andydude Wrote:@Gottfried
I read your paper, and your first theorem is interesting! Although I couldn't quite follow the "proof", it seems plausible, and I might recommend using an umbral-calculus style of proof. From what I understand umbral calculus is making transitions that are not rigorously true, but strictly syntactic substitutions. For example:
\(
\begin{array}{rl}
e^x & = \sum_{k=0}^{\infty} \frac{1}{k!} x^k \\
f(e, x) & = \sum_{k=0}^{\infty} \frac{1}{k!} f(x, k)
\end{align}
\)
which seems similar to a substitution you make in your paper. I think this would make the proof much clearer, and easier to follow, although I think it has gone out of style, and may be viewed as less rigorous than other methods of proof. I don't know if it helps or not, I just thought I'd comment on it.
Andrew Robbins
Hi Andrew -
thanks for the comment!
I know a bit about umbral-calculus from the context of Bernoulli-numbers, where it is often used to prove some identities. However, even with your suggestion I do not know how to apply it here to improve readability (and reliablity) of the proof. But I'll give it a try.
I would like to know, what I am missing. For me it seems easy, but, after a talk with a professor at the math-department here at my university I am a bit discouraged concerning my abilities doing formal proofs...;-) I missed all the classes of formal math education, when I moved from studying computer-science to social-assistance, so I'm doing number-theory only as a hobby in spare time.
Back to the proof. All what I employ is linear combinations and interchanging of order of summation, plus, and this may be the crucial point, to assume, that I can use the values of the eta-function as replacements of the infinite sums of cofactors.
A notation
\( \hspace{24} V(x)\sim * Bs = V(y)\sim \)
is nothing more than a shortcut for the explicite, well known exponential-series for all consecutive powers of y; say in a column c of the result vector
(1) \( \hspace{24} y^c = \sum_{r=0}^{\infty} x^r * log(s)^r * c^r/r! \)
where from the elementary properties of the exponential-series
\( \hspace{24} y^c = (s^x)^c \)
What I am doing then is to apply the linear combination of consecutive x, beginning at x=0 to that formula, expecting, that the result is again the corresponding linear combination:
\( \hspace{24} \begin{eqnarray}
y_0^c &&=&& \sum_{r=0}^{\infty} x_0^r * log(s)^r * c^r/r! \\
y_1^c &&=&& \sum_{r=0}^{\infty} x_1^r * log(s)^r * c^r/r! \end{eqnarray} \)
and
\( \hspace{24} y_0^c - y_1^c = \sum_{r=0}^{\infty} (x_0^r-x_1^r) * log(s)^r * c^r/r! \)
where I expect, that this does not need a special proof (but may be, I'm in error already here)
The crucial point is then to assume, that this is valid for infinite alternating series of (x_0^r - x_1^r + x_2^r - + ... ) , where x_k are the natural numbers, such that
\( \hspace{24} (x_0^r - x_1^r + x_2^r - + ... ) = (0^r - 1^r + 2^r - 3^r...) \)
and that
\( \hspace{24} (0^r - 1^r + 2^r - 3^r...) = eta_0 ( r) \)
is interchangable for the linear combination of x_k.
What we have is then the double-sum for a column c of the result-vector
\( \hspace{24} y^c = \sum_{n=0}^{\infty} \sum_{r=0}^{\infty} ((-1)^n *n^r) * log(s)^r * c^r/r! \)
which is, after interchanging the order of summation
(2) \( \hspace{24} \begin{eqnarray}
y^c &&=&& \sum_{r=0}^{\infty} log(s)^r * c^r/r! * \sum_{n=0}^{\infty} (-1)^n*n^r \\
y^c &&=&& \sum_{r=0}^{\infty} log(s)^r * c^r/r! * eta_0(-r) \\
\end{eqnarray}
\)
The lhs is now the sum
\( \hspace{24} y^c = \sum_{n=0}^{\infty} (-1)^n*(s^n)^c \)
but the interesting result is only in the column where c=1 so
\( \hspace{24} y = \sum_{n=0}^{\infty} (-1)^n*(s^n) = s^0 - s^1 + s^2 ... \)
Then, on the rhs in (2), I use the fact, that each second eta0(-r) = 0, and also I add the remaining eta0(-(2r+1)) with positive and negative signs to zero, which gives then the result (I omit here the other details in my article).
Since powers of Bs are independent of the parameter of x and we can write the serial notation for each power of Bs in a similar form of (1), the reasoning down to (2) is exactly the same for any height of the towers.
The only two possible problems, which I can see here, are the questions, whether the order of summation can be exchanged, and whether the linear combination of V(1)-V(2)+V(3)-V(4)... can be replaced by the eta-values of appropriate exponents.
For the base-parameter s in the range 1/e^e < s < e^(1/e) the series are not too much diverging even for other heights of the towers, and since the sign is alternating, they can be regularly Euler-summed.
This all is nothing else than to rewrite in serial-notation, what is implicite when using the notation of matrix-multiplication.
Hmmm.... If there is something else missing, I would like to learn, what this is (perhaps I could even satisfy my partner of the short discussion here in the math-departement :-))
Regards -
Gottfried
Gottfried Helms, Kassel
