# Fourier transform of hypergeometric function only returns where FT vanishes

I have a probability distribution of a random variable $$X$$ with support between, roughly, $$[-3, 3]$$. I do not know the probability distribution, but I can simulate instances from it approximately, and I have its moments. My aim is to find the probability distribution from the moments.

Odd moments vanish, and even moments are $$\mathbb{E}[X^{2n}] = \frac{\binom{4n}{n}}{3n+1}$$. Using the moments, I constructed the Fourier transform of the probability distribution as $$\mathbb{E}[e^{i k X}] = \sum_{n=0}^{\infty} \frac{(-1)^n k^{2n}}{(2n)!} \frac{\binom{4n}{n}}{3n+1}$$.

Mathematica gives this sum as HypergeometricPFQ[{1/4, 3/4}, {2/3, 1, 4/3}, -((64 k^2)/27)]

Taking the inverse transformation via

  Exp[-I k x] HypergeometricPFQ[{1/4, 3/4}, {2/3, 1, 4/3}, -((64 k^2)/27)], {k, -Infinity, Infinity}]


gives a curious result: ConditionalExpression[0, 1/x^2 < 27/256].

That is, Mathematica is telling me that the distribution vanishes outside $$[-\frac{16}{3 \sqrt{3}}, \frac{16}{3 \sqrt{3}}]$$, which is in strong agreement with my empirical bounds of roughly $$[-3,3]$$. However, there's no information about what the resulting function of $$x$$ looks like within the bounds of $$[-\frac{16}{3 \sqrt{3}}, \frac{16}{3 \sqrt{3}}]$$!

How can I evaluate this inverse Fourier transform for all real $$x$$, not just those where the support vanishes? I would prefer a symbolic solution, but I suspect it's likely that this is the best I can extract from Mathematica symbolically, so I would also accept methods that give the transformation numerically.

f=HypergeometricPFQ[{1/4,3/4},{2/3,1,4/3},-64 k^2/27];

(* this returns a symbolic result *)
g=1/Sqrt[2*Pi]*InverseFourierTransform[f,k,x];

(* check normalization, gives 1 *)
NIntegrate[g,{x,-Infinity,Infinity}]

(* plot *)
Plot[g,{x,-4,4},PlotStyle->Thick] • Really nice to see! Curious that InverseFourierTransform fares better than direct integration, but it likely makes sense, since the former can be more specialized. This solution also helped me realize a curious divergence structure at small arguments; I'd been expecting $x^{-2/3}$, but this diverges as $x^{-1/2}$, confirmed in my approximate sampling histograms. Oct 14, 2022 at 18:02

Amplifying on the answer by user293787

Clear["Global*"]

f[k_] = Sum[
(-1)^n k^(2 n)/(2 n)! Binomial[4 n, n]/(3 n + 1),
{n, 0, Infinity}]

(* HypergeometricPFQ[{1/4, 3/4}, {2/3, 1, 4/3}, -((64 k^2)/27)] *)

(pdf1[x_] = Assuming[Element[x, Reals],
InverseFourierTransform[f[k], k, x,
FourierParameters -> {1, -1}] // FullSimplify])// The total probability is unity

NIntegrate[pdf1[x], {x, -Infinity, Infinity}]

(* 1. *)


The pdf is an even function

Assuming[x != 0, pdf1[-x] == pdf1[x] // FullSimplify]

(* True *)


The support is on the interval

NIntegrate[pdf1[x], {x, -16 Sqrt/9, 16 Sqrt/9}]

(* 1. *)


Given this constraint,

(pdf2[x_] = Piecewise[{{
Assuming[-16 Sqrt/9 < x < 16 Sqrt/9,
pdf1[x] // FullSimplify],
-16 Sqrt/9 < x < 16 Sqrt/9}}, 0])// NIntegrate[pdf2[x], {x, -Infinity, Infinity}]

(* 1. *)


Further, due to the even symmetry, the pdf can be simplified to

(pdf3[x_] = Piecewise[{{
Assuming[0 < x < 16 Sqrt/9,
pdf1[x] // FullSimplify] /. x -> Sqrt[x^2],
-16 Sqrt/9 < x < 16 Sqrt/9}}, 0])// NIntegrate[pdf3[x], {x, -Infinity, Infinity}]

(* 1. *)


EDIT: Following through on the comment by user196574

(ratio = (8*3^(1/4)*Gamma[-(1/12)]*Gamma[1/4]^2*Gamma[7/12])/
(3*3^(3/4)*Gamma[-(1/4)]*Gamma[5/12]*Gamma[3/4]*
Gamma[13/12])) // N

(* 32. *)

Off[N::meprec]

Block[{$MaxExtraPrecision = 500}, ratio - 32 // N[#, 200] &] (* 0.*10^-698 *) (repl = Solve[ (ratio /. Gamma[7/12] -> Inactive[Gamma][7/12]) == 32, Inactive[Gamma][7/12]][] // Activate) // TraditionalForm pdf4[x_] = pdf3[x] /. repl // FullSimplify NIntegrate[pdf4[x], {x, -Infinity, Infinity}] (* 1. *)  The simplification results LeafCount /@ {pdf1[x], pdf2[x], pdf3[x], pdf4[x]} (* {300, 218, 172, 146} *) ` • Smart simplifications! I was working out removing the Heaviside thetas, but you've gotten a much crisper result than I was getting! Nice trick also with writing$x$as$\sqrt{x^2}$to make the evenness manifest. Oct 14, 2022 at 23:41 • I think there's even one more (small) simplification, which is that the numerical prefactors of the first two terms are the same up to a factor of$32\$, that is, N[(8 3^(1/4) Gamma[-1/12] Gamma[1/4]^2 Gamma[7/12])/( 3^(7/4) Gamma[-1/4] Gamma[5/12] Gamma[3/4] Gamma[13/12])] is 32. Oct 15, 2022 at 0:05