Multiple Zeta Function - Two Parameters Case

Two Parameters Case

In the particular case of only two parameters we have (with s>1 and n,m integer):

where are the generalized harmonic numbers.

Multiple zeta functions are known to satisfy what is known as MZV duality, the simplest case of which is the famous identity of Euler:


\sum_{n=1}^\infty \frac{H_n}{(n+1)^2} = \zeta(2,1) = \zeta(3) = \sum_{n=1}^\infty \frac{1}{n^3},
\!

where Hn are the harmonic numbers.

Special values of double zeta functions, with s > 0 and even, t > 1 and odd, but s+t=2N+1 (taking if necessary ζ(0) = 0):

\zeta(s,t)=\zeta(s)\zeta(t)+\tfrac{1}{2}\Big\zeta(s+t)-\sum_{r=1}^{N-1}\Big\zeta(2r+1)\zeta(s+t-1-2r)
s t approximate value explicit formulae OEIS
2 2 0.811742425283353643637002772406  A197110
3 2 0.228810397603353759768746148942
4 2 0.088483382454368714294327839086
5 2 0.038575124342753255505925464373
6 2 0.017819740416835988
2 3 0.711566197550572432096973806086
3 3 0.213798868224592547099583574508
4 3 0.085159822534833651406806018872
5 3 0.037707672984847544011304782294
2 4 0.674523914033968140491560608257
3 4 0.207505014615732095907807605495
4 4 0.083673113016495361614890436542

Note that if we have irriducibles, i.e. these MZVs cannot be written as function of only.

Read more about this topic:  Multiple Zeta Function

Famous quotes containing the words parameters and/or case:

    Men have defined the parameters of every subject. All feminist arguments, however radical in intent or consequence, are with or against assertions or premises implicit in the male system, which is made credible or authentic by the power of men to name.
    Andrea Dworkin (b. 1946)

    [The boss] asked me if I was not interested in a change in my life. I answered that one can never change lives, that in any case all lives were the same, and that I was not at all unhappy with mine.
    Albert Camus (1913–1960)