Probabilistic and Combinatorial Interpretations of the Bernoulli Symbol

Lin Jiu, Diane Yahui Shi

Published 2017-11-19Version 1

The Bernoulli symbol comes from umbral calculus, with simple evaluation rule that identifying super index, i.e., power, and lower index. A recent probabilistic interpretation allows us to view this evaluation rule as the expectation of the Bernoulli random variable, having hyperbolic secant square as its density on the whole real line. Besides further study on the corresponding moment problem, cumulants, and orthogonal polynomial sequence, classical results on continued fractions link Bernoulli polynomials to generalized Motzkin numbers, providing combinatorial interpretations.