function, the generalized gamma function, the modified Bessel function of first and second kinds, $n_{(n)}$ the Pochhammer symbol

$U (n, 2 n, x) = 2^{- n} θ_{- n} (\frac{x}{2})$ (115); $θ$ 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

$\begin{array}{l} \frac{\partial u^{2}}{\partial x^{2}} + (a + 1 - x) \frac{\partial u}{\partial x} \\ + n u = 0 \end{array}$ (119);

$u = L_{n}^{a} (x) = \frac{{(- 1)}^{n}}{n!} U (- n, a + 1, x)$

Special cases: $\begin{array}{l} L_{n}^{a} (x) = \frac{{(- 1)}^{n}}{n!} U (- n, - n + 1, x) \\ ~ \frac{{(- 1)}^{n}}{n!} x^{n} \end{array}$

Step 1: Laplace-Borel Transformation of $\sum_{n = 1}^{n = \infty} \frac{{(- 1)}^{n}}{n!} x^{n} = \sum_{n = 1}^{n = \infty} {(- x)}^{n}$

Step 2: $D_{s} \sum_{n = 1}^{n = \infty} {(- x)}^{n} = - 2^{- s} (2^{s} - 2) ς (s)$

$EGF \equiv f_{3} (- x) = \sum_{n = 1}^{n = \infty} \frac{{(- 1)}^{n}}{n!} x^{n}$

$\begin{array}{l} D_{s} L f_{3} (- x) = D_{s} L \sum_{n} L_{n}^{a} (x) \\ = - 2^{- s} (2^{s} - 2) ς (s) \end{array}$ (120)

Bessel

$\begin{array}{l} x^{2} \frac{\partial u^{2}}{\partial x^{2}} + x (2 p + 1) \frac{\partial u}{\partial x} \\ + (β^{2} + x^{2} a^{2 r}) y = 0 \end{array}$ Math_340#, $β = 0$ , $r = 1$

$y = x^{- p} [c_{1} J_{\frac{q}{r}} (\frac{a}{r} x^{r}) + c_{2} Y_{\frac{q}{r}} (\frac{a}{r} x^{r})]$ ;

$J_{\frac{1}{2}} (k x) = \sqrt{\frac{2}{π}} \frac{\sin (k x)}{x}$

$Y_{\frac{1}{2}} (k x) = \sqrt{\frac{2}{π}} \frac{\cos (k x)}{x}$

J and Y the Bessel function of first and second kinds.

$\begin{array}{l} \sum_{k = 1}^{k = \infty} D_{s} (Y_{\frac{1}{2}} (2 π k x) + i J_{\frac{1}{2}} (2 π k x)) \\ = \frac{1}{Γ (s)} \sqrt{\frac{1}{2 π}} M_{k}^{ξ} [U_{k}^{ξ, r} \sum_{k = 1}^{k = \infty} e^{2 π i k x}] \end{array}$

$\begin{array}{l} \frac{1}{Γ (s)} \sqrt{\frac{1}{2 π}} M_{k}^{ξ} [e^{- 2 π i k x} \sum_{k = 1}^{k = \infty} e^{2 π i k x}] \\ = \frac{1}{Γ (s)} \sqrt{\frac{1}{2 π}} \sum_{k = 1}^{k = \infty} k^{s - 1} d k \\ = \frac{1}{s Γ (s)} \sqrt{\frac{1}{2 π}} ζ (- s) \end{array}$ (121) $s = 2 π i β - 1$ ; $ξ = 0$ ; $a = e^{- x}$ ; $r = - 2$ , $Z (f) = e^{2 π i k f}, f > 0$

$\begin{array}{l} \sum_{k = 1}^{k = \infty} D_{s} (Y_{\frac{1}{2}} (2 π k x) + i J_{\frac{1}{2}} (2 π k x)) \\ = \frac{1}{s Γ (s)} \sqrt{\frac{1}{2 π}} ζ (- s) \end{array}$