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I have the self-referencing recurrence relation $$ d(0) = 0 $$ $$ d(1) = a $$ $$ d(n+1) = d(n) + a*e^{-(\frac{d(n)-n*c}{b})^4} $$ Written as a sum: $$ d(N) = a*\sum_{n=0}^{N}e^{-(\frac{d(n)-n*c}{b})^4} $$ d(N) is a measured variable of a physical system. The constants b and c are given. What I want to do: Measure d(N), then determine the constant a from that. To do so, I need a closed form equation for the recurrence relation. The special case of $c = 0$ would already bring me further.

Here is a graphic to illustrate what I am trying to do: Exemplary plots of d(N) for four different values of a (c = 0, b = 300) and three points representing measured values of d for three different N. I would like to find a from the measurements (in this example, a is approximately 1.5).

I would be very grateful if someone could show me a way to solve the problem, or point me to an existing solution.

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    $\begingroup$ a closed-form solution is unlikely, why not work with a numerical solution? the sequence $d(n)$ converges rapidly for large $n$. $\endgroup$
    – Carlo Beenakker
    Commented Apr 24, 2022 at 10:28
  • $\begingroup$ A numerical solution, gladly! But that's where my limited math skills get me nowhere. For a numerical solution, wouldn’t I still need to re-formulate $$ d(N) = f(N, a, b, c) = a*\sum_{n=0}^{N}e^{-(\frac{d(n)-n*c}{b})^4} $$ somehow to get $$a = f(d(N, b, c))$$ ? $\endgroup$
    – Matte
    Commented Apr 24, 2022 at 12:14
  • $\begingroup$ Why don’t you show me some of your data, and I will see if I can extract this coefficient a from it. $\endgroup$
    – Carlo Beenakker
    Commented Apr 24, 2022 at 13:19
  • $\begingroup$ I think it will help to know the actual values of $b$ and $c$, or at least their signs/ranges. Also, is $a$ positive? Also, what are ranges of values for $N$ and $d(N)$? $\endgroup$
    – Iosif Pinelis
    Commented Apr 24, 2022 at 13:55
  • $\begingroup$ Also, are you sure that $d(1)=a$ rather than $ae^{-(c/b)^4}$ as both the recursion and your summation formula (in which $N$ should probably be $N-1$) suggest? $\endgroup$
    – fedja
    Commented Apr 24, 2022 at 19:22

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