# iterated special functions on complex parameters

Frequently a special function will have many complex parameters: the polylogarithm, for instance: $\mathrm{Li}_{s}(z)$. Usually we’ll iterate on $z$, but we don’t have to. I’m currently rendering $\mathrm{Li}_{\mathrm{Li}_{\mathrm{Li}_{z}(z)}(z)}(z)$

Suppose we have a $latex${}_{5}F_{4}\$ hypergeometric function:

${}_{5}F_{4}\left({t,x,c,v,b\atop a,s,d,f};z\right)$

And we started looking at monstrosities like this:

${}_{5}F_{4}\left[{{}_{5}F_{4}\left({z,x,c,v,b\atop a,s,d,f};t\right),{}_{5}F_{4}\left({t,z,c,v,b\atop a,s,d,f};x\right),{}_{5}F_{4}\left({t,x,z,v,b\atop a,s,d,f};c\right),{}_{5}F_{4}\left({t,x,c,z,b\atop a,s,d,f};v\right),{}_{5}F_{4}\left({t,x,c,v,z\atop a,s,d,f};b\right)\atop {}_{5}F_{4}\left({t,x,c,v,b\atop z,s,d,f};a\right),{}_{5}F_{4}\left({t,x,c,v,b\atop a,z,d,f};s\right),{}_{5}F_{4}\left({t,x,c,v,b\atop a,s,z,f};d\right),{}_{5}F_{4}\left({t,x,c,v,b\atop a,s,d,z};f\right)};z\right]$