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) $ ???
relationship betwen 2 power series with and without a factorial
2
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power-series
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0I 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 $b(n)=\frac{1}{1!},\frac{1}{2!},\frac{1}{3!},...$ – 2017-01-10
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0Hm... 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 $a(n)$... – 2017-01-10
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0@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 1*2*3=6. – 2017-01-16