Product of increasing powers of Gamma values at rational arguments [2, Cor. 18, p. 93]:
with the constants
where γ is Euler's constant [OEIS A001620] and is the Glaisher-Kinkelin constant [OEIS A074962].
Special product ([2, Thms. 12, 13, p. 88] for k = 5):
with the constant
where (1+√5)/2 is the golden ratio [OEIS A001622].
Product of (divided) Bernoulli numbers [2, Thm. 21, p. 95]:
with the constants
where is the product over all Riemann zeta values at even positive integer arguments [OEIS A080729].
Recurrence formula for the divisor function for even n ≥ 8 [Thesis, Satz 1.1.5, p. 8]:
with the convolution function
The first case n = 8 is known as Hurwitz's identity (1881)
Define the polynomials
composed of Stirling numbers of the second kind S_{2}(n,k) and harmonic numbers H_{k} by
The Genocchi numbers G_{n} [OEIS A036968] are related to the Bernoulli numbers by
The second identity of B_{n} is due to Worpitzky (1883):
The central value
is known to be
Involving the harmonic numbers leads to the identities [5, Thm. 1.3, p. 3]:
and
Quotient of Riemann zeta values [4, Thm. 1.3, p. 2687]: If n ≥ 2 is even, then
with
where is a linear functional, which is connected with a special Dirichlet series, and is a certain monic polynomial of degree n. Moreover, if n = p + 1 with p an odd prime, then is an Eisenstein polynomial and therefore irreducible over ℤ[x].
Under the assumption that every p-adic zeta function ζ_{(p,ℓ)}(s) has a unique simple zero ξ_{(p,ℓ)} in case (p,ℓ) is an irregular pair, one has for even n ≥ 2 [1, Thm. 4.9, p. 16]:
where
The denominator can be described by poles (always lying at 0) and the numerator by zeros of p-adic zeta functions. Equivalently, the formula reads for the Bernoulli numbers:
Moreover, the formulas are valid for all irregular pairs (p,ℓ) with
The denominator of B_{n}(x) − B_{n}, the nth Bernoulli polynomial without constant term, is given by the remarkable formula ([7], [8] with Jonathan Sondow, [OEIS A195441])
where s_{p}(n) denotes the sum of the base-p digits of n. The finite product can be written with explicit sharp bounds as
where
If {·} denotes the fractional part and n ≥ 1 is a fixed integer, then there is the surprising relation [11]:
The denominator of the nth Bernoulli polynomial B_{n}(x) can be described by a similar formula ([9] with Jonathan Sondow):