How can known series transformations be applied to product transformations?

[benorin]

Certianly there is a lot of reference material on series transformations: they accelerate convergence, provide analytic continuations and what not. But I have not yet seen a like presentation of product transformations. Given that there are ways to write a product as a series, and vice-versa (see this post), would it be so difficult to translate known series transformations into product transformations?

An example application would be the Riemann zeta fcn, the series definition can be analytically continued to the whole complex plane (except z=1) via some clever series manipulation + a series transformation (see this post), but has anybody ever used similar techniques to analytically continue the Euler product over primes representation of the Riemann zeta? Yeah, I know about the Hadamard Product derived using the Weierstrass formula, but that is not what I'm after.

Just fishing for your ideas,
-Ben

 

[Edwin]

I would be interested in helping you research this concept, as I am interested in infinite product representations; because, one can take the argument of an infinite product term-by-term. Specifically, I am interested in infinite product representations for functions of the form:

h(z) = f(z)*g(z) - e, where e is some positive real valued constant.

To start, perhaps we could look for a way to convert Taylor series, power series, geometric series, and some other well known series into infinite product representations, or see if anyone else has come up with a way to do this. We could then see if there are ways to generalize the technique to any infinite series.

What do you think?

Does this sound feasible?

Inquisitively,

Edwin

 

[benorin]

Edwin, you should look-up the http://planetmath.org/encyclopedia/WeierstrassProductTheorem.html for the infinite product version of the geometric series