"Hypergeometric series" often have forms like (in two variables)
$$\sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \frac{(a_1)_n (a_2)_n}{(b_1)_k (c_1)_{n+k}} \frac{x^n}{n!} \frac{y^k}{k!}$$
And there are functions defined by these series that are used to represent their sums. But why don't there seem to be any functions for a series like
$$\sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \frac{(a_1)_n (a_2)_n}{(b_1)_k (c_1)_{n-k}} \frac{x^n}{n!} \frac{y^k}{k!}$$
which is essentially just like the first series but with a simple sign change? Is there a "hypergeometric function" that can express the sum of the latter series?