Dilcher Karl, Vignat Christophe
1Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 4R2 Canada.
2LSS-Supelec, Université Paris-Sud, Orsay, France.
Res Number Theory. 2017;3(1):1. doi: 10.1007/s40993-016-0065-3. Epub 2017 Jan 5.
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.
我们证明了一个关于受限分拆函数多项式部分（也称为第一西尔维斯特波）的显式公式。这是通过一些高阶伯努利多项式的恒等式来实现的，其中一个恒等式类似于拉贝著名的普通伯努利多项式乘法公式。作为我们主要结果的一个推论，当受限分拆的系数任意增大时，我们得到了第一西尔维斯特波的渐近表达式。
