# Failure of Series[] for hypergeometric functions

I am encountering peculiar errors when asking Mathematica for series expansions of certain hypergeometric functions. To give an example, consider the function $f(x) = {}_{5} F_{4}(3/2,3/2,3/2,2,2; 1,5/2,5/2,5/2; e^{-x})$. I am interested in the small-$x$ behavior, so after defining

f[x_] = HypergeometricPFQ[{3/2, 3/2, 3/2, 2, 2}, {1, 5/2, 5/2, 5/2}, E^(-x)];


I execute the command

Series[f[x], {x, 0, 1}, Assumptions -> {x > 0}]


The claimed result appears to be a bug, and involves lots of internal-looking variables like e.g. SeriesDumps$1186: SeriesData[x, 0, { Rational[27, 8] ((-2) EulerGamma - Log[x] - 2 PolyGamma[0, Rational[3, 2]] + Sum[ Factorial[K$747] Factorial[1 + K$747]^(-1) Pochhammer[ Rational[-1, 2], 1 + K$747] Pochhammer[ Rational[3, 2], 1 + K$747]^(-2) Pochhammer[2, 1 + K$747] Sum[ Factorial[SeriesDumps$1186]^(-1) HypergeometricPFQ[{ Rational[1, 2], Rational[1, 2], -SeriesDumps$1186}, { 1, Rational[1, 2] - SeriesDumps$1186}, 1] Pochhammer[ Rational[1, 2], SeriesDumps$1186] Pochhammer[1, SeriesDumps$1186] Pochhammer[2, SeriesDumps$1186]^(-1) Pochhammer[-1 - K$747, SeriesDumps$1186]/Pochhammer[ Rational[1, 2] - K$747, SeriesDumps$1186], { SeriesDumps$1186, 0, 1 + K$747}], {K$747, 0, DirectedInfinity[1]}]), Rational[27, 32] (21 - 20 EulerGamma - 10 Log[x] - 20 PolyGamma[0, Rational[5, 2]] + 9 Sum[-Factorial[K$747] Factorial[2 + K$747]^(-1) Pochhammer[ Rational[-1, 2], 2 + K$747] Pochhammer[ Rational[3, 2], 2 + K$747]^(-2) Pochhammer[2, 2 + K$747] Sum[ Factorial[SeriesDumps$1588]^(-1) HypergeometricPFQ[{ Rational[1, 2], Rational[1, 2], -SeriesDumps$1588}, { 1, Rational[1, 2] - SeriesDumps$1588}, 1] Pochhammer[ Rational[1, 2], SeriesDumps$1588] Pochhammer[1, SeriesDumps$1588] Pochhammer[2, SeriesDumps$1588]^(-1) Pochhammer[-2 - K$747, SeriesDumps$1588]/Pochhammer[ Rational[-1, 2] - K$747, SeriesDumps$1588], { SeriesDumps$1588, 0, 2 + K$747}], {K$747, 0, DirectedInfinity[1]}])}, 0, 2, 1] All of this happens with Mathematica 10.1.0.0 running on MacOS X 10.11. To reproduce it, the minimal code is f[x_] = HypergeometricPFQ[{3/2, 3/2, 3/2, 2, 2}, {1, 5/2, 5/2, 5/2}, E^(-x)]; Series[f[x], {x, 0, 1}, Assumptions -> {x > 0}]  Is this a bug in Mathematica? If it is not a bug, what am I to make of such a result? ## 1 Answer $Version

(*  "10.2.0 for Mac OS X x86 (64-bit) (July 7, 2015)"  *)


The errors occur because the function is undefined (series does not converge) at x = 0; however, the errors do not occur if the function is rewritten in a different form. However, the function is still undefined at x = 0

f2[x_] = HypergeometricPFQ[{3/2, 3/2, 3/2, 2, 2}, {1, 5/2, 5/2, 5/2},
E^(-x)] // FunctionExpand // FullSimplify

(*  (27/32)*(8*E^x*(-1 +
ArcTanh[Sqrt[E^(-x)]]/
Sqrt[E^(-x)]) -
4*LerchPhi[E^(-x), 2, 3/2] +
LerchPhi[E^(-x), 3, 3/2])  *)

f2[0]

(*  Infinity  *)

Series[f2[x], {x, 0, 1}, Assumptions -> {x > 0}] // Normal // FullSimplify

(*  (27/32)*(-8 - 4*LerchPhi[1 - x, 2,
3/2] + LerchPhi[1 - x, 3,
3/2] + 4*x*(-2 + Log[8]) +
Log[256] - 2*(2 + 3*x)*Log[x])  *)

Limit[%, x -> 0]

(*  Infinity  *)
`
• Thanks for pointing out the usefulness of FunctionExpand in this case! I'm afraid I don't quite follow the comment that the error occurs because the function is not defined at x=0. The function has a singularity at x=0, but one is entirely justified in asking about the behavior of functions near singularities. The answers will involve e.g. Laurent series, asymptotic series, or transseries, and Mathematica usually deals with such things without problems. So to me it still looks like the expressions for Series[] applied to hypergeometric functions involve bugs. Commented Oct 6, 2015 at 21:03