Calculate accurately this hypergeometric function?

Questions
1

Hi,
I have a problem with forcing MATHEMATICA to calculate accurately some hypergeometric function:
s = 10^50;
Block[{$MaxExtraPrecision = 10000},N[Hypergeometric2F1[2, 1 + s, 2 + s, -Exp[10]], 300]].
I am obtaining messages about overflow. The messages disappear when I request 15 digits, but I need 300 digits. It is hard to understand why the overflow arises; judging from the plot of the function, there seems to be a constant limit when s->Infinity. Increasing MaxExtraPrecision does not help. Strangely,
Limit[Hypergeometric2F1[2, 1 + s, 2 + s, -Exp[10]],s->Infinity] does not work and it is also impossible to obtain an asymptotic series for s->Infinity by using the Series command. Is there any way to overcome the problem?
Leslaw

2

Workaround:

At first I convert Hypergeometric2F1 to simpler function like LerchPhi with Maple and using AsymptoticSum to expand series s parameter at infinity.

$Version
(*"12.1.0 for Microsoft Windows (64-bit) (March 14, 2020)"*)

M = 100 (*More terms greater precision *)
Block[{$MaxExtraPrecision = 302}, N[(-1 - s)/(E^10 (1 + E^10)) + (s (1 + s) AsymptoticSum[z^k/(
k + s), {k, 0, Infinity}, {s, Infinity, M}])/E^10 /. z -> -Exp[10] /. s -> 10^50, 300]]

(*2.06096648272347233418959215235994959191908111757491936251422339815271\
8558212616552192199567106427341226560922751917789191834346105243809432\
5329127379111913999269570785982677759971629508381569251261807659636488\
5344917825480306093143427886092553493591682250667234723102601737564361\
910230136964528670935*10^-9*)

Plot[{Hypergeometric2F1[2, 1 + s, 2 + s, -Exp[10]], (-1 - s)/(
   E^10 (1 + E^10)) + (s (1 + s) LerchPhi[-E^10, 1, s])/E^10}, {s, 0, 
  10}, PlotStyle -> {Red, {Dashed, Black}}](*Looks the same*)

10293Bez tytułu.png
10293Bez tytułu.png1078×134 6.97 KB

I found a Issue in AsymptoticSum:

  AsymptoticSum[ z^k/(k + s), {k, 0, Infinity}, {s, Infinity, 1}](*Works fine*)
  AsymptoticSum[z^k/(k + s) /. z -> -Exp[10], {k, 0, Infinity}, {s, Infinity, 1}](* Gives input !!! *)

Regards M.I.

3

We have:

Sum[(z^k Pochhammer[2, k] Pochhammer[1 + s, k])/(
 k! Pochhammer[2 + s, k]), {k, 0, Infinity}]
 (*Hypergeometric2F1[2, 1 + s, 2 + s, z]*)

then:

 Limit[(z^k Pochhammer[2, k] Pochhammer[1 + s, k])/(k! Pochhammer[2 + s, k]), s -> Infinity]
 (*(z^k Pochhammer[2, k])/k!*) 
 N[Sum[(z^k Pochhammer[2, k])/k!, {k, 0, Infinity}] /. z -> -Exp[10], 300]

 (*2.06096648272347233418959215235994959191908111757487814505583864436199\
 9184683402296197924473085819095047447295634942724479179145245736116114\
 9156752936212085037037409690812216803036585395729723051186738618366655\
 5146641627747571896005193529868156699033219139667312892465093410814678\
 504338970396392816996*10^-9*)
\frac{1}{\left(1+e^{10}\right)^2}

Symbolic value what You are looking for.