De Moivre and Bell polynomials

The Pascal-de Moivre Triangles*

The coefficients of the Pascal triangle were generalized in 1756 by de Moivre [5]. Each row of a Pascal triangle contains a sequence of numbers that are the coefficients of the power series expansion for the binary expression (l + x)^. The de Moivre formula [2], [4], [5], [6] derives the coefficients of the power series for the generalized expansion of (1 + x + x + • • • + x^"^). Thus, for inte...

Laguerre-type Bell polynomials

We develop an extension of the classical Bell polynomials introducing the Laguerre-type version of this well-known mathematical tool. The Laguerre-type Bell polynomials are useful in order to compute the nth Laguerre-type derivatives of a composite function. Incidentally, we generalize a result considered by L. Carlitz in order to obtain explicit relationships between Bessel functions and gener...

Bell polynomials and Fibonacci numbers

Bell Polynomials of Iterated Functions

Physics can be modeled by PDEs or recursively as iterated functions. A method is presented to decompose iterated functions into Bell polynomials and then into the combinatoric structure Schroeder’s Fourth Problem. Consistency with complex dynamics is shown by deriving the classification of fixed points in complex dynamics.

Dyck paths and partial Bell polynomials

In the present paper, we consider two kinds of statistics “number of usegments” and “number of internal u-segments” in Dyck paths. More precisely, using Lagrange inversion formula we present the generating function for the number of Dyck paths according to semilength and our new statistics by the partial Bell polynomials, namely, ∑ D∈Dn ∏ i≥1 t αi(D) i = n ∑ i=1 1 (n− i+ 1)!n,i ( 1!t1, 2!t2, · ...