Orthogonal Polynomials - 18.23 Hahn Class: Generating Functions

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
18.23.E5	${\displaystyle{\displaystyle e^{z}\left(1-\frac{z}{a}\right)^{x}=\sum_{n=0}^{\infty}\frac{C_{n}\left(x;a\right)}{n!}z^{n}}}$ e^{z}\left(1-\frac{z}{a}\right)^{x} = \sum_{n=0}^{\infty}\frac{\CharlierpolyC{n}@{x}{a}}{n!}z^{n}		Error	Exp[x + y*I]*(1 -Divide[x + y*I,a])^(x) == Sum[Divide[HypergeometricPFQ[{-(n), -(x)}, {}, -Divide[1,a]],(n)!]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Missing Macro Error	-	-
18.23.E6	${\displaystyle{\displaystyle{{}_{1}F_{1}}\left({a+\mathrm{i}x\atop 2\Re a};-\mathrm{i}z\right){{}_{1}F_{1}}\left({\overline{b}-\mathrm{i}x\atop 2\Re b};\mathrm{i}z\right)=\sum_{n=0}^{\infty}\frac{p_{n}\left(x;a,b,\overline{a},\overline{b}\right)}{{\left(2\Re a\right)_{n}}{\left(2\Re b\right)_{n}}}z^{n}}}$ \genhyperF{1}{1}@@{a+\iunit x}{2\realpart@@{a}}{-\iunit z}\genhyperF{1}{1}@@{\conj{b}-\iunit x}{2\realpart@@{b}}{\iunit z} = \sum_{n=0}^{\infty}\frac{\contHahnpolyp{n}@{x}{a}{b}{\conj{a}}{\conj{b}}}{\Pochhammersym{2\realpart@@{a}}{n}\Pochhammersym{2\realpart@@{b}}{n}}z^{n}		Error	HypergeometricPFQ[{a + I*x}, {2*Re[a]}, - I*(x + y*I)]*HypergeometricPFQ[{Conjugate[b]- I*x}, {2*Re[b]}, I*(x + y*I)] == Sum[Divide[I^(n)*Divide[Pochhammer[a + Conjugate[a], n]*Pochhammer[a + Conjugate[b], n], (n)!] * HypergeometricPFQ[{-(n), n + 2*Re[a + b] - 1, a + I*(x)}, {a + Conjugate[a], a + Conjugate[b]}, 1],Pochhammer[2*Re[a], n]*Pochhammer[2*Re[b], n]]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Missing Macro Error	-	-