Relations for Bernoulli–Barnes numbers and Barnes zeta functions

Abstract

ABSTRACT. The Barnes ζ -function is ζ n (z, x; a) := ∑ m∈Z n ≥0 1 (x + m 1 a 1 + · · · + m n a n ) z defined for Re(x) > 0 and Re(z) > n and continued meromorphically to C. Specialized at negative integers −k, the Barnes ζ -function giveswhere B k (x; a) is a Bernoulli-Barnes polynomial, which can be also defined through a generating function that has a slightly more general form than that for Bernoulli polynomials. Specializing B k (0; a) gives the BernoulliBarnes numbers. We exhibit relations among Barnes ζ -functions, Bernoulli-Barnes numbers and polynomials, which generalize various identities of Agoh, Apostol, Dilcher, and Euler.

On the multidimensional zeta functions associated with theta functions, and the multidimensional Appell polynomials

“…Example Let g be a holomorphic function with g (0)=0 and g ′ (0)≠0 and an arbitrary function. Bayad and Komatsu introduce the notion of Appell polynomials of type ( g , ϕ ), that is a sequence of polynomials satisfying where f is a holomorphic function on with f (0)≠0. When f admits a Laurent expansion for 0< t ≪1 and an exponential decay at + ∞ , and g is analytic near 0 and | g ( t )| ≤ − ct for t ≫1 with c is a positive constant, then the function t → f ( t ) e xg ( t ) is a theta function for any 0 ≤ x ≪1. Let .…”

Representations for the derivative at zero and finite parts of the Barnes zeta function

“…We omit the superscript as we will use it shortly with a different meaning, that of derivative of order d. 3 This is the definition used for example in [27,28,29,2,23]. Some authors, namely in [14,26], use a different convention, where these polynomials differ by a factor of d i=1 w i . B n (a|w) is a polynomial of order n in a.…”

“…The representations here obtained, given below in Eqs. (26), (35) and (36) (together with Eq. (3.13) of [14]), in turn provide line integral representations for the Barnes' functions just mentioned, alternative to the ones given by Barnes. In contrast to the series and limit representations of the previous section, whose validity is absolutely general, the integral representations of the present section are valid when ℜ(a) > 0 and ℜ(w i ) > 0 (corresponding to θ = 0 in the definition of the half-plane H).…”

“…This latter procedure allows the treatment of higher dimensional cases (Dowker [7] did some work on the 3-dimensional case) and is, in fact, extensible to the case where the w i are rational numbers. However, the expressions soon become extremely complicated and not very helpful (see [26] in this connection).…”

On the asymptotic distribution of cranks and ranks of cubic partitions

“…where ζ r (s, x; (a 1 , a 2 , · · · , a r )) is a Barnes zeta function defined by ζ r (s, x; (a 1 , a 2 , · · · , a r )) = (m 1 ,··· ,m r )∈Z r ≥0 1 (x + m 1 a 1 + · · · + m r a r ) s for Re(x) > 0 and Re(s) > r. By [4,Theorem 3], the Barnes zeta function can be expressed in terms of Bernoulli-Barnes polynomials, Hurwitz zeta functions, and Fourier-Dedekind sums:…”