• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 A support for Andy's (P.Walker's) slog-matrix-method Gottfried Ultimate Fellow Posts: 757 Threads: 116 Joined: Aug 2007 11/14/2011, 04:01 AM (This post was last modified: 11/14/2011, 01:15 PM by Gottfried.) Hi - in a selfstudy of the possibility of defining a "bernoulli-polynomial"-like solution for the problem of summing of like powers of logarithms $s_p(a,b)=\log(1+a)^p + \log(2+a) ^p+ \ldots + \log(b)^p$ and its generalizations to arbitrary lower and upper summation-bounds a and b I tried the method of indefinite summation. There I had to find an infinite-sized matrix-reciprocal (inverse) in the same spirit as the matrix-reciprocal which occurs in the slog-ansatz of Andy Robbins(also used earlier by P.Walker). Interestingly the matrix-reciprocal, which can be defined in the same way, gives not only meaningfully approximated values. That would be nice enough, but we might not be able to check, whether the computed values are always true approximations to the expected values. Actually we get even more: we seem to get exactly the coefficients of the most meaningful closed-form-function for this sums-of-like-powers-problem, namely involving the lngamma-function. This occurence of the lngamma here is interesting in twofold manner: a) it supports Andy's/P.Walker's matrix-ansatz for the solution of the tetration/slog b) it supports the meaningfulness of the choice of the L. Euler's gamma-definition for the interpolation of the factorial besides of the criterion of log convexity (maybe this has then a similar effect for the solution of tetration). I began to write a small article about that for my "mathematical miniatures" website, but am a bit distracted currently by my teaching duties and my weak health, and do not know when I'll have time to polish it up fully for presentation. However I thought it might already be useful/interesting to be accessible here in the current state; I think it should be readable, be selfcontained enough and understandable so far. If not, I'd like to answer/elaborate on specific questions. I uploaded the *.pdf to this forum, see attachment Gottfried P.s.: this is very near to that first-time observation in the thread http://math.eretrandre.org/tetrationforu...hp?tid=632 where I used that slog-matrix-computation rather as a curiosity, where here I'm focusing specifically on it.   BernoulliForLogSums.pdf (Size: 123.46 KB / Downloads: 444) Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Possibly Related Threads... Thread Author Replies Views Last Post Half-iterates and periodic stuff , my mod method [2019] tommy1729 0 84 09/09/2019, 10:55 PM Last Post: tommy1729 2 fixpoints , 1 period --> method of iteration series tommy1729 0 1,271 12/21/2016, 01:27 PM Last Post: tommy1729 Tommy's matrix method for superlogarithm. tommy1729 0 1,474 05/07/2016, 12:28 PM Last Post: tommy1729 IOS 9.3 does not support Latex ?!? tommy1729 0 1,226 03/27/2016, 06:26 PM Last Post: tommy1729 [split] Understanding Kneser Riemann method andydude 7 7,101 01/13/2016, 10:58 PM Last Post: sheldonison Some slog stuff tommy1729 15 10,969 05/14/2015, 09:25 PM Last Post: tommy1729 [2015] New zeration and matrix log ? tommy1729 1 2,883 03/24/2015, 07:07 AM Last Post: marraco Kouznetsov-Tommy-Cauchy method tommy1729 0 1,871 02/18/2015, 07:05 PM Last Post: tommy1729 Problem with cauchy method ? tommy1729 0 1,707 02/16/2015, 01:51 AM Last Post: tommy1729 Regular iteration using matrix-Jordan-form Gottfried 7 7,889 09/29/2014, 11:39 PM Last Post: Gottfried

Users browsing this thread: 1 Guest(s)