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)

$\begin{array}{l}u\left(x\right)=x^{\lambda }\cos \left(\mu \ln x\right)+i{x}^{\lambda }\sin \left(\mu \ln x\right)\\ =x^{\lambda }\exp \left(i\mu \ln x\right)\end{array}$

$\begin{array}{l}{D}_s{\displaystyle {\sum }_{\mu =1}^{\mu =\infty }\underset{x^{\lambda +i\mu }}{\underbrace{x^{\lambda }\exp \left(i\mu \ln x\right)}}}\\ =\frac{1}{\Gamma \left(s\right)}{\mathcal{M}}_x{\displaystyle {\sum }_{\mu =1}^{\mu =\infty }{\text{e}}^{-\lambda x}\exp \left(i\mu \ln {\text{e}}^{-x}\right)}\\ ={\left(-i\right)}^{-s}\varsigma \left(s,-i\lambda \right)=\Phi \left(1,s,-i\lambda \right)i^s\end{array}$ (99)

$\Phi$ is Lerch Transcedent function

${\left(-i\right)}^{-s}\varsigma \left(s,-i\lambda \right)$ . $\varsigma \left(s,\lambda \right)={\displaystyle {\sum }_{n=0}^{n=\infty }\frac{1}{{\left(n+\lambda \right)}^s}}$ (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

$\begin{array}{l}\frac{\partial u^2}{\partial x^2}-\left(\frac{n}{2}+x-1\right)\frac{\partial u}{\partial x}\\ +nu=0\end{array}$ (101)

${\tilde{H}}_{e_n}\left(z\right)=\stackrel{{\left|z\right|}^n}{\overbrace{U\left(\frac{-n}{2},\underset{-\frac{n}{2}+1}{\underbrace{p+\frac{1}{2}}},\frac{z^2}{2}\right)}}$

$\varsigma \left(-s\right)=\stackrel{2^{\frac{s}{2}}{2}^{-\frac{s}{2}}{\displaystyle {\sum }_{n=1}^{n=\infty }{n}^s}}{\overbrace{2^{\frac{s}{2}}{\displaystyle {\sum }_{n=1}^{n=\infty }U\left(-\frac{s}{2},-\frac{s}{2}+1,\frac{n^2}{2}\right)}}}$

${\displaystyle {\sum }_{n=1}^{n=\infty }{D}_s{\tilde{H}}_{e_n}\left(z\right)}=\varsigma \left(-s\right)$ (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)

$\begin{array}{l}-1^{-s}\Gamma \left(s+1\right)\varsigma \left(s\right)\\ ={\displaystyle {\sum }_{n=1}^{n=\infty }{}_{1}F{}_1\left(s,s+1,n\right)}\end{array}$ (105), since

$\varsigma \left(s\right)={\displaystyle {\sum }_{n=1}^{n=\infty }U\left(s,s+1,n\right)}$ .

$\phi \left(s\right)={\displaystyle {\sum }_{n=1}^{n=\infty }{a}_n{n}^{-s}}$ (110); $a_n={\displaystyle {\sum }_{n=1}^{n=\infty }{\displaystyle {\sum }_{k=0}^n{a}_{n,k}}}$ (111) with $a_{n,k}$ OIES A113025. See Table B2.

$\Gamma \left(s\right)\phi \left(s\right)=\Gamma \left(s\right){\displaystyle {\sum }_{n=1}^{n=\infty }{a}_n{n}^{-s}}$ (114); $a_n=\frac{n_{\left(n\right)}}{n_{\left(2n\right)}}$ (115);

$\phi \left(s\right)={\displaystyle {\sum }_{n=1}^{n=\infty }{a}_n{n}^{-s}}$ (116);

$\begin{array}{l}{D}_s{\displaystyle {\sum }_{n=1}^{n=\infty }{}_{1}F{}_1\left(n,n+1,x\right)}\\ =-1^{-s}\Gamma \left(s+1\right)\varsigma \left(s\right)\end{array}$ (106)

$D_s{\displaystyle {\sum }_{n=1}^{n=\infty }U\left(n,n+1,x\right)}=\varsigma \left(s\right)$ (108)

$D_s{\displaystyle {\sum }_{n=1}^{n=\infty }U\left(n,2n,x\right)}=\phi \left(s\right)$ (112)

$\phi \left(s\right)={\displaystyle {\sum }_{n=1}^{n=\infty }{a}_n{n}^{-s}}$ (110)

$D_s{\displaystyle {\sum }_{n=1}^{n=\infty }{}_{1}F{}_1\left(n,2n,x\right)}=\phi \left(s\right)\Gamma \left(s\right)$ (116)

$\Gamma \left(s\right)\phi \left(s\right)=\Gamma \left(s\right){\displaystyle {\sum }_{n=1}^{n=\infty }{a}_n{n}^{-s}}$ (114)

$D_s{\displaystyle {\sum }_{n=1}^{n=\infty }U\left(n,2n,x\right)}=\phi \left(s\right)$ (118)