The triangle algorithm for Bernoulli polynomials (Q6546724)

Let \(\{B_n(x)\}\) be the Bernoulli polynomials defined by \(\frac{te^{xt}}{e^t-1}=\sum_{n=0}^{\infty}B_n(x)\frac{t^n}{n!}\). For non-negative integers \(m\) and \(n\) let \(b_{0,m}(x)=\frac 1{m+1}\) and \(b_{n+1,m}(x)=(m+x)b_{n,m}(x)-(m+1)b_{n,m+1}(x)\). The authors prove that \(B_n(x)=b_{n,0}(x)\) and give complicated generalizations for the so-called poly-Bernoulli polynomials and multi-poly Bernoulli polynomials.