Series similar to hypergeometric function

Question

I have a series like this

$$\sum_{n=0}^\infty a^{(n)}b^{(n)} \frac{x^n}{n!}$$

Here $a^{(n)}$ is a rising Pochhammer symbol.

I wonder if there is any function under the above form which is studied before.

The series is quite similar to the hypergeometric function in the following form

$$\sum_{n=0}^\infty \frac{a^{(n)}b^{(n)}}{c^{(n)}} \frac{x^n}{n!}$$

Thanks.