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"?



Name for series $\sum f_n x^n / (n! (n+k)!)$
No, there is no name for this kind of generating function, except in the case $k=0$, when they are called “doubly exponential generating functions”. I do not know of any applications for $k>0$. 


