Clear[ x, n, α, β, nSquared, gaussianExponential, gaussianExponentialSq, f, hankel, powers, hankellist, integrand , nSquaredList, series, range]  
nSquared[ n_ ] := α /(2^n n! Sqrt[π])
gaussianExponential[x_] := Exp[ -α^2 x^2/2]
gaussianExponentialSq[x_] := gaussianExponential[x] * gaussianExponential[x]
f[x_] := Sqrt[β] Exp[ - β Sqrt[x^2] ]
(* http://stackoverflow.com/questions/7669498/mathematica-exponential-nth-derivative-treated-as-an-unknown-function *)
hankel[n_] := FullSimplify[((-1)^n)/(gaussianExponentialSq[x] α^n) Derivative[n][gaussianExponentialSq][x]]

(* http://stackoverflow.com/questions/7674973/mathematica-function-of-two-parameters-becomes-listable-by-wrong-parameter 

hankellist := { hankel[0], hankel[1], hankel[2], hankel[3], hankel[4]} 
nSquaredList = { nSquared[0], nSquared[1], nSquared[2],  nSquared[3],  nSquared[4]  }
*)
SetAttributes[hankel,Listable]
(*range := {0, 2, 4, 6, 8, 10} *) 
range := Range[0, 20, 2] (* all the even indexes end up with zero contributions *)
hankellist := hankel[range]
nSquaredList := nSquared[range]

integrand := hankellist gaussianExponential[x] f[x] 

Assuming[ β ∈ Reals && β>0 && α ∈ Reals && α>0, FullSimplify[powers := Integrate[integrand, {x, -Infinity, Infinity}]]]
(* Assuming[ β ∈ Reals && β>0 && α ∈ Reals && α>0, FullSimplify[powers  -  Integrate[integrandHack, {x, -Infinity, Infinity}]]] *)

Assuming[ β ∈ Reals && β>0 && α ∈ Reals && α>0, FullSimplify[series = hankellist nSquaredList powers gaussianExponential[x] ]]

Plot[Block[{ α = 1, β = 1} , f[x]], {x, -8, 8}, PlotRange->{-0.1,1}]
Plot[Block[{ α = 1, β = 1} , series], {x, -8, 8}, PlotRange->{-0.1,1}]
Plot[ Block[{ α = 1, β = 1} , {Total[series], f[x]}], {x, -8, 8}, PlotRange->{-0.1,1}]

Plot[UnitStep[x],{x,-3,3},PlotStyle->Thick]