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A sum of products ? - tommy1729 - 08/11/2013

Inspired by the famous Weierstrass I wondered about the following.

Let f(z) be an entire function.

how to find f(z) = A(z) + B(z) =
(z^a exp(a(z)) product_n [(1-z/a_n)^(a*_n)]) +
(z^a exp(b(z)) product_n [1-z/b_n]^(b*_n)])

with a,a*_n the multiplicities of the zero's of A : 0,a_n.
with a,b*_n the multiplicities of the zero's of B : 0,b_n.
with a(z) and b(z) entire functions.

Of course nontrivial ways , not such as f(z) = A(z) + A(z) for some f(z).