Generalized hypergeometric Bernoulli numbers

Abstract

We introduce generalized hypergeometric Bernoulli numbers for Dirichlet characters. We study their properties, including relations, expressions and determinants. At the end in Appendix we derive first few expressions of these numbers.

Appendix

We derive first few values of $B_{N,n,\chi }$:

$$\begin{aligned} B_{N,0,\chi }&=\frac{S_0}{f^N}\,,\\ B_{N,1,\chi }&=-\frac{S_0}{(N+1)f^{N-1}}+\frac{S_1}{f^N}\,,\\ B_{N,2,\chi }&=\frac{2 S_0}{(N+1)^2(N+2)f^{N-2}}-\frac{2 S_1}{(N+1)f^{N-1}}+\frac{S_2}{f^N}\,,\\ B_{N,3,\chi }&=\frac{3!(N-1)S_0}{(N+1)^3(N+2)(N+3)f^{N-3}}+\frac{3! S_1}{(N+1)^2(N+2)f^{N-2}}-\frac{3 S_2}{(N+1)f^{N-1}}+\frac{S_3}{f^N}\,,\\ B_{N,4,\chi }&=\frac{4!(N^3-N^2-6 N+2)S_0}{(N+1)^4(N+2)^2(N+3)(N+4)f^{N-4}}+\frac{4!(N-1)S_1}{(N+1)^3(N+2)(N+3)f^{N-3}}\\&\qquad +\frac{12 S_2}{(N+1)^2(N+2)f^{N-2}}-\frac{4 S_3}{(N+1)f^{N-1}}+\frac{S_4}{f^N}\,. \end{aligned}$$

Now

$$\begin{aligned} B_{N,0}&=1\,,\\ B_{N,1}&=-\frac{1}{N+1}\,,\\ B_{N,2}&=\frac{2}{(N+1)^2(N+2)}\,,\\ B_{N,3}&=\frac{3!(N-1)}{(N+1)^3(N+2)(N+3)}\,,\\ B_{N,4}&=\frac{4!(N^3-N^2-6 N+2)}{(N+1)^4(N+2)^2(N+3)(N+4)}\,. \end{aligned}$$

From [2] we can see that,

$$\begin{aligned} B_{N,n,\chi }=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) \frac{S_{n-k}B_{N,k}}{f^{N-k}}\,, \end{aligned}$$

as shown in Corollary 1.

When $N=1$ and $\chi$ is not the trivial character, as $S_0=0$, we find that

$$\begin{aligned} B_{1,0,\chi }&=0\,,\\ B_{1,1,\chi }&=\frac{S_1}{f}\,,\\ B_{1,2,\chi }&=-S_1+\frac{S_2}{f}\,,\\ B_{1,3,\chi }&=\frac{f S_1}{2}-\frac{3 S_2}{2}+\frac{S_3}{f}\,,\\ B_{1,4,\chi }&=f S_2-2 S_3+\frac{S_4}{f}\,. \end{aligned}$$