Old Quesne.

Problem. Write (∑_(n=0)^∞)((x^n)/((Subscript[[n], q])!)) at q=(e^ϵ) as a power series in ϵ.

Simplify[(∏_(k=1)^n)Series[((k(q-1))/((q^k)-1)) /. q->(E^ϵ),{ϵ,0,5}]]

First find (Subscript[G, d])[n], the coefficient of (ϵ^d) in the above product.

(Subscript[G, 0])[_] = 1;
(Subscript[G, d_])[n_] := Simplify@Sum[
Sum[(Subscript[G, d-j])[m-1]SeriesCoefficient[((m(q-1))/((q^m)-1)) /. q->(E^ϵ),{ϵ,0,j}],{m,1,n}],
{j,1,d}
];

Table[(Subscript[G, d])[n], {d,0,5}]

(Subscript[Q, d_])[x_] := Sum[(((Subscript[G, d])[n](x^n))/(n!)),{n,0,∞}]

Table[(Subscript[Q, d])[x], {d,0,5}]

Comparing with Quesne’s arXiv:math-ph/0305003:

(Subscript[c, k_])[q_] := ((((1-q))^k)/(k(1-(q^k))));
With[{n=5}, Collect[
(E^(-x))Normal@Series[Exp[Sum[(Subscript[c, k])[(E^ϵ)](x^k),{k, 1,n+1}]], {ϵ,0,n}],
ϵ,Simplify
]]

Inverse Quesne.

Problem. Write (∑_(n=0)^∞)(((Subscript[[n], q])!(x^n))/(((n!))^2)) at q=(e^ϵ) as a power series in ϵ.

Simplify[(x^n)(∏_(k=1)^n)Series[(((q^k)-1)/((k^2)(q-1))) /. q->(E^ϵ),{ϵ,0,4}]]

Simplify[(∏_(k=1)^n)Series[(((q^k)-1)/(k(q-1))) /. q->(E^ϵ),{ϵ,0,5}]]

First find (Subscript[F, d])[n], the coefficient of (ϵ^d) in the above product.

SeriesCoefficient[(((q^k)-1)/(k(q-1))) /. q->(E^ϵ),{ϵ,0,5}]

Sum[(((-k+5 (k^3)-6 (k^4)+2 (k^5)))/1440), {k, n1,n2}]

(Subscript[F, 0])[_] = 1;
(Subscript[F, d_])[n_] := Simplify@Sum[
Sum[(Subscript[F, d-j])[m-1]SeriesCoefficient[(((q^m)-1)/(m(q-1))) /. q->(E^ϵ),{ϵ,0,j}],{m,1,n}],
{j,1,d}
];

Table[(Subscript[F, d])[n], {d,0,5}]

(Subscript[IQ, d_])[x_] := Sum[(((Subscript[F, d])[n](x^n))/(n!)),{n,0,∞}]

Table[(Subscript[IQ, d])[x], {d,0,5}]

Table[(Subscript[Q, d])[x], {d,0,5}]