A summation on Bernoulli numbers

Abstract

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.

Extended Zeilberger's algorithm for identities on Bernoulli and Euler polynomials

“…It should be mentioned that the above identity (7.1) is the special case s = r = 3 of an identity of K.-W. Chen [3]: It can be seen that our approach also applies to the above identity (7.2). Notice that when m = 1, the above identity becomes Kaneko's identity.…”

Hankel Determinants of sequences related to Bernoulli and Euler Polynomials

“…4. Among other generalizations of Bernoulli and Euler numbers for which Hankel determinants have been computed are the q-Bernoulli-Carlitz numbers [3], the median Bernoulli numbers [4], and some character analogues (Corollaries 3.5 and 4.4 above). We refer the interested reader to three very extensive studies by Krattenthaler [13,14] and Milne [15] for numerous other Hankel determinant evaluations.…”

Shifted Euler-Seidel matrices

“…For moment sequences, they gave an integral representation of the Euler-Seidel matrix. Chen [2] investigated the summation form of Bernoulli numbers which can form an Euler-Seidel matrix. The upper diagonal elements of this Euler-Seidel matrix are called "the median Bernoulli numbers".…”