# Finding a mapping between two types of (generalized) hypergeometric series

I am given two functions, one is of the form $$2F1(a,b,c;z)$$, where $$2F1$$ is a hypergeometric series. The other one is a generalized hypergeometric series $$3F2(d,e,f;g,h;w)$$, where the characters are certain (specific) parameters. Is it possible to use mathematica to find the mapping between those two functions?

Thanks in advance!

• Depends on the parameters. Commented Jun 22, 2021 at 17:02
• Thank you for the response. Do you know of any mathematica tool which can be used for this? To be specific the functions are given by $2F_1\left(\frac{1}{4}, \frac{1}{2}, \frac{5}{4} ;-\frac{1}{x^{4}}\right)$ and $_3F_2\left(\frac{1}{6},\frac{1}{3},\frac{2}{3},\frac{7}{6},\frac{3}{2};\frac{-27}{2x^6}\right)$.
– john
Commented Jun 23, 2021 at 14:19
• What do you mean by mapping? What kind of answer would you consider satisfactory? Commented Jul 23, 2021 at 18:16

## 1 Answer

\$Version

(* "12.3.0 for Mac OS X x86 (64-bit) (May 10, 2021)" *)

Clear["Global*"]

(f21[x_] =
Hypergeometric2F1[1/4, 1/2, 5/4, -1/x^4]) // TraditionalForm


(f32[x_] =
HypergeometricPFQ[{1/6, 1/3, 2/3}, {7/6,
3/2}, -27/(2 x^6)]) // TraditionalForm


Both functions have even symmetry

f21[x] == f21[-x] && f32[x] == f32[-x]

(* True *)


Both functions asymptotically approach 1 from below

Asymptotic[#, {x, Infinity, 6}] & /@ {f21[x], f32[x]}

(* {1 - 1/(10 x^4), 1 - 2/(7 x^6)} *)


Graphically,

xmax = 2;

Plot[{f21[x], f32[x]}, {x, 0, xmax},
PlotLegends -> Placed["Expressions", {0.6, 0.4}]]


You can use ParametricPlot to map from one function to the other

Legended[
ParametricPlot[{f21[x], f32[x]}, {x, 0, xmax},
Frame -> True,
FrameLabel -> (Style[#, 12, Bold] & /@
{HoldForm[f21[x]],
HoldForm[f32[x]]}),
PlotRange -> All,
ColorFunction -> Function[{f21, f32, x},
ColorData["Rainbow"][x/xmax]],
ColorFunctionScaling -> False,
Epilog -> {Lighter[Gray, 0.5], Dashed,
Line[{{0, 0}, {1, 1}}]}],
Placed[
BarLegend[{"Rainbow", {0, xmax}},
LegendMarkerSize -> 175,
LegendLabel -> Style[x, 14, Bold]],
{0.2, 0.625}]]
`