Let $(f_n)_{n\ge0}$ to be a real sequence.

  Then $\sum f_n {x^n \over n!}$ is called the exponential generating function of $(f_n)$.

Let $k\ge0$ be a nonnegative integer. If we add another factorial $(n+k)!$ to the denominator and obtain

$$\sum_{n\ge0} f_n {x^n \over n!(n+k)!},$$

is there a name for this kind of generating functions? "Bessel generating function"?

Let $(f_n)_{n\ge0}$ be a real sequence. Then $\sum f_n {x^n \over n!}$ is called the exponential generating function of $(f_n)$.

Let $k\ge0$ be a nonnegative integer. If we add another factorial $(n+k)!$ to the denominator and obtain $$\sum_{n\ge0} f_n {x^n \over n!(n+k)!} , $$ is there a name for this kind of generating functions? Perhaps "Bessel generating function"?