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?
Best Answer
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, $$f_{abc} = \frac{1}{a!b!c!} \sum_{m=0}^\infty \frac{(-a)_m(-c)_m}{(b+1)_m}\frac{x^m}{m!} = \frac{1}{a!b!c!}\,{_2F_1}(-a,-c;b+1;x).$$