hi, my problem is to evaluate the n-th difference to be able to approximate the n-th derivative of a function f(x) at the point x=0(adsbygoogle = window.adsbygoogle || []).push({});

as for small 'h' then [tex] \frac{\Delta ^{n}}{h^{n} \sim \frac{d^{n}f(0)}{dx^{n}} [/tex]

i think that the n-th difference (with step h) has the representation:

[tex] \Delta ^{n} f(0) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty}ds \frac{f(s)}{(s(s-h)(s-2h)(s-3h).....(s-nh)}x^{s} [/tex]

and that evaluating this we could obtain an (approximate) asymptotic expansion for n-->infinity.

the intention is given a generating function

[tex] f(x)= a(0)+a(1)x+a(2)x^{2}+..................... [/tex]

then to calculate a(n) you need to know the n-th derivative, we can approximate this derivative by the n-th forward difference at x=0 with an small step 'h'

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# Evaluating the n-th difference

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