oBellPol: Ordinary Bell polynomials

In kStatistics: Unbiased Estimators for Cumulant Products and Faa Di Bruno's Formula

Ordinary Bell polynomials

Description

The function generates a complete or a partial ordinary Bell polynomial.

Usage

oBellPol(n = 1, m = 0)

Arguments

n	integer, the degree of the polynomial
m	integer, the fixed degree of each monomial in the polynomial

Details

Faa di Bruno's formula gives the coefficients of the exponential formal power series obtained from the composition f[g()] of the exponential formal power series f with g. The partial ordinary Bell polynomials B[n,m] can be expressed in the terms of the partial exponential Bell polynomials B(n,m)(y[1],...,y[n-m+1]) using the following formula:

B[n,m](y[1],...,y[n-m+1])=k!/n!B(n,m)(y[1],...,y[n-m+1]).

The complete ordinary Bell polynomials are given by B[n]=B[n,1]+B[n,2]+...B[n,n], where B[n,m] is the partial ordinary Bell polynomial of order (n,m) for m from 1 to n.

Value

string

the expression of the polynomial

Warning

The value of the first parameter is the same as the MFB function in the univariate with univariate composition.

Note

This function calls the MFB function in the kStatistics package.

Author(s)

Elvira Di Nardo [e-mail],
Giuseppe Guarino [e-mail]

References

C.A. Charalambides (2002) Enumerative Combinatoris, Chapman & Haii/CRC.

E. Di Nardo, G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286-6295. (download from https://arxiv.org/abs/1012.6008)

See Also

MFB

Examples

# Return the complete ordinary Bell Polynomial for n=5, that is 
# (y1^5) + 20(y1^3)(y2) + 30(y1)(y2^2) + 60(y1^2)(y3) + 120(y2)(y3) + 120(y1)(y4) + 120(y5)
oBellPol(5)
#
# OR (same output)
#
oBellPol(5,0)

# Return the partial ordinary Bell polynomial for n=5 and m=3, that is 
# 30(y1)(y2^2) + 60(y1^2)(y3)
oBellPol(5,3)

kStatistics documentation built on June 8, 2022, 5:05 p.m.