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:
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):
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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