# Transformation of a Taylor series: “doubling” the derivative order

Suppose a function $f(z)$ has a convergent Taylor expansion:

$$f(z)=\sum_{n=0}^{\infty} c_n \frac{z^n}{n!}$$

Are there general tools to compute $$g(z) = \sum_{n=0}^{\infty} c_{2n} \frac{z^n}{n!}=\sum_{n=0}^{\infty} \, f^{(2n)}(0)\frac{z^n}{n!} \text{ ?}$$

I came across this general problem in a more specific context explained in this question.

One possible way is to do an integral transform $$\dfrac{1}{s} \int_0^{\infty} e^{-z/s} f(z) dz$$ to remove the factorials, then replace $s \to \sqrt{s}$ and do the inverse transform. However, this easily leads to very complicated integrals, and does not seem to work for my particular problem.

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