Binomial[n, p]) to an algebraic format ?.Probably you want:
FullSimplify[Binomial[n, p] // FunctionExpand, Assumptions -> {n, p} \[Element] PositiveIntegers]
(*n!/(p! Gamma[1 + n - p])*)
AiryAi[z] // FunctionExpand(*Dosen't work ! *)
For integral where: Im[z] == 0 that's mean,formula works only for:
$z\in \mathbb{R}$
You can plot the integral form, see code below:
AIR[z_?NumericQ] := (1/Pi) *NIntegrate[Cos[t^3/3 + z t], {t, 0, Infinity}]
Plot[{AIR[z], AiryAi[z]}, {z, -2, 2}, PlotStyle -> {Red, {Black, Dashed}}, PlotLegends -> "Expressions"]
Regards M.I.