Type D.e. Solution Dirichlet Transform of superposition of (discretized) solutions with explanation Dirichlet Transform of superposition of (discretized) solutions EULER $\begin{array}{l}{x}^{2}\frac{\partial {u}^{2}}{\partial {x}^{2}}+x\alpha \frac{\partial u}{\partial x}\\ +\beta u=0\end{array}$ (1) $\begin{array}{l}u\left(x\right)={x}^{\lambda }\mathrm{cos}\left(\mu \mathrm{ln}x\right)+i{x}^{\lambda }\mathrm{sin}\left(\mu \mathrm{ln}x\right)\\ ={x}^{\lambda }\mathrm{exp}\left(i\mu \mathrm{ln}x\right)\end{array}$ $\begin{array}{l}{D}_{s}{\sum }_{\mu =1}^{\mu =\infty }\underset{{x}^{\lambda +i\mu }}{\underbrace{{x}^{\lambda }\mathrm{exp}\left(i\mu \mathrm{ln}x\right)}}\\ =\frac{1}{\Gamma \left(s\right)}{\mathcal{M}}_{x}{\sum }_{\mu =1}^{\mu =\infty }{\text{e}}^{-\lambda x}\mathrm{exp}\left(i\mu \mathrm{ln}{\text{e}}^{-x}\right)\\ ={\left(-i\right)}^{-s}\varsigma \left(s,-i\lambda \right)=\Phi \left(1,s,-i\lambda \right){i}^{s}\end{array}$ (99) $\Phi$ is Lerch Transcedent function ${\left(-i\right)}^{-s}\varsigma \left(s,-i\lambda \right)$ . $\varsigma \left(s,\lambda \right)={\sum }_{n=0}^{n=\infty }\frac{1}{{\left(n+\lambda \right)}^{s}}$ (100) is the Hurwitz Zeta function as generalization of $\varsigma \left(s\right)$ with $\varsigma \left(s,1\right)=\varsigma \left(s\right)$ . variant modified Hermite $\begin{array}{l}\frac{\partial {u}^{2}}{\partial {x}^{2}}-\left(\frac{n}{2}+x-1\right)\frac{\partial u}{\partial x}\\ \text{ }+nu=0\end{array}$ (101) ${\stackrel{˜}{H}}_{{e}_{n}}\left(z\right)=\stackrel{{|z|}^{n}}{\overbrace{U\left(\frac{-n}{2},\underset{-\frac{n}{2}+1}{\underbrace{p+\frac{1}{2}}},\frac{{z}^{2}}{2}\right)}}$ $\varsigma \left(-s\right)=\stackrel{{2}^{\frac{s}{2}}{2}^{-\frac{s}{2}}{\sum }_{n=1}^{n=\infty }{n}^{s}}{\overbrace{{2}^{\frac{s}{2}}{\sum }_{n=1}^{n=\infty }U\left(-\frac{s}{2},-\frac{s}{2}+1,\frac{{n}^{2}}{2}\right)}}$ ${\sum }_{n=1}^{n=\infty }{D}_{s}{\stackrel{˜}{H}}_{{e}_{n}}\left(z\right)=\varsigma \left(-s\right)$ (102) Confluent hypergeometric $\begin{array}{l}\frac{\partial {u}^{2}}{\partial {x}^{2}}+\left(c-x\right)\frac{\partial u}{\partial x}\\ -au=0\end{array}$ $u={b}_{1}{}_{1}F{}_{1}\left(a;c;x\right)+{b}_{2}U\left(a,c,x\right)$ ; Special cases: $\begin{array}{l}{}_{1}F{}_{1}\left(n,n+1,x\right)\\ ~\Gamma \left(1+n\right)\Gamma \left(1-n\right){L}_{-n}^{n}\left(x\right)\end{array}$ (103) with ${L}_{i}^{n}\left(x\right)$ the associated Laguerre polynomials [21] $U\left(n,n+1,x\right)~{x}^{-n}$ (107) $\begin{array}{l}U\left(n,2n,x\right)\\ ~\frac{{\text{e}}^{\frac{x}{2}}}{\sqrt{\pi }}{x}^{\frac{1}{2}-n}{K}_{n-\frac{1}{2}}\left(\frac{x}{2}\right)~{a}_{n}{x}^{-n}\end{array}$ (109) $\begin{array}{l}{}_{1}F{}_{1}\left(n,2n,x\right)\\ ~\frac{{\text{e}}^{\frac{x}{2}}}{\sqrt{\pi }}{\frac{x}{4}}^{\frac{1}{2}-n}\Gamma \left(n+\frac{1}{2}\right){I}_{n-\frac{1}{2}}\left(\frac{x}{2}\right)\\ \sim \frac{{n}_{\left(n\right)}}{{n}_{\left(2n\right)}}\frac{{x}^{n}}{n!}\end{array}$ (113) ; With Γ, Γ(; ;), I, K, resp. the gamma $\begin{array}{l}\Gamma \left(1+s\right)\Gamma \left(1-s\right){L}_{-x}^{x}\left(n\right)\\ =\Gamma \left(1+s\right)\Gamma \left(1-s\right)\frac{-{1}^{-s}U\left(s,s+1,n\right)}{\left(-s\right)!}\\ =-{1}^{-s}U\left(s,s+1,n\right)\Gamma \left(1+s\right)\end{array}$ (104) $\begin{array}{l}-{1}^{-s}\Gamma \left(s+1\right)\varsigma \left(s\right)\\ ={\sum }_{n=1}^{n=\infty }{}_{1}F{}_{1}\left(s,s+1,n\right)\end{array}$ (105), since $\varsigma \left(s\right)={\sum }_{n=1}^{n=\infty }U\left(s,s+1,n\right)$ . $\phi \left(s\right)={\sum }_{n=1}^{n=\infty }{a}_{n}{n}^{-s}$ (110); ${a}_{n}={\sum }_{n=1}^{n=\infty }{\sum }_{k=0}^{n}{a}_{n,k}$ (111) with ${a}_{n,k}$ OIES A113025. See Table B2. $\Gamma \left(s\right)\phi \left(s\right)=\Gamma \left(s\right){\sum }_{n=1}^{n=\infty }{a}_{n}{n}^{-s}$ (114); ${a}_{n}=\frac{{n}_{\left(n\right)}}{{n}_{\left(2n\right)}}$ (115); $\phi \left(s\right)={\sum }_{n=1}^{n=\infty }{a}_{n}{n}^{-s}$ (116); $\begin{array}{l}{D}_{s}{\sum }_{n=1}^{n=\infty }{}_{1}F{}_{1}\left(n,n+1,x\right)\\ =-{1}^{-s}\Gamma \left(s+1\right)\varsigma \left(s\right)\end{array}$ (106) ${D}_{s}{\sum }_{n=1}^{n=\infty }U\left(n,n+1,x\right)=\varsigma \left(s\right)$ (108) ${D}_{s}{\sum }_{n=1}^{n=\infty }U\left(n,2n,x\right)=\phi \left(s\right)$ (112) $\phi \left(s\right)={\sum }_{n=1}^{n=\infty }{a}_{n}{n}^{-s}$ (110) ${D}_{s}{\sum }_{n=1}^{n=\infty }{}_{1}F{}_{1}\left(n,2n,x\right)=\phi \left(s\right)\Gamma \left(s\right)$ (116) $\Gamma \left(s\right)\phi \left(s\right)=\Gamma \left(s\right){\sum }_{n=1}^{n=\infty }{a}_{n}{n}^{-s}$ (114) ${D}_{s}{\sum }_{n=1}^{n=\infty }U\left(n,2n,x\right)=\phi \left(s\right)$ (118)