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Thanks for the comment. Yes, let us assume that $g$ is not identically zero.
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Todor
Todor
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If $f(z)=\sum_{n=0}^\infty a_n z^n$ is a formal power series with complex coefficients, then its Borel transform is defined by $$B(f)(z)=\sum_{n=0}^\infty a_n \tfrac{z^n}{n!}.$$ Suppose that $f$ and $g$$g\neq 0$ are formal power series such that $B(g)$ and $B(fg)$ are entire functions, i.e., analytic for all $z\in \mathbb{C}$. Is it true that $B(f)$ is an entire function?

If $f(z)=\sum_{n=0}^\infty a_n z^n$ is a formal power series with complex coefficients, then its Borel transform is defined by $$B(f)(z)=\sum_{n=0}^\infty a_n \tfrac{z^n}{n!}.$$ Suppose that $f$ and $g$ are formal power series such that $B(g)$ and $B(fg)$ are entire functions, i.e., analytic for all $z\in \mathbb{C}$. Is it true that $B(f)$ is an entire function?

If $f(z)=\sum_{n=0}^\infty a_n z^n$ is a formal power series with complex coefficients, then its Borel transform is defined by $$B(f)(z)=\sum_{n=0}^\infty a_n \tfrac{z^n}{n!}.$$ Suppose that $f$ and $g\neq 0$ are formal power series such that $B(g)$ and $B(fg)$ are entire functions, i.e., analytic for all $z\in \mathbb{C}$. Is it true that $B(f)$ is an entire function?

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Todor
Todor
  • 139
  • 3

Borel summation

If $f(z)=\sum_{n=0}^\infty a_n z^n$ is a formal power series with complex coefficients, then its Borel transform is defined by $$B(f)(z)=\sum_{n=0}^\infty a_n \tfrac{z^n}{n!}.$$ Suppose that $f$ and $g$ are formal power series such that $B(g)$ and $B(fg)$ are entire functions, i.e., analytic for all $z\in \mathbb{C}$. Is it true that $B(f)$ is an entire function?