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Given a sequence of complex numbers $a_n$ with $n\in\mathbb{N}$, is it possible to find an analytic (or meromorphic) function that interpolates this sequence in the sense that $f(n)=a_n$?

If this is not always possible, what sort of conditions on the sequence could guarantee the existence of such an $f$?

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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.

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  • $\begingroup$ There are other references easily available on the web, hopefully a search for Ramanujan's master formula--maybe on MathSciNet--will help you find anything you need. $\endgroup$
    – Ben Weiss
    Commented Dec 1, 2009 at 1:26
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    $\begingroup$ It seems Google of Ramanujan "master theorem" works better than Ramanujan "master formula" $\endgroup$
    – Gerald Edgar
    Commented Dec 1, 2009 at 13:09

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