On the symmetric properties for the generalized twisted Bernoulli polynomials (Q962411)

The authors consider generalized twisted Bernoulli polynomials and numbers, and obtain symmetric identities and two others between them. The generalized twisted Bernoulli polynomials \(B_{n,\chi,\xi}(x)\) for a Dirichlet character \(\chi\) with conductor \(d\) and a \(p^m\)th root of unity \(\xi\) for some positive integer \(m\) are defined by \[ \sum^{d-1}_{a=0} {\chi(a)\xi^a e^{at} t\over \xi^d e^{dt}- 1} e^{xt}= \sum^\infty_{n= 0} B_{n,\chi,\xi}(x){t^n\over n!}, \] and the generalized twisted Bernoulli numbers for \(\chi\) and \(\xi\) are defined by \(B_{n,\chi,\xi}= B_{n,\chi,\xi}(0)\). These are generalizations of the twisted Bernoulli polynomials and numbers. They can be interpreted as the following \(p\)-adic integral \[ \int_X \chi(y) \xi^y e^{(x+ y)t}\,dy= \sum^\infty_{n=0} B_{n,\chi,\xi}(x){t^n\over n!},\quad\text{where }X= X_d= \lim_{\leftarrow} \mathbb Z/dp^N\mathbb Z. \] >From this expression and the \(p\)-adic functional defined in terms of the double integral on \(X\), they obtain symmetric relations such as \[ \begin{multlined} \sum^l_{i=0} {l\choose i} B_{i,\chi,\xi^{\omega_1}}(\omega_2 x) K(\chi, \xi^{\omega_2}, l-i: d\omega_1- 1) \omega^{i-1}_1 \omega^{l-i}_2=\\ \sum^l_{i=0} {l\choose i} B_{i,\chi, \xi^{\omega_2}}(\omega_1 x)K(\chi,\xi^{\omega_1}, l-i: d\omega_2- 1)\omega^{i-1}_2 \omega^{l-i}_1\end{multlined} \] with \(K(\chi,\xi,k: n)= \sum^n_{t=0} \chi(l)\xi^l l^k\).