# Any name for this special function?

We know $$\sum_{m=0}^\infty \frac{x^m}{(a-m)!m!} = \frac{1}{a!}(1+x)^m$$ where we understand the factorial as Gamma function $$\Gamma(x)$$ such that it is divergent if the argument is negative integer. We also know $$\sum_{m=0}^\infty \frac{x^m}{(b+m)!m!} \sim \,_0F_1(b,x)$$ as hypergeometric function while this can be generalized to any $$p,q$$ for $$_pF_q(\cdots,x)$$.

Now I want to study $$f_{abc}(x)=\sum_{m=0}^\infty \frac{x^m}{m!(a-m)!(b+m)!(c-m)!}, \qquad a,b,c \in \mathbb{Z}^+$$ Although the function is defined to sum over infinite numbers of integers, but it is effectively truncated at min$$(a,c)$$. I want to ask whether I can find any reference studying the polynomial $$f_{abc}(x)$$? Is there any special name for it?

I can imagine one possible manipulation is to use the reflection formula of Gamma function $$\Gamma(z) \Gamma(1-z) = \frac{\pi}{\sin \pi z}$$ to change the sum over $$(c-m)!$$ to $$(m-c)! \sin(\pi(m-c))$$. And you might then call this function some kind of $$_2F_1$$ type hypergeometric function. However, it is not that good. First of all, the relection formula is better used if arguments $$z$$ are not integer, which is different from my purpose. Second, even after doing that, the function is not fully hypergeometric as there are extra sine functions as coefficients.

I was expecting whether any literature has ever studied these kinds of functions and name them?

This is a standard hypergeometric function. Note that $$\frac{1}{(a-m)!} = (-1)^m \frac{(-a)_m}{a!}\quad\text{and}\quad \frac{1}{(b+m)!} = \frac{1}{b!\,(b+1)_m}$$ in terms of the rising Pochhammer symbol $$(q)_m = q(q+1)\cdots(q+m-1)$$. Hence, $$f_{abc} = \frac{1}{a!b!c!} \sum_{m=0}^\infty \frac{(-a)_m(-c)_m}{(b+1)_m}\frac{x^m}{m!} = \frac{1}{a!b!c!}\,{_2F_1}(-a,-c;b+1;x).$$