${\stackrel{˜}{p}}_{0}\left(x\right)$ $\sqrt{\frac{\left(2s+1\right)\left(p\left(s\right)\right)}{{d}_{0}^{2}}}$ ${\stackrel{˜}{p}}_{1}\left(x\right)$ $\frac{{p}_{1}\left(s\right)-p\left(s-1\right)}{x\left(s+\frac{1}{2}\right)-x\left(s-\frac{1}{2}\right)}\frac{1}{p\left(s\right)}\ast \sqrt{\frac{{\text{e}}^{-\mu }{\mu }^{x+1}}{x!}}$ ${A}_{n}$ $\frac{n\left(\alpha +\beta +n\right)}{\left(\alpha +\beta +2n-1\right)\left(\alpha +\beta +2n\right)}$ ${B}_{n}$ $\begin{array}{l}x-\frac{{a}^{2}+{b}^{2}+{\left(a-\beta \right)}^{2}{\left(b+\alpha \right)}^{2}-2\left(\alpha +\beta +2n-2\right)\left(\alpha +\beta -2\right)}{4}\\ +\frac{\left(\alpha +\beta +2n-2\right)-\left(\alpha +\beta +2n\right)}{8}-\frac{\left({\beta }^{2}-{\alpha }^{2}\right)\left[{\left(b+\frac{\alpha }{2}\right)}^{2}-{\left(a-\beta /2\right)}^{2}\right]}{2\left(\alpha +\beta +2n-2\right)\left(\alpha +\beta +2n\right)}\end{array}$ ${C}_{n}$ $\begin{array}{l}-\frac{\left(\alpha +n-1\right)\left(\beta +n-1\right)}{\left(\alpha +\beta +2n-2\right)\left(\alpha +\beta +2n\right)}\ast \left[{\left(a+b+\frac{\alpha -\beta }{2}\right)}^{2}{\left(n-1+\frac{\alpha +\beta }{2}\right)}^{2}\right]\\ \ast \left[{\left(b+a+\frac{\alpha +\beta }{2}\right)}^{2}{\left(n-1+\frac{\alpha +\beta }{2}\right)}^{2}\right]\end{array}$ ${d}_{n}$ $\frac{\text{Γ}\left(\alpha +n+1\right)\text{Γ}\left(\beta +n+1\right)\text{Γ}\left(b-a+\alpha +\beta +n+1\right)\text{Γ}\left(a+b+\alpha +n+1\right)}{\left(\alpha +\beta +2n+1\right)n!\left(b-a-n-1\right)\text{Γ}\left(\alpha +\beta +n+1\right)\text{Γ}\left(a+b-\beta -n\right)}$ ${\rho }_{n}\left(s\right)$ $\frac{\text{Γ}\left(\alpha +s+n+1\right)\text{Γ}\left(s-a+\beta +n+1\right)\text{Γ}\left(N+\alpha -s\right)\text{Γ}\left(N+\alpha +s+n+1\right)}{\text{Γ}\left(a-\beta +s+1\right)\text{Γ}\left(s-a+1\right)\text{Γ}\left(N-s-n\right)\text{Γ}\left(N+s+1\right)}$