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 ∑_n=1^∞ H_n^3(ζ(2)-∑_k=1^n1/k^2-1/n)= -11/2ζ(4)+ζ(3)+3ζ(2)-6. We also prove analogous identities involving alternating sums of harmonic numbers and their powers.

We give an explicit formula for the Galois descent expressing multiple t-values of maximal height in terms of classical multiple zeta values, making precise Murakami's earlier motivic result. Our results rely on the theory of iterated beta integrals. We apply this formula to obtain evaluations of various multiple zeta-half values.

For a composition I whose last part exceeds 1, we can define the multiple t-value t(I) as the sum of all the terms in the series for the multiple zeta value ζ (I) whose denominators are odd. In this paper we show that if I is composition of n≥ 3, then t(I)=(-1)^n-1t(I̅) mod products, where I̅ is the reverse of I, and both sides are suitably regularized when I starts or ends in 1. This result is not true for multiple zeta values, though there is an argument-reversal result that does hold for them (and for multiple t-values as well). We actually prove a more general version of this result, and then use it to establish explicit formulas for several classes of multiple t-values and interpolated multiple t-values.

We establish an identity amongst certain differential operators applied to a formal power-series. As a corollary we obtain an explicit depth reduction result for alternating MZV's of the form ζ(1,…,1,2m), which resolves a conjecture posed earlier by the third author.

We study logarithmic integrals of the form $$\int _0^1 x^i\ln ^n(x)\ln ^m(1-x)dx$$ . They are expressed as a rational linear combination of certain rational numbers $$(n,m)_{i}$$ , which we call tiered binomial coefficients, and products of the zeta values $$\zeta (2)$$ , $$\zeta (3)$$ ,.... Various properties of the tiered binomial coefficients are established. They involve, amongst others, the binomial transform, truncated multiple zeta and multiple zeta star values, as well as special functions. We present extensions to generalized Nielsen polylogarithms. As an application we revisit the limit law of the number of comparisons of the Quicksort algorithm: we reprove that the moments of the limit law are rational polynomials in the zeta values. Properties of the cumulants of the Quicksort limit law are also discussed.

We obtain an asymptotic series $$\sum _{j=0}^\infty \frac{I_j}{n^j}$$ for the integral $$\int _0^1[x^n+(1-x)^n]^{\frac{1}{n}}\mathrm{{d}}x$$ as $$n\rightarrow \infty $$, and compute $$I_j$$ in terms of alternating (or “colored”) multiple zeta values. We also show that $$I_j$$ is a rational polynomial in the ordinary zeta values, and give explicit formulas for $$j\le 12$$. As a by-product, we obtain precise results about the convergence of norms of random variables and their moments. We study $$Z_n=\Vert (U,1-U)\Vert _n$$ as n tends to infinity and we also discuss $$W_n=\Vert (U_1,U_2,\dots ,U_r)\Vert _n$$ for standard uniformly distributed random variables.

Quasi-shuffle algebras have been a useful tool in studying multiple zeta values and related quantities, including multiple polylogarithms, finite multiple harmonic sums, and $q$-multiple zeta values. Here we show that two ideas previously considered only for multiple zeta values, the interpolated product of S. Yamamoto and the symmetric sum theorem, can be generalized to any quasi-shuffle algebra.

For positive integers i_1,...,i_k with i_1 > 1, we define the multiple t-value t(i_1,...,i_k) as the sum of those terms in the usual infinite series for the multiple zeta value ζ(i_1,...,i_k) with odd denominators. Like the multiple zeta values, the multiple t-values can be multiplied according to the rules of the harmonic algebra. Using this fact, we obtain explicit formulas for multiple t-values of repeated arguments analogous to those known for multiple zeta values. Multiple t-values can be written as rational linear combinations of the alternating or "colored" multiple zeta values. Using known results for colored multiple zeta values, we obtain tables of multiple t-values through weight 7, suggesting some interesting conjectures, including one that the dimension of the rational vector space generated by weight-n multiple t-values has dimension equal to the nth Fibonacci number. We express the generating function of the height one multiple t-values t(n,1,...,1) in terms of a generalized hypergeometric function. We also define alternating multiple t-values and prove some results about them.

The zeros of the digamma function are known to be simple and real, but up to now few identities involving them have appeared in the literature. By establishing a Weierstrass infinite product for a particular regularization of the digamma function, we are able to find interesting formulas for the sums of the nth powers of the reciprocals of its zeros, for n >= 2. We make a parallel study of the zeros of the logarithmic derivative of the Barnes G-function. We also compare asymptotic estimates of the zeros of the digamma function and those of its Barnes G-function analogue.

For [Formula: see text], let [Formula: see text] be the sum of all multiple zeta values with even arguments whose weight is [Formula: see text] and whose depth is [Formula: see text]. Of course [Formula: see text] is the value [Formula: see text] of the Riemann zeta function at [Formula: see text], and it is well known that [Formula: see text]. Recently Shen and Cai gave formulas for [Formula: see text] and [Formula: see text] in terms of [Formula: see text] and [Formula: see text]. We give two formulas for [Formula: see text], both valid for arbitrary [Formula: see text], one of which generalizes the Shen–Cai results; by comparing the two we obtain a Bernoulli-number identity. We also give explicit generating functions for the numbers [Formula: see text] and for the analogous numbers [Formula: see text] defined using multiple zeta-star values of even arguments.

Recently G. Louchard obtained an asymptotic series $\sum_{j=0}^\infty\frac{I_j}{n^j}$ for the integral $\int_0^1[x^n+(1-x)^n]^{\frac1n}dx$ as $n\to\infty$, and computed $I_j$ for $j\le 5$ in terms of values of the Riemann zeta function. An interesting feature of the computation is that the $I_j$ are first obtained in terms of alternating multiple zeta values, but then everything except products of ordinary zeta values cancels out. We obtain similar formulas for $I_n$, $6\le n\le 9$, and conjecture a general formula for $I_n$ in terms of alternating multiple zeta values. We also conjecture that $I_n$ is a rational polynomial in the ordinary zeta values.

We show how infinite series of a certain type involving generalized harmonic numbers can be computed using a knowledge of symmetric functions and multiple zeta values. In particular, we prove and generalize some identities recently conjectured by Choi, and give several more families of identities of a similar nature.

A recent paper of Furdui and Vălean proves some results about sums of products of “tails” of the series for the Riemann zeta function. We show how such results can be proved with weaker hypotheses using multiple zeta values, and also show how they can be generalized to products of three or more such tails.

A poset can be regarded as a category in which there is at most one morphism between objects, and such that at most one of the sets Hom(c,c′) and Hom(c′,c) is nonempty for distinct objects c, c′. Retaining the latter axiom but allowing for more than one morphism between objects gives a sort of generalized poset in which there are multiplicities attached to the covering relations, and possibly nontrivial automorphism groups of objects. An updown category is such a category with an appropriate grading on objects. In this paper we give a precise definition of updown categories and develop a theory for them, including two types of associated generating functions and a notion of universal covers. We give a detailed account of ten examples, including updown categories of sets, graphs, necklaces, integer partitions, integer compositions, planar rooted trees, and rooted trees.

We present a number of results about (finite) multiple harmonic sums modulo a prime, which provide interesting parallels to known results about multiple zeta values (i.e. infinite multiple harmonic series). In particular, we prove a 'duality' result for mod p harmonic sums similar to (but distinct from) that for multiple zeta values. We also exploit the Hopf algebra structure of the quasi-symmetric functions to perform calculations with multiple harmonic sums mod p, and obtain, for each weight n through nine, a set of generators for the space of weight-n multiple harmonic sums mod p. When combined with recent work, the results of this paper offer significant evidence that the number of quantities needed to generate the weight-n multiple harmonic sums mod p is the nth Padovan number (OEIS sequence A000931).

The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of planar rooted trees are related to each other and to the well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair of commutative diagrams. We show how this point of view can simplify computations in the Connes Kreimer Hopf algebra and its dual, particularly for combinatorial Dyson-Schwinger equations.

Recent work in perturbative quantum field theory has led to much study of the Connes-Kreimer Hopf algebra. Its (graded) dual, the Grossman-Larson Hopf algebra of rooted trees, had already been studied by algebraists. L. Foissy introduced a noncommutative version of the Connes-Kreimer Hopf algebra, which turns out to be self-dual. Using some homomorphisms defined by the author and W. Zhao, we describe a commutative diagram that relates the aforementioned Hopf algebras to each other and to the Hopf algebras of symmetric functions, noncommutative symmetric functions, and quasi-symmetric functions.

We define a homomorphism $\zeta$ from the algebra of quasi-symmetric functions to the reals which involves the Euler constant and multiple zeta values. Besides advancing the study of multiple zeta values, the homomorphism $\zeta$ appears in connection with two Hirzebruch genera of almost complex manifolds: the $\Gamma$-genus (related to mirror symmetry) and the $\hat{\Gamma}$-genus (related to an $S^1$-equivariant Euler class). We decompose $\zeta$ into its even and odd factors in the sense of Aguiar, Bergeron, and Sottille, and demonstrate the usefulness of this decomposition in computing $\zeta$ on the subalgebra of symmetric functions (which suffices for computations of the $\Gamma$- and $\hat{\Gamma}$-genera).