Some Identities on the Generalized Higher-Order Euler and Bernoulli Numbers

Abstract

By the classical method for obtaining the values of the Riemann zeta-function at even positive integral arguments, we shall give some functional equational proof of some interesting identities and recurrence relations related to the generalized higher-order Euler and Bernoulli numbers attached to a Dirichlet character $\chi$ with odd conductor $d$, and shall show an identity between generalized Euler numbers and generalized Bernoulli numbers. Finally, we remark that any weighted short-interval character sums can be expressed as a linear combination of Dirichlet $L$-function values at positive integral arguments, via generalized Bernoulli (or Euler) numbers.