eBellPol {kStatistics} | R Documentation |
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 elvira.dinardo@unito.it,
Giuseppe Guarino giuseppe.guarino@rete.basilicata.it
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
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)