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Please, can the following Meijer G-function

$$G_{1,3}^{3,0}\left(a^2\Bigg| \begin{array}{c} n+2 \\ -\dfrac{1}{2},1,\dfrac{3}{2} \\ \end{array} \right)$$

MeijerG[{{},{2+n}},{{-(1/2),1,3/2},{}},a^2]

be expressed more explicitly in terms of other special functions, but involving no infinite integrals?

$a\;$ is positive real number and $\;n=0,1,2\;$ is an integer.

Gallagher
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  • @Claude Leibovici, this result comes from an integral, this is the MA syntax MeijerG[{{},{2+n}},{{-(1/2),1,3/2},{}},a^2] – Gallagher Oct 17 '22 at 07:27
  • @Claude Leibovici, it comes from this integral $\frac{a^2\int_1^{\infty}\frac{\left (x^2 - 1 \right)^{n + \frac{3}{2}}\exp(-a x)}{x^{2 n + 1}}, dx} {K_2(a)} $ (a^2/BesselK[2,a]) Integrate[(x^2-1)^(n+(3/2))/x^(2n+1) Exp[-a x],{x,1,Infinity}] – Gallagher Oct 17 '22 at 08:52
  • I don't think anything is known. There are known formulas for $G^{3,0}_{1,3}$ but none of them match your specific case. – K.defaoite Oct 17 '22 at 11:24
  • @K.defaoite, yes but the issue is that b3-b1 = 3/2-(-1/2) = 2 is an integer in my expression, which is excluded from the formula on the functions page. – Gallagher Oct 17 '22 at 11:32
  • Precisely why I said "none of them match your specific case". – K.defaoite Oct 17 '22 at 13:05

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