given the 2 series
$ f(x)= \sum_{n=0}^{\infty} a(n) x^{n} $
amd $ g(x)= \sum_{n=0}^{\infty} \frac{a(n)}{n!} x^{n} $
is there a method to obtain the value of $ g(x) $ if we know the value of $ f(x) $ ???
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I doubt that there is a way that always works because to me this looks like element wise multiplication of two sequences $a(n)=a_1,a_2,a_3,...$ and $b(n)=\frac{1}{1!},\frac{1}{2!},\frac{1}{3!},...$ – Mats Granvik Jan 10 '17 at 19:16
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Hm... what if I know some more information? Like what is $f(x)$ in a neighborhood of $0$? With that much information, I can directly find $a(n)$... – Simply Beautiful Art Jan 10 '17 at 19:19
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@JoseGarcia You should look into power series with Greatest Common Divisors (GCD) as coefficients. In those you can multiply elementwise numerators of Dirichlet series of two prime number columns and get a composite column as a result. Might work for power series also. Remember to take into account the first column in the GCD matrix also. Like 123=6. – Mats Granvik Jan 16 '17 at 14:19