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: 2,235)
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: 2,235)
Gottfried Helms, Kassel