function, the generalized gamma function, the modified Bessel function of first and second kinds, ${n}_{\left(n\right)}$ the Pochhammer symbol $U\left(n,2n,x\right)={2}^{-n}{\theta }_{-n}\left(\frac{x}{2}\right)$ (115); $\theta$ is a (reverse) Bessel polynomial [22] when n is a non-negative integer ${a}_{n}={\sum }_{n=1}^{n=\infty }{\sum }_{k=0}^{n}{a}_{n,k}$ (117); with ${a}_{n,k}$ OIES A001497 Generalized or associated Laguerre $\begin{array}{l}\frac{\partial {u}^{2}}{\partial {x}^{2}}+\left(a+1-x\right)\frac{\partial u}{\partial x}\\ +nu=0\end{array}$ (119); $u={L}_{n}^{a}\left(x\right)=\frac{{\left(-1\right)}^{n}}{n!}U\left(-n,a+1,x\right)$ Special cases: $\begin{array}{l}{L}_{n}^{a}\left(x\right)=\frac{{\left(-1\right)}^{n}}{n!}U\left(-n,-n+1,x\right)\\ ~\frac{{\left(-1\right)}^{n}}{n!}{x}^{n}\end{array}$ Step 1: Laplace-Borel Transformation of ${\sum }_{n=1}^{n=\infty }\frac{{\left(-1\right)}^{n}}{n!}{x}^{n}={\sum }_{n=1}^{n=\infty }{\left(-x\right)}^{n}$ Step 2: ${D}_{s}{\sum }_{n=1}^{n=\infty }{\left(-x\right)}^{n}=-{2}^{-s}\left({2}^{s}-2\right)\varsigma \left(s\right)$ $\text{EGF}\equiv {f}_{3}\left(-x\right)={\sum }_{n=1}^{n=\infty }\frac{{\left(-1\right)}^{n}}{n!}{x}^{n}$ $\begin{array}{l}{D}_{s}\mathcal{L}{f}_{3}\left(-x\right)={D}_{s}\mathcal{L}{\sum }_{n}{L}_{n}^{a}\left(x\right)\\ =-{2}^{-s}\left({2}^{s}-2\right)\varsigma \left(s\right)\end{array}$ (120) Bessel $\begin{array}{l}{x}^{2}\frac{\partial {u}^{2}}{\partial {x}^{2}}+x\left(2p+1\right)\frac{\partial u}{\partial x}\\ +\left({\beta }^{2}+{x}^{2}{a}^{2r}\right)y=0\end{array}$ Math_340#, $\beta =0$ , $r=1$ $y={x}^{-p}\left[{c}_{1}{J}_{\frac{q}{r}}\left(\frac{a}{r}{x}^{r}\right)+{c}_{2}{Y}_{\frac{q}{r}}\left(\frac{a}{r}{x}^{r}\right)\right]$ ; ${J}_{\frac{1}{2}}\left(kx\right)=\sqrt{\frac{2}{\pi }}\frac{\mathrm{sin}\left(kx\right)}{x}$ ${Y}_{\frac{1}{2}}\left(kx\right)=\sqrt{\frac{2}{\pi }}\frac{\mathrm{cos}\left(kx\right)}{x}$ J and Y the Bessel function of first and second kinds. $\begin{array}{l}{\sum }_{k=1}^{k=\infty }{D}_{s}\left({Y}_{\frac{1}{2}}\left(2\pi kx\right)+i{J}_{\frac{1}{2}}\left(2\pi kx\right)\right)\\ =\frac{1}{\Gamma \left(s\right)}\sqrt{\frac{1}{2\pi }}{\mathcal{M}}_{k}^{\xi }\left[{U}_{k}^{\xi ,r}{\sum }_{k=1}^{k=\infty }{\text{e}}^{2\pi ikx}\right]\end{array}$ $\begin{array}{l}\frac{1}{\Gamma \left(s\right)}\sqrt{\frac{1}{2\pi }}{\mathcal{M}}_{k}^{\xi }\left[{\text{e}}^{-2\pi ikx}{\sum }_{k=1}^{k=\infty }{\text{e}}^{2\pi ikx}\right]\\ =\frac{1}{\Gamma \left(s\right)}\sqrt{\frac{1}{2\pi }}{\sum }_{k=1}^{k=\infty }{k}^{s-1}dk\\ =\frac{1}{s\Gamma \left(s\right)}\sqrt{\frac{1}{2\pi }}\zeta \left(-s\right)\end{array}$ (121) $s=2\pi i\beta -1$ ; $\xi =0$ ; $a={\text{e}}^{-x}$ ; $r=-2$ , $Z\left(f\right)={\text{e}}^{2\pi ikf},f>0$ $\begin{array}{l}{\sum }_{k=1}^{k=\infty }{D}_{s}\left({Y}_{\frac{1}{2}}\left(2\pi kx\right)+i{J}_{\frac{1}{2}}\left(2\pi kx\right)\right)\\ =\frac{1}{s\Gamma \left(s\right)}\sqrt{\frac{1}{2\pi }}\zeta \left(-s\right)\end{array}$