Table 1 Values of A, B, C, D, and E for each of the polynomial set Tchebichef \({{t}}_{{n}}\left({x};{ N}\right)\), Krawtchouk \({k}_{n}\left(x;P,N\right)\), Charlier \({c}_{n}\left(x\right)\), Hahn \({h}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)\) and Meixner \({m}_{n}^{\left(a,b\right)}\left(x\right)\)
From: An efficient computation of discrete orthogonal moments for bio-signals reconstruction
| Â | \({{\varvec{t}}}_{{\varvec{n}}}\left({\varvec{x}};{\varvec{N}}\right)\) | \({{\varvec{k}}}_{{\varvec{n}}}\left({\varvec{x}};{\varvec{P}},{\varvec{N}}\right)\) | \({{\varvec{c}}}_{{\varvec{n}}}\left({\varvec{x}}\right)\) | \({{\varvec{h}}}_{{\varvec{n}}}^{(\boldsymbol{\alpha },{\varvec{\beta}})}\left({\varvec{x}}\right)\) | \({{\varvec{m}}}_{{\varvec{n}}}^{({\varvec{a}},{\varvec{b}})}\left({\varvec{x}}\right)\) |
|---|---|---|---|---|---|
A | \(\frac{n}{2(2n-1)}\) | \(n\) | \(-{{\varvec{a}}}_{1}\) | \(\begin{array}{c}\frac{n}{(\alpha +\beta +2n-1)}\\ \times \frac{(\alpha +\beta +n)}{(\alpha +\beta +2n)}\end{array}\) | \(\frac{{\varvec{b}}}{{\varvec{b}}-1}\) |
B | \(x-\frac{N-1}{2}\) | \(x-n+1-p(N-2n+2)\) | \({\varvec{x}}-{\varvec{n}}+1-{{\varvec{a}}}_{1}\) | \(\begin{array}{c}x-\frac{\alpha -\beta +2N-2}{4}\\ -\frac{\left({\beta }^{2}-{\alpha }^{2}\right)(\alpha +\beta +2N)}{4(\alpha +\beta +2n-2)(\alpha +\beta +2n)}\end{array}\) | \(\frac{{\varvec{x}}-{\varvec{x}}{\varvec{b}}-{\varvec{n}}+1-{\varvec{b}}{\varvec{n}}+{\varvec{b}}-{\varvec{a}}{\varvec{b}}}{1-{\varvec{b}}}\) |
C | \(-\frac{(n-1)\left[{N}^{2}-(n-1{)}^{2}\right]}{2(2n-1)}\) | \(-p(1-p)(N-n+2)\) | \({\varvec{n}}-1\) | \(\begin{array}{c}-\frac{(\alpha +n-1)(\beta +n-1)}{(\alpha +\beta +2n-2)}\\ \times \frac{(\alpha +\beta +N+n-1)(N-n+1)}{(\alpha +\beta +2n-1)}\end{array}\) | \(\frac{({\varvec{n}}-1)({\varvec{n}}-2+{\varvec{a}})}{1-{\varvec{b}}}\) |
D | \(\sqrt{\frac{(2n+1)}{\left({N}^{2}-{n}^{2}\right)(2n-1)}}\) | \(\sqrt{\frac{n}{p(1-p)(N-n+1)}}\) | \(\sqrt{\frac{{{\varvec{a}}}_{1}}{{\varvec{n}}}}\) | \(\sqrt{\frac{n(\alpha +\beta +n)(\alpha +\beta +2n+1)}{(\alpha +n)(\beta +n)(\alpha +\beta +n+N)(N-n)(\alpha +\beta +2n-1)}}\) | \(\sqrt{\frac{{\varvec{b}}}{{\varvec{n}}({\varvec{a}}+{\varvec{n}}-1)}}\) |
E | \(\sqrt{\frac{2n+1}{\left({N}^{2}-{n}^{2}\right)\left[{N}^{2}-(n-1{)}^{2}\right](2n-3)}}\) | \(\sqrt{\frac{n(n-1)}{(p(1-p){)}^{2}(N-n+2)(N-n+1)}}\) | \(\sqrt{\frac{{{\varvec{a}}}_{1}^{2}}{{\varvec{n}}({\varvec{n}}-1)}}\) | \(\begin{array}{c}\sqrt{\frac{n(n-1)(\alpha +\beta +n)}{(\alpha +n)(\alpha +n-1)(\beta +n)(\beta +n-1)(N-n+1)(N-n)}}\\ \times \sqrt{\frac{(\alpha +\beta +n-1)(\alpha +\beta +2n+1)}{(\alpha +\beta +2n-3)(\alpha +\beta +n+N)(\alpha +\beta +n+N-1)}}\end{array}\) | \(\sqrt{\frac{{{\varvec{b}}}^{2}}{{\varvec{n}}({\varvec{n}}-1)({\varvec{a}}+{\varvec{n}}-2)({\varvec{a}}+{\varvec{n}}-1)}}\) |