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Results tagged with reference-request
Search options answers only
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user 90413
This tag is used if a reference is needed in a paper or textbook on a specific result.
1
vote
Accepted
A Vandermonde like determinant with exponentials
After slight renumbering $(m,n)\mapsto (m,n)-1$ and horizontal reflection $j\mapsto m+1-j$, let's call the $m\times m$ matrix in the OP's determinant
$$
A=\left[\sum_{k=i}^{i+n-1}s_k^{j-1}e^{-s_k}\pro …
- 3,671
2
votes
The diophantine equation $ \sum_{n=1}^{N} \frac{1}{x_{n}} = \prod_{k=1}^{N} \left(1-\frac{1}...
A short answer related to OP's Q3, derived from Gareth's answer and my comments: Define the (characteristic) polynomial of the sequence $x=\{x_1,...,x_N\}$,
$$ \tag{1}\label{eq:1}
P_{x}(\xi)=\prod_{k= …
- 3,671
6
votes
Possible new series for $\pi$
This is a non-answer-followup to the non-answer of @HenriCohen:
Noting that
$$\tag{*}\label{eq:*}
1-\lambda+\frac{(\lambda-s_1)(\lambda-s_2)}{n+\lambda}=
\frac{(n+s_1) (n+s_2)}{n+\lambda} -n-s_1-s_2+1 …
- 3,671
6
votes
Solving a three-parameter recursive sequence
I assume a typo as suggested in my comments, such that
\begin{align}
f(\alpha,\beta,\gamma)
&=(2\alpha+8\beta+12\gamma-1) \, f(\alpha-1,\beta,\gamma) \\
{}&- 2(\alpha+1) \, f(\alpha+1,\bet …
- 3,671