Let's consider $U(s,z)=s^z\sum_{n=0}^\infty \binom{z}{n}u_ns^n.$ Which of the following phrases are correct grammatically?
- The function $U(s,z)$ is the product of an entire function and a series of an entire functions, or
- The function $U(s,z)$ is the product of an entire function with a series of an entire functions, or
- The function $U(s,z)$ is the product of an entire function by a series of an entire functions.
This isn't about grammar, but about idiom (which is much more important than grammar if you want to be understood - your three sentences are actually all grammatically correct, but they don't all convey the intended idea).
Firstly, when you say "denote", it would be much more idiomatic to say "write" or "set". "Denote" is quite a difficult term to pin down except in the case when we are talking about what a symbol means (as opposed to what we mean by a symbol).
If $z = x \times y$, then $z$ is the product of $x$ and $y$. Neither $x$ by $y$ nor $x$ with $y$ works very well in this sense. It is idiomatic to say that $z$ is the result of multiplying $x$ by $y$.
In all your examples, when you say "series", you should say "sum of a series". I can see that this is very confusing: we distinguish between sequences $s_n$ and series $\sum_n s_n$, but when we talk about the sum of the series we have to say we mean the sum (I think when we are talking about the series, we are implicitly talking about its partial sums).
The phrase "series of entire functions" doesn't read very well to me. I would have written something more explicit: like "sum of a series of terms, each of which is an entire function of $z$" (or whatever the parameter is).