Some identities of higher-order Bernoulli, Euler and Hermite polynomials arising from umbral calculus

In this paper, we study umbral calculus to have alternative ways of obtaining our results. That is, we derive some interesting identities of the higher-order Bernoulli, Euler and Hermite polynomials arising from umbral calculus to have alternative ways.Comment: 10 page

On some applications of a symbolic representation of non-centered L\'evy processes

By using a symbolic technique known in the literature as the classical umbral calculus, we characterize two classes of polynomials related to L\'evy processes: the Kailath-Segall and the time-space harmonic polynomials. We provide the Kailath-Segall formula in terms of cumulants and we recover simple closed-forms for several families of polynomials with respect to not centered L\'evy processes, such as the Hermite polynomials with the Brownian motion, the Poisson-Charlier polynomials with the Poisson processes, the actuarial polynomials with the Gamma processes, the first kind Meixner polynomials with the Pascal processes, the Bernoulli, Euler and Krawtchuk polynomials with suitable random walks

Some aspects of Fibonacci polynomial congruences

This paper formulates a definition of Fibonacci polynomials which is slightly different from the traditional definitions, but which is related to the classical polynomials of Bernoulli, Euler and Hermite. Some related congruence properties are developed and some unanswered questions are outlined. Keywords: Congruences, recurrence relations, Fibonacci sequence, Lucas sequences, umbral calculus