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Generating Functions

Roman Grigorovich
Posted 9 years ago

Hello. I have a generating function G(t) = E^(-2*t^2)

How I can find using wolfram generate term of sequence which generates this function?
POSTED BY: Roman Grigorovich
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Dana DeLouis
Dana DeLouis
Posted 9 years ago

... How I can find using wolfram generate term of sequence

Hi. There is also CoefficientList ... (which eliminates the table command) :>)

CoefficientList[Series[g[t],{t,0,20}],t]
{1,0,-2,0,2,0,-(4/3),0,2/3,... etc}
POSTED BY: Dana DeLouis

Hi Roman,

the sequence which generates the function is just the Taylor serie. The general formula for the coefficients are the well known terms $\left.\frac{1}{n!}\frac{\partial}{\partial x} g(x)\right|_{x=0}$. You can calculate a bunch of those terms and than use FindSequenceFunction to find in a somewhat empirical way a more specific formula:

g[t_] := Exp[-2*t^2]
gterms = Table[(D[g[t], {t, n}]/n!) /. t -> 0, {n, 0, 20}]
(* or likewise: *)
gterms = Table[SeriesCoefficient[g[t], {t, 0, n}], {n, 0, 20}]
(* find a specific formula for the terms: *)
FindSequenceFunction[gterms, n]

resulting in:

enter image description here

Isn't Mathematica just great?!? Regards -- Henrik
POSTED BY: Henrik Schachner

Perhaps you can share your findings along with an example. Such as to make this discussion useful for future reference.
POSTED BY: Sander Huisman

Thx for help. I got it now! c:
POSTED BY: Roman Grigorovich

Please have a look at GeneratingFunction, FindGeneratingFunction, FindSequenceFunction, SeriesCoefficient and Series there are a plethora of examples!
POSTED BY: Sander Huisman

Table[SeriesCoefficient[e^(-2t^2), {t, 0, n}], {n, 0, 10}]

It gives me table of values for each n but I want to get a formula for a_n depends on n.
POSTED BY: Roman Grigorovich 
Table[SeriesCoefficient[1/(1 - t), {t, 0, n}], {n, 1, 10}]

You can query specific coefficients of the series using SeriesCoefficient. What have you tried yourself?
POSTED BY: Sander Huisman

I have generating function G(t) I want to know how looks a_n

for example if i have G(t) = 1/(1-t) a_n = 1 for any n
POSTED BY: Roman Grigorovich

Welcome to the Wolfram Community,

Have you looked at the functions Series and InverseSeries and their help? There must be an example that you can use.

I'm not sure if your question is posed correctly; you want to generate terms that generates the function or generate terms that the funtion generates?
POSTED BY: Sander Huisman
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