Touchard polynomials {countDM}

Touchard polynomials

Description

The function allows to compuate the Touchard polynomial. It is mathematically defined by

T_{x}\left(\theta\right)=\frac{1}{e^{\theta}}\sum_{k=0}^{\infty}\frac{k^{x}}{k!}\theta^{k}.

The first few Touchard polynomials are as follows:

\begin{cases} \begin{array}{ccccccccc} T_{0}\left(\theta\right) & = & 1\\ T_{1}\left(\theta\right) & = & \theta\\ T_{2}\left(\theta\right) & = & \theta^{2} & + & \theta\\ T_{3}\left(\theta\right) & = & \theta^{3} & + & 3\theta^{2} & + & \theta\\ T_{4}\left(\theta\right) & = & \theta^{4} & + & 6\theta^{3} & + & 7\theta^{2} & + & \theta. \end{array}\end{cases}.

Usage

TP (x, theta)

Arguments

x

A vector of (non-negative integer) discrete values.

theta

A vector of (non-negative integer) values.

Details

The function allows to provide the Touchard polynomials.

Value

TP gives the Touchard polynomials after specifying parameteric value.

Author(s)

Muhammad Imran and M.H. Tahir.

R implementation and documentation: Muhammad Imran [e-mail] and M.H. Tahir <[e-mail]>.

References

Castellares, F., Lemonte, A. J., & Moreno–Arenas, G. (2020). On the two-parameter Bell–Touchard discrete distribution. Communications in Statistics-Theory and Methods, 49(19), 4834-4852.

Examples

TP (2,3)