauchy numbers, harmonic numbers and zeta values. Ramanujan Journal (The), 2012, 27, pp.305-328. ⟨hal-00495767v9⟩