I'm extending a note, which I posted to sci.math.research today.
I hope to find a solution of the problem by a reconsideration of the structure of the matrix of stirlingnumbers 1'st kind, which may contain "Infinitesimals" which become significant if infinite summing of its powers are assumed.
The error may essentially be due to a misconception about the matrix of Stirling-numbers 1'st kind and the infinite series of its powers.
Assuming for an entry \( p_{r,r+1} \) of the first upper subdiagonal in row r (r beginning at zero) in the pascal matrix P
\( \hspace{24} p_{r,r+1} = binomial(r,r+1) \)
which is an infinitesimal quantity and appears as zero in all usual applications of P. But if applied in a operation including infinite series of consecutive powers, then in the matrix Z, containing this sum of all consecutive powers of P we get the entries
\( \hspace{24} z_{r,r+1} = binomial(r,r+1)* \sum_{k=1}^{\infty} \frac1k \)
By defining
\( \hspace{24}
binomial(r,r+1) = \frac{r!}{(r+1)!(-1)!} = \frac1{(r+1)*(-1)!}
\)
assuming \( \frac{\zeta(1)}{(-1)!} = - 1 \) this leads to the non-neglectable rational quantities in that subdiagonal of the sum-matrix
\( \hspace{24} \begin{eqnarray}
z_{r,r+1}& = &\frac1{(r+1)* (-1)!} * \sum_{k=1}^{\infty} \frac1k \\
& = &\frac1{(r+1)} * \frac{\zeta(1)}{(-1)!} \\
& = & - \frac1{(r+1)} \end{eqnarray}
\)
With that correction the infinite series of powers of the pascal-matrix leads then to a correct matrix,
\( \hspace{24} z_{r,c} = binomial(r,c)* \zeta(c-r) \)
including the above definitions for the first upper subdiagonal, which provides the coefficients of the (integrals of) the bernoulli-polynomials, and can be used to express sums of like powers as expected and described by H.Faulhaber and J.Bernoulli.
(for more details see powerseries of P, page 13 ff )
This suggests then to reconsider the matrix of Stirling-numbers of 1'st kind with the focus of existence of a similar structure in there.
Does this sound reasonable? It would require a description of the matrix of Stirling-numbers 1'st kind, which allows such an infinitesimal quantity.
But there is one important remark: this matrix contains the coefficients of the series for logarithm and powers of logarithms. A modification of this matrix would then introduce an additional term in the definition of these series. Something hazardeous...
I hope to find a solution of the problem by a reconsideration of the structure of the matrix of stirlingnumbers 1'st kind, which may contain "Infinitesimals" which become significant if infinite summing of its powers are assumed.
------------------------------- (text is a bit edited) -----------------
Due to a counterexample by Prof. Edgar (see sci.math) I have to withdraw this conjecture.The error may essentially be due to a misconception about the matrix of Stirling-numbers 1'st kind and the infinite series of its powers.
----
It is perhaps similar to the problem of the infinite series of powers of the pascal-matrix P, which could be cured by assuming a non-neglectable infinitesimal in the first upper subdiagonalAssuming for an entry \( p_{r,r+1} \) of the first upper subdiagonal in row r (r beginning at zero) in the pascal matrix P
\( \hspace{24} p_{r,r+1} = binomial(r,r+1) \)
which is an infinitesimal quantity and appears as zero in all usual applications of P. But if applied in a operation including infinite series of consecutive powers, then in the matrix Z, containing this sum of all consecutive powers of P we get the entries
\( \hspace{24} z_{r,r+1} = binomial(r,r+1)* \sum_{k=1}^{\infty} \frac1k \)
By defining
\( \hspace{24}
binomial(r,r+1) = \frac{r!}{(r+1)!(-1)!} = \frac1{(r+1)*(-1)!}
\)
assuming \( \frac{\zeta(1)}{(-1)!} = - 1 \) this leads to the non-neglectable rational quantities in that subdiagonal of the sum-matrix
\( \hspace{24} \begin{eqnarray}
z_{r,r+1}& = &\frac1{(r+1)* (-1)!} * \sum_{k=1}^{\infty} \frac1k \\
& = &\frac1{(r+1)} * \frac{\zeta(1)}{(-1)!} \\
& = & - \frac1{(r+1)} \end{eqnarray}
\)
With that correction the infinite series of powers of the pascal-matrix leads then to a correct matrix,
\( \hspace{24} z_{r,c} = binomial(r,c)* \zeta(c-r) \)
including the above definitions for the first upper subdiagonal, which provides the coefficients of the (integrals of) the bernoulli-polynomials, and can be used to express sums of like powers as expected and described by H.Faulhaber and J.Bernoulli.
(for more details see powerseries of P, page 13 ff )
---
This suggests then to reconsider the matrix of Stirling-numbers of 1'st kind with the focus of existence of a similar structure in there.
-------------------------------------------------------
Does this sound reasonable? It would require a description of the matrix of Stirling-numbers 1'st kind, which allows such an infinitesimal quantity.
But there is one important remark: this matrix contains the coefficients of the series for logarithm and powers of logarithms. A modification of this matrix would then introduce an additional term in the definition of these series. Something hazardeous...
Gottfried Helms, Kassel
