An explicit form of the polynomial part of a restricted partition function

Karl Dilcher (1) , Christophe Vignat (2)

1 Dalhousie University [Halifax]
2 L2S - Laboratoire des signaux et systèmes

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.

Domaines: Probabilités [math.PR]

Karl Dilcher, Christophe Vignat. An explicit form of the polynomial part of a restricted partition function. Research in Number Theory, 2017, 3 (1), ⟨10.1007/s40993-016-0065-3⟩. ⟨hal-01448646⟩