My goal is to solve a recursive summation without iteration (a lot to ask for I know). Is there a technique for converting a recursive summation into an integral?

For example, these two recursive summations: $$ F_n = \sum_{k=n-1}^{n} F_k $$

$$ L_{x} = \sum_{t=0}^{180} \sum_{k=x+t}^{1000} L_k $$

How can I turn either of these into an integral or a closed form equation? I'm specifically interested in the second (double summation) example.

Better yet, is there a universal solution for turning any arbitrary recursive summation into integral form?

  • $\begingroup$ You can get an integral representation for the Fibonacci numbers by following the path (recurrence relation) > (generating function) > (Cauchy integral formula). Is that the kind of thing you're looking for? $\endgroup$ – Antonio Vargas Apr 10 '17 at 3:02
  • $\begingroup$ Possibly. Can I do this for a double summation such as in the second example? What are the limitations of such summations/functions I can do this for? $\endgroup$ – KobeSystem Apr 10 '17 at 3:11
  • $\begingroup$ In principle, if you can find a generating function then you can find an integral representation. $\endgroup$ – Antonio Vargas Apr 10 '17 at 4:17
  • $\begingroup$ Is there a basic example you can direct me to (or briefly illustrate yourself) that shows the process of the three steps you listed? (Recurrence relation) > (Generating function) > (Cauchy integral formula) $\endgroup$ – KobeSystem Apr 10 '17 at 4:27
  • $\begingroup$ See here for the steps to obtain the generating function $s(x)$ for the Fibonacci numbers. Then, since $F_n = s^{(n)}(0)/n!$, apply the Cauchy integral formula for higher derivatives (called Cauchy's differentiation formula on wikipedia) using an appropriately small circle $\gamma$, say the circle $|x| = 1/2$, to obtain an integral representation for $s^{(n)}(0)$ and hence $F_n$. $\endgroup$ – Antonio Vargas Apr 10 '17 at 12:58

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