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In the paper [1] below, a monotonicity rule for the ratio of two Maclaurin power series was presented as follow. Let $a_k$ and $b_k$ for $k\in\{0\}\cup\mathbb{N}$ be real numbers and the power series ...

Has the following operation $*$ on formal power series $f,g$ been studied before? $$[X^n](f*g) = n! \cdot [X^n]f \cdot [X^n]g$$ where $n$ is a nonnegative integer? This is the typical Hadamard ...

The tetration is denoted $^n a$, where $a$ is called the base and $n$ is called the height, and is defined for $n\in\mathbb N\cup\{-1,\,0\}$ by the recurrence $$ {^{-1} a} = 0, \quad {^{n+1} a} = a^{\...