Truncated Multiple Zeta Values

We generalize the definition of truncated multiple zeta values by allowing arbitrary integers as arguments. This leads to interesting identities, particularly with the argument 0. Truncated multiple zeta values satisfy the same quasi-shuffle algebraic identities as multiple zeta values, but we need to extend the algebra QSym of quasi-symmetric functions to a larger algebra. Using this algebra, we are able to sum systematically powers of harmonic and generalized harmonic numbers. This leads to summation identities such as $$\sum_{n=1}^\infty H_n^3\bigg(\zeta(2)-\sum_{k=1}^n\frac{1}{k^2}-\frac{1}{n}\bigg)= -\frac{11}2\zeta(4)+\zeta(3)+3\zeta(2)-6.$$ We also prove analogous identities involving alternating sums of harmonic numbers and their powers.