/*   10-7-2023   h-star coeff for t_p  */

ftp = Gamma[(p+1)/2] / ( Sqrt[p Pi] Gamma[p/2] ) (1+x^2/p)^(-(p+1)/2)

Integrate[ ftp, {x,-Infinity,Infinity}, Assumptions -> p>2 ]

m = Integrate[ x ftp, {x,-Infinity,Infinity}, Assumptions -> p>2 ]

v = Integrate[ (x-m)^2 ftp, {x,-Infinity,Infinity}, Assumptions -> p>2 ]

rfp = Integrate[ D[ftp,x]^2, {x,-Infinity,Infinity}, Assumptions -> p>2 ]

hs = ( 6 / rfp )^(1/3)   (* without n  *)

hstar = hs / Sqrt[v]

log$hstar = PowerExpand[ Log[ hstar ] ]

LogLogPlot[ {Exp[log$hstar], 3.49083}, {p, 3, 2050}, PlotRange -> {2, 3.5}]