Bell polynomials and differential equations of Freud-type polynomials

Higher order bell polynomials and the relevant integer sequences

“…We will limit ourselves to present here the very first of them. Putting, for shortness: x = ϕ (1) , y = ϕ (2) , z = ϕ (3) , u = ϕ (4) , v = ϕ (5) , and denoting by indices the order of derivatives, we have found:…”

“…According to the above reference we have found, using the recurrence relation (19)- (20) and by means of the computer algebra program Mathematica c , the following sequences for the higher order Bell numbers b [2] n , b [3] n , b [4] n , b [5] n , (n = 1, 2, . .…”

“…Several applications of the classical Bell polynomials have been considered in [5], [7], [9] (in connection with [22]), [13], [14].…”

Bell polynomials and modified Bessel functions of half-integral order

“…They have been also applied in many different situations, such as the Blissard problem (see [3], p. 46), the representation of Lucas polynomials of the first and second kind [4,5], the construction of recurrence relations for a class of Freud-type polynomials [6], etc. However, in our opinion, the most important of their applications is connected with the possibility to represent, by using such a powerful tool, the symmetric function of a countable set of numbers.…”

Laguerre‐type Bell polynomials

“…They are often used in combinatorial analysis [20], and even in statistics [14], although without explicit references. Moreover these polynomials have been applied even in many other contexts, such as the Blissard problem (see [20, page 46]), the representation of Lucas polynomials of the first and second kinds [4,9], the representation formulas of Newton sum rules for polynomials' zeros [12,13], the recurrence relations for a class of Freud-type polynomials [3], the representation of symmetric functions of a countable set of numbers, generalizing the classical algebraic Newton-Girard formulas [15]. Consequently they were also used [6] in order to find reduction formulas for the orthogonal invariants of a strictly positive compact operator, deriving in a simple way the so-called Robert formulas [21].…”