What is the best way to generate this power series expansion?

Question

f[m_, z_] := (1/m)*Sum[Exp[z*Exp[2*Pi*I/m]^k], {k, 0, m - 1}]
g[t_] := 1/(2 - f[5, t^(1/5)])
Series[g[t], {t, 0, 10}]

When I tried to compute this on Wolfram Programming Cloud I got the message: "The Wolfram Engine has been terminated because the evaluation time limit was reached."

Perhaps the function g can be rewritten in a simpler form (a sum of trigonometric terms)?

Answer 1

It should be noted that the Mittag-Leffler function is built-in:

f[m_, z_] := MittagLefflerE[m, z^m]
g[t_] := 1/(2 - f[5, t^(1/5)])
Series[g[t], {t, 0, 10}]

which yields the same series as in the Wizard's answer. If you are using an older version that does not yet support MittagLefflerE[], here is an alternative:

f[m_, z_] := HypergeometricPFQ[{}, Range[m - 1]/m, (z/m)^m]
g[t_] := 1/(2 - f[5, t^(1/5)])
Series[g[t], {t, 0, 10}]

Answer 2

My best guess as to what you want:

f[m_, z_] := (1/m)*Sum[Exp[z*Exp[2*Pi*I/m]^k], {k, 0, m - 1}]

g[t_] := 1/(2 - f[5, t^(1/5)]);

Series[ExpToTrig @ g[t], {t, 0, 10}]

1 + t/120 + (253 t^2)/3628800 + (762763 t^3)/1307674368000 + (
 43173223 t^4)/8846916393369600 + (633287284180541 t^5)/15511210043330985984000000 + (
 633594892177711781 t^6)/1854915103581755654799360000000 + (
 29529277377602939454694793 t^7)/10333147966386144929666651337523200000000 + (
 118226228593807528558241820049 t^8)/4944941110593319602094613755127975116800000000 + \
(1259341493633888212897976517963115369 t^9)/
  6295906361341062871682271657666195529704407040000000000 + \
(107604770495341349966501435996477524098581 t^10)/
  64300408460281983178885006693583020812637720018944000000000000

(I used Normal on the output for the sake of copying.)