ca.analysis and odes - Interpolation of sequences by analytic functions

There is a classic result of Ramanujan known as his Master Formula which Wolfram has here:

http://mathworld.wolfram.com/RamanujansInterpolationFormula.html

To summarize briefly (and coarsely): if the values you want to interpolate do not grow faster than the gamma function, things will be ok. If they do grow faster than that, his formula doesn't converge, so you'll have to find another technique.