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Background

Consider the $(n \times n)$ Hessenberg matrix

$$ A_{n} := \begin{pmatrix} 1/2 & 1/3 & 1/4 & 1/5 & \dots & \dots & 1/(n+1) & \dots \\ 1 & 1/2 & 1/3 & 1/4 & 1/5 & \dots & 1/n \\ 0 & 1 & 1/2 & 1/3 & 1/4 & 1/5 & \dots \\ 0 & 0 & 1 & 1/2 & 1/3 & 1/4 & \dots \\ \vdots & \vdots & & \ddots & \ddots & \ddots \end{pmatrix} . $$

Then it appears to be the case that $$ \det (A_n) = -G_{n+1} \tag{1} \label{1} \ \ ,$$ where $G_n$ is the $n$'th Gregory coefficient. I believe this is proved in the paper Some Properties of the Bernoulli Numbers of the Second Kind and their Generating Function by Qi and Zhao (link paper , p. 3, Theorem 3), though it seems to be stated in a slightly different form. Also, note that the authors employ the notion of "Bernoulli numbers of the second kind," which is another name for the Gregory coefficients.

A Variant

Recently, I also looked into $(n \times n)$ Hessenberg matrices of the form

$$ B_{n} := \begin{pmatrix} 1/4 & 1/9 & 1/16 & 1/25 & \dots & \dots & 1/(n+1)^2 & \dots \\ 1 & 1/4 & 1/9 & 1/16 & 1/25 & \dots & 1/n^2 \\ 0 & 1 & 1/4 & 1/9 & 1/16 & 1/25 & \dots \\ 0 & 0 & 1 & 1/4 & 1/9 & 1/16 & \dots \\ \vdots & \vdots & & \ddots & \ddots & \ddots \end{pmatrix} . $$

The denominators of $\det(B_n)$ seem to align with the coefficients of the expansion of $\operatorname{PolyLog}(-2, x)/\operatorname{PolyLog}(2, x)$, where $\operatorname{PolyLog}(m, x)$ is the polylogarithm function of order $m$. These numbers are tabulated in the OEIS sequence A273698. The pattern appears to break somewhat for $n=5$, as $\det(B_5) = \frac{6151}{691200} $. We do have $ \frac{691200}{\text{A273698}(5)} = \frac{691200}{518400} = 4/3$.

For the numerators, I have not yet been able to discern some kind of pattern. The first five are $1, 7, 13, 6911, 6151$. With the exception of the first number of the sequence, these are prime numbers.

Questions

  1. What is the relationship between $\det(B_n)$ and the coefficients of the expansion of $\operatorname{PolyLog}(-2, x)/\operatorname{PolyLog}(2, x)$, if any?
  2. What is the closed form of the numerators of $\det(B_n)$ ?
  3. Has $\det(B_n)$ and variants thereof (for instance, with $1/n^k$ entries for $k \geq 3$) been considered in the literature? If so, do you have pointers to relevant papers?
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This is a comment on Fred Hucht's answer which doesn't fit in a comment box: Let $a_0$, $a_1$, $a_2$ ... be any sequence of real numbers with $a_0 = 1$ and let $b_0$, $b_1$, $b_2$ ... be defined by $$\sum b_k x^k = \left( \sum a_k x^k \right)^{-1}.$$ Then $$b_k = (-1)^k \det \begin{bmatrix} a_1 & a_2 & a_3 & \cdots & a_k & a_{k+1} \\ 1 & a_1 & a_2 & \cdots & a_{k-1} & a_k \\ 0 & 1& a_1 & \cdots & a_{k-2} & a_{k-1} \\ 0 & 0 & 1 & \cdots & a_{k-3} & a_{k-2} \\ \vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\ 0& 0& 0 & \cdots & 1 & a_1 \\ \end{bmatrix}.$$

Proof sketch: The condition $\left( \sum a_k x^k \right)\left( \sum b_k x^k \right)=1$ translates into an infinite matrix condition $$\begin{bmatrix} 1 &a_1 & a_2 & a_3 & \cdots \\ 0 &1 & a_1 & a_2 & \cdots \\ 0& 0 & 1& a_1 & \cdots \\ 0& 0 & 0 & 1 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{bmatrix} \begin{bmatrix} b_0 \\ b_1 \\ b_2 \\ b_3 \\ b_4 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}. $$ Solve these equations using Cramer's rule. $\square$

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I don't have a pointer to relevant papers, but I found (without proof) the generating function for the general form $$\label{eq:1}\tag{1} A_{k,n}:= \begin{pmatrix} 1/2^k & 1/3^k & 1/4^k & 1/5^k & \dots & \dots & 1/(n+1)^k \\ 1 & 1/2^k & 1/3^k & 1/4^k & 1/5^k & \dots & 1/n^k \\ 0 & 1 & 1/2^k & 1/3^k & 1/4^k & 1/5^k & \dots \\ 0 & 0 & 1 & 1/2^k & 1/3^k & 1/4^k & \dots \\ \vdots & \vdots & & \ddots & \ddots & \ddots \end{pmatrix} $$ using Mathematica: Defining the matrix (and fixing Det[])

Clear[A]; A[k_,0]:={{}}; A[k_,1]:={{2^-k}}; 
A[k_,n_]:=ToeplitzMatrix[Join[{1/2,1}^k,ConstantArray[0, n-2]], 1/Range[2,n+1]^k]
Unprotect[Det];Det[{{}}]=1;Protect[Det];

we consider the formal power series $$\label{eq:2}\tag{2} f_k(z) = \sum_{n=0}^{m} \det A_{k,n} z^n $$

Clear[f]; 
f[k_,max_] := Sum[Det@A[k,n] z^n, {n,0,max}] + O[z]^(max + 1)

and verify the generating function for the Gregory coefficients with $k=1$:

f[1, 30]/(z/Log[1 + z])

Out[] = 1 + O[z]^31

Motivated by the OP's discussion of the polylogarithm, and noting that $$ \operatorname{Li}_1(-z)=-\log(1+z), $$ an educated guess for the general generating function is $$\label{eq:3}\tag{3} g_k(z) = \frac{-z}{\operatorname{Li}_k(-z)}. $$ And indeed,

Clear[g]; g[k_] := -z/PolyLog[k, -z]
Union@Table[f[k,30]/g[k], {k, -10, 10}]

Out[] = {1 + O[z]^31}

is always one, even for nonpositive $k$.

Therefore, I conjecture that equation \eqref{eq:3} is the generating function of $\det A_{k,n}$. Note that the relation even holds for noninteger $k$, e.g.:

Union@Simplify@Table[f[k, 10]/g[k], {k, 0, 4, 1/3}]

Out[] = {1 + O[z]^11}
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    $\begingroup$ Your conjectured formula for the generating function is correct; see my answer. $\endgroup$ Commented Jan 30 at 19:18
  • $\begingroup$ For those like me with less Wolfram savvy, what's happening with Det[]? $\endgroup$ Commented Jan 31 at 0:30
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    $\begingroup$ @LSpice: $\det()$ of the $(0\times 0)$ matrix is undefined in Mathematica, I define it to be $1$ as usual. As a system function, Det[] is protected and must be unprotected before redefinition. $\endgroup$ Commented Jan 31 at 6:07

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