Abstract. We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.

We study the moments of the distance traveled by a walk in the plane with unit steps in random directions. While this historically interesting random walk is well understood from a modern probabilistic point of view, our own interest is in determining explicit closed forms for the moment functions and their arithmetic values at integers when only a small number of steps is taken. As a consequence of a more general evaluation, a closed form is obtained for the average distance traveled in three steps. This evaluation, as well as its proof, rely on explicit combinatorial properties, such as recurrence equations of the even moments (which are lifted to functional equations). The corresponding general combinatorial and analytic features are collected and made explicit in the case of 3 and 4 steps. Explicit hypergeometric expressions are given for the moments of a 3-step and 4-step walk and a general conjecture for even length walks is made.

S. Ramanujan introduced a technique, known as Ramanujan's Master Theorem, which provides an analytic expression for the Mellin transform of a function. The main identity of this theorem involves the extrapolation of the sequence of coefficients of the integrand, defined originally as a function on N to C. The history and proof of this result are reviewed. Applications to the evaluation of a variety of definite integrals is presented.

A special case of an elegant result due to Anderson proves that the number of (s, s + 1)-core partitions is finite and is given by the Catalan number Cs. Amdeberhan recently conjectured that the number of (s, s + 1)-core partitions into distinct parts equals the Fibonacci number Fs+1. We prove this conjecture by enumerating, more generally, (s, ds − 1)-core partitions into distinct parts. We do this by relating them to certain tuples of nested twin-free sets.As a by-product of our results, we obtain a bijection between partitions into distinct parts and partitions into odd parts, which preserves the perimeter (that is, the largest part plus the number of parts minus 1). This simple but curious analog of Euler's theorem appears to be missing from the literature on partitions. *

We investigate the moments of 3-step and 4-step uniform random walk in the plane. In particular, we further analyse a formula conjectured in [BNSW09] expressing 4-step moments in terms of 3-step moments. Diverse related results including hypergeometric and elliptic closed forms for W 4 (±1) are given and two new conjectures are recorded.

We study generalized log-sine integrals at special values. At π and multiples thereof explicit evaluations are obtained in terms of Nielsen polylogarithms at ±1. For general arguments we present algorithmic evaluations involving Nielsen polylogarithms at related arguments. In particular, we consider log-sine integrals at π/3 which evaluate in terms of polylogarithms at the sixth root of unity. An implementation of our results for the computer algebra systems Mathematica and SAGE is provided.

We provide evaluations of several recently studied higher and multiple Mahler measures using logsine integrals. This is complemented with an analysis of generating functions and identities for logsine integrals which allows the evaluations to be expressed in terms of zeta values or more general polylogarithmic terms. The machinery developed is then applied to evaluation of further families of multiple Mahler measures.2010 Mathematics subject classification: primary 33E20; secondary 33F10, 11R06.

We prove two-term supercongruences for generalizations of recently discovered sporadic sequences of Cooper. We also discuss recent progress and future directions concerning other types of supercongruences.