What would be good features to have in an analytical function which expresses the behaviour of national production? Here are a few quick suggestions:
1. The main accumulating determinants of growth are included
Accumulating determinants are things like physical capital and education.
2. It allows for different elasticities of substitution
The elasticity of substitution measures how much of one input is substituted for another if the first input becomes better value. It measures market flexibility.
3. It allows for different income distributions
4. It allows for different innovation potentials
The Cobb-Douglas production function is widely used, and meets criterion one. The CES production function generalises Cobb-Douglas and meets criterion two. Further generalisations might meet the other criteria.
Once a production function starts getting really complex, it is probably time to move into a full macroeconomic model, since the assumptions and interactions can be spelt out more exactly.