I want to express $e^{x^2}$ as MeijerG function?
it would be possible? or what?
can i use $e^x$ MeijerG expression for this one?
1 Answer
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start from the identity $$G^{2,0}_{0,2}(z,\tfrac{1}{2}|b,c)=2z^{b+c}K_{c-b}(2z),$$ substitute $b=0$ and $c=1/2$, $$G^{2,0}_{0,2}(z,\tfrac{1}{2}|0,\tfrac{1}{2})=2z^{1/2}K_{1/2}(2z)=\sqrt{\pi}e^{-2z},$$ and for $z=-x^2/2$ you have the desired relation.
answered Feb 28, 2018 at 13:50
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$\begingroup$ thanks in advance. what is the reference book or article? would you guide me? $\endgroup$ Commented Mar 3, 2018 at 10:53
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$\begingroup$ I gave a link to my reference. $\endgroup$ Commented Mar 3, 2018 at 12:31
Exp[x^2] // MeijerGReduce[#, x] &$\frac{\text{MeijerG}\left(\{\{\},\{\}\},\left\{\left\{0,\frac{1}{2}\right\},\{\}\right\},-\frac{x^2}{2},\frac{1}{2}\right)}{\sqrt{\pi }}$ $\endgroup$