We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order geometric polynomials, particularly for p-Bernoulli numbers.

Recurrences and Congruences for Higher order Geometric Polynomials and Related Numbers

In this paper, by the umbral calculus method, we give a remarkable congruence involving Appell polynomials. Some applications on derangement polynomials are also presented.

A remarkable congruence involving Appell polynomials

Abstract The paper investigates the asymptotic behavior of the geometric polynomials, when the polynomial degree tends to infinity. Using the contour integration technique, we obtain an asymptotic formula, given explicitly in terms of the polynomial degree and variable. This type of asymptotics will be applied to derive limit theorems for combinatorial numbers.

/pdf/the-asymptotics-of-the-geometric-polynomials-7bzaiv6z.pdf

The Asymptotics of the Geometric Polynomials

Semiorthogonality of Geometric Polynomials

We obtain new recurrence relations, an explicit formula, and convolution identities for higher-order geometric polynomials. These relations generalize known results for geometric polynomials and lead to congruences for higher-order geometric polynomials and, in particular, for p-Bernoulli numbers. 

/pdf/recurrences-and-congruences-for-higher-order-geometric-3s6ptoxd.pdf

Recurrences and Congruences for Higher-Order Geometric Polynomials and Related Numbers

/pdf/identities-and-congruences-involving-the-geometric-1011zj6vkj.pdf

Identities and congruences involving the geometric polynomials

In this papier, by the classical umbral calculus method, we establish identities involving the Appell polynomials and extend some existing identities.

Some identities involving Appell polynomials

In this paper, by the classical umbral calculus method, we establishidentities involving the Appell polynomials and extend some existing identities.Mathematics Subject Classication (2010): 05A40, 11B68, 70H03.Key words: Classical umbral calculus, Appell polynomials, binomial sequences, identities.

By counting on the multiset partitions, we give in this paper a combinatorial interpretation of a class of the generalized Stirling numbers generalizing the Stirling, r-Stirling, Jacobi–Stirling, r-Jacobi–Stirling and two classes of p-Stirling numbers of the second kind.

The multiset partitions and the generalized Stirling numbers

In this paper, we investigate the umbral representation of the Fubini polynomials $F_{x}^{n}:=F_{n}(x)$ to derive some properties involving these polynomials. For any prime number $p$ and any polynomial $f$ with integer coefficients, we show $(f(F_{x}))^{p}\equiv f(F_{x})$ and we give other curious congruences.

Said Taharbouchet

Papers

Identities and congruences involving the geometric polynomials

Some identities involving Appell polynomials

Some identities involving Appell polynomials

The multiset partitions and the generalized Stirling numbers

Identities and congruences involving the Fubini polynomials