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: $$\begin{cases} \begin{array}{ccccccccc} T_{0}\left(\theta\right) & = & 1\\ T_{1}\left(\theta\right) & = & \theta\\ T_{2}\left(\theta\right) & = & \theta^{2} & + & \theta\\ T_{3}\left(\theta\right) & = & \theta^{3} & + & 3\theta^{2} & + & \theta\\ T_{4}\left(\theta\right) & = & \theta^{4} & + & 6\theta^{3} & + & 7\theta^{2} & + & \theta. \end{array}\end{cases}. $$
TP (x, theta)TP gives the Touchard polynomials after specifying parameteric value.
A vector of (non-negative integer) discrete values.
A vector of (non-negative integer) values.
Muhammad Imran and M.H. Tahir.
R implementation and documentation: Muhammad Imran imranshakoor84@yahoo.com and M.H. Tahir <mht@iub.edu.pk>.
The function allows to provide the Touchard polynomials.
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.