We prove an explicit formula for the polynomial part of a restricted

partition function, also known as the first Sylvester wave. This is achieved by way of some identities for higher-order Bernoulli polynomials, one of which is analogous to Raabe's well-known multiplication formula for the ordinary Bernoulli polynomials. As a consequence of our main result we obtain an asymptotic expression of the first Sylvester wave as the coefficients of the restricted partition grow arbitrarily large.

(Joint work with Christophe Vignat).

### S³ seminar : On the polynomial part of a restricted partition function

Séminaire le 21 Décembre 2017, 10h30 à CentraleSupelec (Gif-sur-Yvette) Salle du conseil du L2S - B4.40

**Karl Dilcher, (Dalhousie University, Halifax, Canada)**