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

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