Inverse relations and the products of Bernoulli polynomials
In this article, the authors obtain two inverse pairs of identities involving the Bernoulli polynomials and the Bernoulli polynomials of the second kind. More precisely, the following are given: Theorem 1. For \(y=x_1+x_2+\cdots +x_N\) and \(n\geq N\), \[ \begin{multlined} \sum_{k_1+\cdots+k_N=n, k_1,\dots,k_N\geq 0}{n \choose {k_1,\dots,k_n}}B_{k_1}(x_1)\cdots B_{k_N}(x_N)=(-1)^{N-1}N{n \choose N}\\ \times \sum_{j=0}^{N-1}(-1)^j\frac{B_{n-j}(y)}{n-j}\sum_{k=0}^j{{N+k-j-1} \choose k}s(N,N-j+k)y^k,\end{multlined} \] and for \(n\geq N+1\), \[ \begin{multlined} (n-N)\sum_{k=0}^{N}{N \choose k}(-y)^k \frac{B_{n-k}(y)}{n-k}=(-1)^N\sum_{k=0}^N{{n-N+k} \choose k}^{-1}S(N+1,k+1)\\ \times \sum_{l_0+\cdots + l_k=n-N+k,l_0,\dots,l_k\geq 0}{{n-N+k} \choose {l_0,\dots,l_k}}B_{l_0}(y-\sum_{i=1}^k x_i)\prod_{i=1}^kB_{l_i}(x_i),\end{multlined} \] where \(s(n,k)\) and \(S(n,k)\) are the Stirling numbers of the first kind and the second kind, respectively. Also similar identities for the Bernoulli polynomials of the second kind are given as Theorem 2.
Bernoulli polynomials of the second kind
Stirling number of the first kind and the second kind
scientific article; zbMATH DE number 5797147