Good morning. I am working with higher powers of the Generalized Euler Number generating function
$$\left[\frac{n}{\sum_{j=0}^{n-1}\exp{\left(w_n^jx\right)}}\right]^\alpha$$
where $w_n=\exp{\left(2i\pi/n\right)}$. This can be written in terms of what are called Olivier functions, where
$$\Phi_{n,k}(x)=\frac{1}{n}\sum_{j=0}^{n-1}w_n^{-jk}\exp{\left(w_n^jx\right)}$$
and thus I can rewrite the exponential generating function as
$$\left[\frac{1}{\Phi_{n,0}(x)}\right]^\alpha=\sum_{j=0}^\infty{A_{n,0}^{(\alpha)}\frac{x^j}{j!}}$$
My goal is to generate the first fifty or so numbers for varying $\alpha$. So I use the following definitions to help in Mathematica:
w[n_] := E^((2*i*\Pi)/n)
Phi[n_, k_, x_] := (1/n)*Sum[(w[n]^(-j*k)*E^(x*w[n]^j)), {j, 0, n - 1}]
NEuler[a_, n_, x_] := (1/Phi[n, 0, x])^a
NENumber[a_, n_, r_, M_] :=
FullSimplify[Coefficient[r! Normal[Series[NEuler[a, n, x], {x, 0, M}]], x, r]]
My problem here is when I go to evaluate say
Table[NENumber[2, 3, j, M], {j, 0, M}]
If $M>20$ it seems that the calculation never actually evaluates, or if it will, it takes way too long. Is there another method in order to generate these numbers much more quickly and efficiently on my computer? My math skills are better than my Mathematica skills, as I have definitely never had formal instruction using Mathematica, and everything that I've learned I've learned by myself. Thank you for your help.
I
andi
are two very different things. Thus:OlivierPhi[n_, k_, x_] := Sum[Exp[x Exp[2 π I j/n] - 2 π I j k/n], {j, 0, n - 1}]/n; With[{n = 2, a = 3}, Table[j! SeriesCoefficient[(1/OlivierPhi[n, 0, x])^a, {x, 0, j}] // RootReduce, {j, 0, 20}]]
$\endgroup$ – J. M. will be back soon♦ Jul 24 '16 at 14:05I
is the imaginary unit inInputForm[]
; the "fancy i" should have automatically pasted asI
here if you copied things right. $\endgroup$ – J. M. will be back soon♦ Jul 24 '16 at 14:14