David M. Bradley

Multiple q-zeta values

We introduce a q-analog of the multiple harmonic series commonly referred to as multiple zeta values. The multiple q-zeta values satisfy a q-stuffle multiplica tion rule analogous to the stuffle multiplication rule arising from the series representation of ordinary multiple zeta values. Additionally, multiple q-zeta values can be viewed as special values of the multiple q-polylogarithm, which admits a multiple Jackson q-integral representation whose limiting case is the Drinfel’d simplex integral for the ordinary multiple polylogarithm when q = 1. The multiple Jackson q-integral representation for multiple q-zeta values leads to a second multiplication rule satisfied by them, referred to as a q-shuffle. Despite this, it appears that many numerical relations satisfied by ordinary multiple zeta values have no interesting q-extension. For example, a suitable q-analog of Broadhurst’s formula for ζ( {3, 1} n ), if one exists, is likely to be rather complicated. Nevertheless, we show that a number of infinite classes of relations, inc luding Hoffman’s partition identities, Ohno’s cyclic sum identities, Granville’s sum formula, Euler’s convolution formula, Ohno’s generalized duality relation, and the derivation relations of Ihara and Kaneko extend to multiple q-zeta values.

On the Distribution of the Sum of n Non-Identically Distributed Uniform Random Variables

The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the characteristic function, we derive explicit formulae for the distribution of the sum of n non-identically distributed uniform random variables in both the continuous and the discrete case. The results, though involved, have a certain elegance. As examples, we derive from our general formulae some special cases which have appeared in the literature.

THIRTY-TWO GOLDBACH VARIATIONS

We give thirty-two diverse proofs of a small mathematical gem — the fundamental Euler sum identity We also discuss various generalizations for multiple harmonic (Euler) sums and some of their many connections, thereby illustrating both the wide variety of techniques fruitfully used to study such sums and the attraction of their study.

The Algebra and Combinatorics of Shuffles andMultiple Zeta Values

The algebraic and combinatorial theory of shuffles, introduced by Chen and Ree, is further developed and applied to the study of multiple zeta values. In particular, we establish evaluations for certain sums of cyclically generated multiple zeta values. The boundary case of our result reduces to a former conjecture of Zagier.

A Class of Series Acceleration Formulae for Catalan's Constant

In this note, we develop transformation formulae and expansions for the log tangent integral, which are then used to derive series acceleration formulae for certain values of Dirichlet L-functions, such as Catalans constant. The formulae are characterized by the presence of an infinite series whose general term consists of a linear recurrence damped by the central binomial coefficient and a certain quadratic polynomial. Typically, the series can be expressed in closed form as a rational linear combination of Catalans constant and π times the logarithm of an algebraic unit.

Resolution of Some Open Problems Concerning Multiple Zeta Evaluations of Arbitrary Depth

We prove some new evaluations for multiple polylogarithms of arbitrary depth. The simplest of our results is a multiple zeta evaluation one order of complexity beyond the well-known Broadhurst–Zagier formula. Other results we provide settle three of the remaining outstanding conjectures of Borwein, Bradley, and Broadhurst. A complete treatment of a certain arbitrary depth class of periodic alternating unit Euler sums is also given.

Limiting behaviour of the mean residual life

In survival or reliability studies, the mean residual life or life expectancy is an important characteristic of the model. Here, we study the limiting behaviour of the mean residual life, and derive an asymptotic expansion which can be used to obtain a good approximation for large values of the time variable. The asymptotic expansion is valid for a quite general class of failure rate distributions—perhaps the largest class that can be expected given that the terms depend only on the failure rate and its derivatives.

Parametric Euler sum identities

We consider some parametrized classes of multiple sums first studied by Euler. Identities between meromorphic functions of one or more variables in many cases account for reduction formulae for these sums.

A q-ANALOG OF EULER'S DECOMPOSITION FORMULA FOR THE DOUBLE ZETA FUNCTION

The double zeta function was first studied by Euler in response to a letter from Goldbach in 1742. One of Eulers results for this function is a decomposition formula, which expresses the product of two values of the Riemann zeta function as a finite sum of double zeta values involving binomial coefficients. Here, we establish a q-analog of Eulers decomposition formula. More specifically, we show that Eulers decomposition formula can be extended to what might be referred to as a “double q-zeta function” in such a way that Eulers formula is recovered in the limit as q tends to 1.

Duality for finite multiple harmonic q-series

Abstract We define two finite q -analogs of certain multiple harmonic series with an arbitrary number of free parameters, and prove identities for these q -analogs, expressing them in terms of multiply nested sums involving the Gaussian binomial coefficients. Special cases of these identities—for example, with all parameters equal to 1—have occurred in the literature. The special case with only one parameter reduces to an identity for the divisor generating function, which has received some attention in connection with problems in sorting theory. The general case can be viewed as a duality result, reminiscent of the duality relation for the ordinary multiple zeta function.