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 closed form for regular superfunction expressed as a periodic function mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 08/31/2010, 08:04 AM (This post was last modified: 08/31/2010, 08:32 AM by mike3.) (08/31/2010, 06:51 AM)mike3 Wrote: The product of exp is the exponential of a formal power series. This can be expressed using the Bell polynomials: $\prod_{n=0}^{\infty} \exp(a_n y^n) = \exp\left(\sum_{n=0}^{\infty} a_n y^n\right) = \exp(a_0) \exp\left(\sum_{n=1}^{\infty} a_n y^n\right)$. This then becomes $\exp(a_0) \exp\left(\sum_{n=1}^{\infty} a_n y^n\right) = \exp(a_0) \sum_{n=1}^{\infty} \frac{\sum_{k=1}^n B_{n,k}(1! a_1, ..., (n-k+1)! a_{n-k+1})}{n!} y^n = \exp(a_0) \sum_{n=1}^{\infty} \frac{B_n(1! a_1, ..., n! a_n)}{n!} y^n$. Thus the equations to solve are $a_n L^n = \exp(a_0) \frac{B_n(1! a_1, ..., n! a_n)}{n!}$. Since $a_0 = L$ and $a_1 = 1$, this is $a_n L^n = L \frac{B_n(1, 2! a_2, ..., n! a_n)}{n!}$. This is derived from Faà di Bruno's formula, see http://en.wikipedia.org/wiki/Fa%C3%A0_di...7s_formula for details. Doing some tests, it appears that $B_n(1, 2! a_2, ..., n! a_n)$ has only one occurrence of $n! a_n$, and no higher powers of it, and it never seems to be multiplied by any sort of n-dependent coefficient. This means that $B_n(1, a_2, ..., a_n) - n! a_n = B_n(1, 2! a_2, ..., (n-1)! a_{n-1}, 0)$. I don't have a proof at this point, but we can then solve this: $a_n n! L^n = L B_n(1, 2! a_2, ..., n! a_n)$ $a_n n! L^{n-1} = B_n(1, 2! a_2, ..., n! a_n)$ $a_n n! L^{n-1} - n! a_n = B_n(1, 2! a_2, 3! a_3, ..., (n-1)! a_{n-1}, 0)$ $n! (L^{n-1} - 1) a_n = B_n(1, 2! a_2, 3! a_3, ..., (n-1)! a_{n-1}, 0)$ $a_n = \frac{B_n(1, 2! a_2, 3! a_3, ..., (n-1)! a_{n-1}, 0)}{n! (L^{n-1} - 1)}$. And this is the recurrent formula for the general coefficients. Together with $a_0 = L$ and $a_1 = 1$, this completes the as-close-to-explicit-as-possible-so-far formula for the coefficients of the Fourier series for the regular iteration. EDIT: posts corrected to include factorials on terms $a_n$ in Bell polynomials « Next Oldest | Next Newest »

 Messages In This Thread closed form for regular superfunction expressed as a periodic function - by sheldonison - 08/27/2010, 02:09 PM RE: regular superfunction expressed as a periodic function - by sheldonison - 08/28/2010, 08:44 PM RE: regular superfunction expressed as a periodic function - by tommy1729 - 08/28/2010, 11:21 PM RE: regular superfunction expressed as a periodic function - by sheldonison - 08/30/2010, 03:09 AM RE: regular superfunction expressed as a periodic function - by Gottfried - 08/30/2010, 09:22 AM RE: regular superfunction expressed as a periodic function - by tommy1729 - 08/30/2010, 09:41 AM RE: regular superfunction expressed as a periodic function - by tommy1729 - 08/30/2010, 09:46 AM RE: regular superfunction expressed as a periodic function - by Gottfried - 08/31/2010, 08:34 PM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/03/2010, 08:19 PM RE: closed form for regular superfunction expressed as a periodic function - by sheldonison - 09/05/2010, 05:36 AM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/05/2010, 04:45 PM RE: closed form for regular superfunction expressed as a periodic function - by sheldonison - 09/07/2010, 03:54 PM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/07/2010, 07:46 PM RE: closed form for regular superfunction expressed as a periodic function - by bo198214 - 09/08/2010, 06:03 AM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/08/2010, 06:55 PM RE: closed form for regular superfunction expressed as a periodic function - by bo198214 - 09/09/2010, 10:12 AM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/09/2010, 10:18 PM

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