e_eBellPol: Evaluation of exponential Bell polynomials

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

Evaluation of exponential Bell polynomials

Description

The function evaluates a complete or a partial exponential Bell polynomial (output of the eBellPol function) when its variables are substituted with numerical values.

Usage

e_eBellPol(n=1,m=0,v=c(rep(1,n)))

Arguments

n	integer, the degree of the polynomial
m	integer, the fixed degree of each monomial in the polynomial
v	vector, the numerical values in place of the variables of the polynomial

Details

The eBellPol function generates a complete or a partial exponential Bell polynomial in the variables y[1],..., y[n-m+1]. The e_eBellPol function computes the value assumed by this polynomial when its variables are substituted with numerical values.

Value

numerical value

the value assumed by the polynomial.

Warning

By default, the function returns the Stirling numbers of second kind.

Note

This function calls the eBellPol 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

eBellPol

Examples

# Return S(5,3) = 25 (where S=Stirling number of second kind)  
e_eBellPol(5,3) 
#
# OR (same output) 
#
e_eBellPol(5,3,c(1,1,1,1,1))

# Return  B5=52 (where B5 is the 5-th Bell number)   
e_eBellPol(5) 
#
# OR (same output) 
#  
e_eBellPol(5,0) 
#
# OR (same output) 
# 
e_eBellPol(5,0,c(1,1,1,1,1))

# Return  s(5,3) = 35 (where s=Stirling number of first kind)  
e_eBellPol(5,3,c(1,-1,2,-6,24))

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