In this paper, our aim is to investigate the summation form of Bernoulli numbers B n , such as n k=0 n k B k+m . We derive some basic identities among them. These numbers can form a Seidel matrix. The upper diagonal elements of this Seidel matrix are called "the median Bernoulli numbers". We determine the prime divisors of their numerators and denominators. And we characterize their ordinary generating function as the unique solution of some functional equation. At last, we also obtain the continued fraction representation of their ordinary generating function and their value of Hankel determinant.

In this paper, we investigate the zeta functionwhere a i ≥ 0, χ i is a Dirichlet character with conductor N i , and P is a polynomial satisfying certain conditions. Its special values at nonpositive integers are closely related to generalized Bernoulli polynomials. Using this fact we can easily get sums of products of Euler polynomials and generalized Bernoulli polynomials.

In this paper, we investigate two kinds of Euler sums that involve the generalized harmonic numbers with arbitrary depth. These sums establish numerous summation formulas including the special values of Arakawa–Kaneno zeta functions and a new formula of multiple zeta values of height one as examples.

A sequence b n is the binomial transform of the sequence a n if b n = n k=0 n k a k . We derive a general identity for such pairs of sequences. Various known identities are obtained as particular cases.

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