Algorithm 1 The algorithm used to compute the Charlier polynomials

Input:N = Polynomial size, p = Polynomial parameter.

Output:$\widehat{C}$ = Charlier polynomials.

1: Initialize $\widehat{C}$ with a size of $N\times N$

2: ${\widehat{C}}_p(0;p)\leftarrow {\left(e^{-\left(log\Gamma \left(p\right)+p+(1-p)log\left(p\right)\right)}\right)}^{0.5}$ {Compute initial value}.

3: $n\leftarrow [p-2,p-1,\cdots ,0]$ {Set the range of n}.

4:for i in range n do

5:   ${\widehat{C}}_{i-1}(0;p)\leftarrow \sqrt{\frac{i}{p}}{\widehat{C}}_i(0;p)$

6: end for

7: $n\leftarrow [n=p+1,p+2,\cdots ,N-1,]$ {Set the range of n}.

8: for i in range n do

9:   ${\widehat{C}}_i(0;p)\leftarrow \sqrt{\frac{p}{n}}{\widehat{C}}_{i-1}(0;p)$

10:end for

11:for $n=0$ to $N-1$ do

12:   ${\widehat{C}}_n(1;p)\leftarrow \frac{(p-n)}{\sqrt{n}}{\widehat{C}}_n\left(0;p\right)$

13:end for

14:{Compute the coefficients in “Part 1”}

15:for $x=1$ to $N-1$ do

16:    for $n=x$ to $N-1$ do

17:      $A\leftarrow \frac{a-n+x}{\sqrt{a(x+1)}}$

18:      $B\leftarrow -\sqrt{\frac{x}{(x+1)}}$

19:      ${\widehat{C}}_n\left(x+1;p\right)\leftarrow A{\widehat{C}}_n\left(x;p\right)+B{\widehat{C}}_n\left(x-1;p\right)$

20:    end for

21: end for

22: {Compute the coefficients in “Part 2”}

23:for $n=1$ to $N-1$ do

24:    for $x=0$ to $n-1$ do

25:       ${\widehat{C}}_n(x;p)\leftarrow {\widehat{C}}_x(n;p)$

26:    end for

27: end for

28: return {$\widehat{C}$ Note that $H_d$ in Equation (27) is equal to $\widehat{C}$.}