LOOP DECOMPOSITIONS OF RANDOM WALKS AND NONTRIVIAL IDENTITIES OF BERNOULLI AND EULER POLYNOMIALS

Let a and b, with a < b, be two level sites of a general random walk. If we partition [a, b] into n arbitrary subintervals with endpoints a < a1 < a2 < · · · < an−1 < b, then the hitting time from a to b can also be decomposed by the hitting times between adjacent pairs of sites. As the walk moves back and forth between these endpoints, loops are created. Hence, we can express the generating function of the hitting time as a loop decomposition for an arbitrary number of consecutive sites. By applying this decomposition to a 1-dimensional reflected Brownian motion with equally distributed sites, we derive identities of Bernoulli and Euler polynomials in terms of their higher-order generalizations. Similar results from a 3-dimensional Bessel process are also obtained.

Duke Scholars

Author Lin Jiu DKU Faculty