• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 matrix function like iteration without power series expansion andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 07/09/2008, 02:57 PM I guess I should be more specific. The key observation to show the equivalence is that $n!\left({s \atop n}\right) = \prod_{k=0}^{n-1}(s-k)$ which means Woon's formula simplifies to: $ A^s = w^s \left( 1 + \sum_{n=1}^{\infty} (-1)^n \left({s \atop n}\right) \left[ 1 + \sum_{m=1}^{n} (-1)^m w^{-m} \left({n \atop m}\right) A^m \right] \right)$ and letting $w=1$ this simplifies to $ A^s = \left( 1 + \sum_{n=1}^{\infty} (-1)^n \left({s \atop n}\right) \left[ 1 + \sum_{m=1}^{n} (-1)^m \left({n \atop m}\right) A^m \right] \right)$ and assuming $0^0 = 1$ this simplifies to $ A^s = \left( \sum_{n=0}^{\infty} (-1)^n \left({s \atop n}\right) \left[ \sum_{m=0}^{n} (-1)^m \left({n \atop m}\right) A^m \right] \right)$ and combining (-1) terms, this simplifies to $ A^s = \sum_{n=0}^{\infty} \left({s \atop n}\right) \sum_{m=0}^{n} (-1)^{n+m} \left({n \atop m}\right) A^m$ and that is, as you call it, "Henryk's formula" from Woon's formula. Andrew Robbins « Next Oldest | Next Newest »

 Messages In This Thread matrix function like iteration without power series expansion - by bo198214 - 06/30/2008, 03:01 PM RE: matrix function like iteration without power series expansion - by Gottfried - 06/30/2008, 09:23 PM RE: matrix function like iteration without power series expansion - by bo198214 - 07/01/2008, 06:48 AM RE: matrix function like iteration without power series expansion - by Gottfried - 07/01/2008, 02:18 PM RE: matrix function like iteration without power series expansion - by Gottfried - 07/02/2008, 11:23 AM RE: matrix function like iteration without power series expansion - by Gottfried - 07/08/2008, 06:56 AM RE: matrix function like iteration without power series expansion - by andydude - 07/08/2008, 06:01 PM RE: matrix function like iteration without power series expansion - by andydude - 07/08/2008, 09:35 PM RE: matrix function like iteration without power series expansion - by Gottfried - 07/08/2008, 06:23 PM RE: matrix function like iteration without power series expansion - by Gottfried - 07/09/2008, 08:58 AM RE: matrix function like iteration without power series expansion - by andydude - 07/09/2008, 02:38 PM RE: matrix function like iteration without power series expansion - by andydude - 07/09/2008, 02:57 PM RE: matrix function like iteration without power series expansion - by bo198214 - 07/09/2008, 06:18 PM RE: matrix function like iteration without power series expansion - by andydude - 07/14/2008, 06:21 PM RE: matrix function like iteration without power series expansion - by bo198214 - 07/14/2008, 09:55 PM RE: matrix function like iteration without power series expansion - by Gottfried - 07/10/2008, 07:01 AM

 Possibly Related Threads... Thread Author Replies Views Last Post New mathematical object - hyperanalytic function arybnikov 4 230 01/02/2020, 01:38 AM Last Post: arybnikov Perhaps a new series for log^0.5(x) Gottfried 0 183 12/05/2019, 04:35 PM Last Post: Gottfried Is there a function space for tetration? Chenjesu 0 460 06/23/2019, 08:24 PM Last Post: Chenjesu A Notation Question (raising the highest value in pow-tower to a different power) Micah 8 2,711 02/18/2019, 10:34 PM Last Post: Micah Degamma function Xorter 0 904 10/22/2018, 11:29 AM Last Post: Xorter Taylor series of i[x] Xorter 12 11,818 02/20/2018, 09:55 PM Last Post: Xorter Functional power Xorter 0 1,344 03/11/2017, 10:22 AM Last Post: Xorter Complaining about MSE ; attitude against tetration and iteration series ! tommy1729 0 1,632 12/26/2016, 03:01 AM Last Post: tommy1729 2 fixpoints , 1 period --> method of iteration series tommy1729 0 1,689 12/21/2016, 01:27 PM Last Post: tommy1729 2 fixpoints related by power ? tommy1729 0 1,480 12/07/2016, 01:29 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)