HypergeometricPFQ[{a, b}, {c}, x]

HypergeometricPFQ[{}, {}, x]

HypergeometricPFQ[{-α}, {}, -x]

x HypergeometricPFQ[{1, 1}, {2}, x]

x HypergeometricPFQ[{}, {(3/2)}, -((x^2)/4)]

HypergeometricPFQ[{}, {(1/2)}, -((x^2)/4)]

x HypergeometricPFQ[{(1/2), 1}, {(3/2)}, -(x^2)]

HypergeometricPFQ[{-5, a}, {b}, x]

Needs["SpecialFunctions`"]

SumToHypergeometric[
      sum[(1/(k!)) (x^k), {k, 0, ∞}]]

SumToHypergeometric[
      sum[((((-1))^k)/((2  k + 1)!)) (x^(2  k + 1)), {k, 
          0, ∞}]]

F[n_, k_] := Binomial[n, k]/2^n

SumToHypergeometric[sum[F[n, k], {k, 0, ∞}]]

SumToHypergeometric[
      sum[F[n + 1, k] - F[n, k], {k, 0, ∞}]]

SumToHypergeometric[
      sum[ ((n! (((-1))^k))/(k! (n - 
                      2  k)!)) (((2  x))^(n - 2  k)), {k, 
          0, ∞}]]

s = FPS[(((ArcSin[(Sqrt[x])]/(Sqrt[x])))^2), x]

SumToHypergeometric[s]

s = FPS[ArcSin[x]^2, x]

DEtoRE[HolonomicDE[ArcSin[x]^2, F[x]], F[x], a[k]]