Multiple Zeta Values online tools

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Usage

  1. Click the “Activate” button
  2. Type your code at the above box.
  3. Click “Evaluate” button.
  4. Repeat 2 and 3.
The followings are examples of usage. The author of this page is Minoru Hirose. See github page for the source code.

Symbolic expression for multiple zeta(-star) Value


>>> mzv(1,2)
ζ(3)

>>> mzv(2,3)
1/2*ζ(3)*π^2 - 11/2*ζ(5)

>>> ζ(2,3)
1/2*ζ(3)*π^2 - 11/2*ζ(5)

>>> mzsv(1,2)
2*ζ(3)

>> mzv(1,2,t=1)
2*ζ(3)

Harmonic product


>>> stuffle(index(2), index(5))
index(2,5) + index(5,2) + index(7)

>>> stuffle(index(2,3), index(5))
index(2,3,5) + index(2,5,3) + index(2,8) + index(5,2,3) + index(7,3)

>>> stuffle(-e1*e0, -e1*e0*e0*e0*e0)
-e1*e0^6 + e1*e0^4*e1*e0 + e1*e0*e1*e0^4

Shuffle product 


>>> shuffle(e1*e0, e1 )
e1*e0*e1 + 2*e1^2*e0

>>> shuffle(index(2), index(1))
index(2,1) + 2*index(1,2)

Dual of index 


>>> dual(e1*e0*e0*e1*e0)
-e1*e0*e1^2*e0

>>> dual(index(3,2))
index(2,1,2)

Hoffman-dual of index


>>> Hoffman_dual( index(3,4) )
index(1,1,2,1,1,1)

Shuffle regularization 


>>> shuffle_regularization( index(5,1) )
-index(4,2) - index(3,3) - index(2,4) - 2*index(1,5)

>>> shuffle_regularization( shuffle(index(1,1), index(2,3) ) )
0

>>> shuffle_regularized_mzv(2,1)
-2*ζ(3)

Stuffle regularization 


>>> stuffle_regularization( index(5,1) )
-index(1,5) - index(6)

>> stuffle_regularized_mzv(1,1)
-1/12*π^2

Symbolic expression for symmetric multiple zeta(-star) value 


>>> shuffle_regularized_symmetric_mzv(3,8)
-4/315*ζ(5)*π^6 - 1/3*ζ(7)*π^4 - 12*ζ(9)*π^2 + 165*ζ(11)

>>> stuffle_regularized_symmetric_mzv(3,8)
-4/315*ζ(5)*π^6 - 1/3*ζ(7)*π^4 - 12*ζ(9)*π^2 + 165*ζ(11)

>>> symmetric_mzv(3,8)
165*ζ(11)

>>> shuffle_regularized_symmetric_mzv(1,1)
0

>>> shuffle_regularized_symmetric_mzsv(1,1) #star-value
1/3*π^2

>>> stuffle_regularized_symmetric_mzv(1,1)
-1/6*π^2

>>> shuffle_regularized_symmetric_mzsv(1,1) # star-value
1/6*π^2