eBellPol: Exponential Bell polynomials

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

Exponential Bell polynomials

Description

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

Usage

eBellPol(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 composition f[g()] obtained from the composition of the exponential formal power series f with g. Complete exponential Bell polynomials in the variables y[1],...,y[n] are generated by setting f[i]=1 and g[i]=y[i], for each i from 1 to n. Partial exponential Bell polynomials are polynomials in the variables y[1],...,y[n-m+1] with fixed degree m for each of the involved monomials. Partial exponential Bell polynomials are recovered from Faa di Bruno's formula by setting g[i]=y[i] for each i from 1 to n and f[i]=1 if i=m, f[i]=0 otherwise.

Value

string

the expression of the exponential Bell 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 (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from https://arxiv.org/pdf/math/0607623.pdf)

E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis. 52(11), 4909-4922. (download from https://arxiv.org/abs/0806.0129)

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 exponential Bell Polynomial for n=5, that is 
# (y1^5) + 10(y1^3)(y2) + 15(y1)(y2^2) + 10(y1^2)(y3) + 10(y2)(y3) + 5(y1)(y4) + (y5)
eBellPol(5) 
# OR (same output)
eBellPol(5,0)

# Return the partial exponential Bell Polynomial for n=5 and m=3, that is 
# 15(y1)(y2^2) + 10(y1^2)(y3)
eBellPol(5,3)

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