1 Introduction

1.1 Notations

• [ $F\left( n,x\right)$]inert
• [ $G\left( n,x\right)$]inert ... $F_{n}\left( x\right) =Lcoe\left( \alpha ,\, n\right) \, x^{n}\, G_{n}\left( 1/x\right)$
• [ $H\left( n,x\right)$]the nth Hermite polynomial.
• [ $L\left( n,x\right)$]the nth Laguerre polynomial.

1.2 About the Gamma function

Define, for $n\in \mathbb{N}\setminus 0$ and $a>0$, $\Gamma \left( n+a\right) =\left( a+n-1\right) \, \cdots \, \left( a+1\right) \, a\, \Gamma \left( a\right)$, and you obtain :

$$\Gamma \left( n+\frac{1}{2}\right) =\frac{1}{2^{2n}}\Gamma \left(... ...ac{1}{2}\right) \frac{\Gamma \left( 2\, n+1\right) }{\Gamma \left( n+1\right) }$$ (1)

Define $\Gamma$ in a more complicated way, and you obtain : $\Gamma \left( \frac{1}{2}\right) =\sqrt{\pi }$, the validity of EQ. 1 for all $n>0$ (integer or not) and therefore ($n=-1/4$) $\Gamma \left( \frac{3}{4}\right) \Gamma \left( \frac{1}{4}\right) =\pi \, \sqrt{2}$

1.3 Revisiting the hypergeom function

$\mathrm{hypergeom}\left( [1,\, b,\, a,\, -2],\, [3,\, -2,\, 1],\, z\right) =\mathrm{hypergeom}\left( [-2,\, a,\, b],\, [-2,\, 3],\, z\right)$

$\mathrm{hypergeomV}6\left( [1,\, b,\, a,\, -2],\, [3,\, -2,\, 1],\, z\right) =\mathrm{hypergeomV}6\left( [a,\, b],\, [3],\, z\right)$