Unanswered Questions
7,519 questions with no upvoted or accepted answers
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Solving a second-order recurrence relation / Series expansion of a confluent Heun equation
I would like to know whether it is possible to solve (in "closed form") either one of the following two second-order recurrence relations, which are closely related to each other. The first ...
- 1,145
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42
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Can P-recursive functions assume only prime values?
A function $f\colon \{0,1,\dots\}\to \mathbb{R}$ is P-recursive if
it satisfies a recurrence $$
P_d(n)f(n+d)+P_{d-1}(n)f(n+d-1)+\cdots+P_0(n)f(n)=0,\ n\geq 0, $$
where each $P_i(n)\in \mathbb{R}[n]$ ...
- 49.9k
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30
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Restriction of the local Artin map on the valuation ring of a local field
Let $F$ be a local field, in particular a finite extension of $\mathbb{Q}_p$ for some prime $p$ and let $Art_{L}: L^\times \to Gal(F^{ab}/F)$ be its local Artin map. We know that if $L/F$ is a finite ...
- 263
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Why is $ULU=NU$ (a refinement of $|N|=q^{n^2-n}$)?
Let $G=GL_n(\mathbb{F}_q)$, $U$, $L$, $N$ the subsets of upper-triangular unipotent, lower-triangular unipotent, all unipotent matrices respectively. Then $ULU=NU$ means that for any $g\in G$ the ...
- 3,632
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How many Tverberg partition are in cloud of points?
Tverberg's Theorem: A collection of $(d+1)(r-1) +1$ points in $\mathbb{R}^d$ can always be partitioned into $r$ parts whose convex hulls intersect.
For example, $d=2$, $r=3$, 7 points:
Let $p_1, p_2,...
- 189
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Closed form for the A357990 using A329369 and generalised A373183
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor, \\
\ell(0) = -1
$$
Let
$$
f(n) = \ell(n) - \ell(n-2^{\ell(n)}) - 1
$$
Here $f(n)$ is A290255.
Let $A(n,k)$ be a square array such that
$$
A(n,k)...
- 4,024
1
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118
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Property of a sequence on $\mathbb Q[\sqrt m~]$
Given that $a_{1} = \sqrt m$ in which $m$ is a integer that is not the square of any integer. And $$a_{n+1}=\frac{[a_{n}]}{\{a_{n}\}}$$where $[~ ]$ and $\{~ \}$ respectively represent the integer part ...
- 12.3k
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38
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Isotropic vectors of quadratic forms over number fields
By Meyer's theorem, an indefinite quadratic form $Q$ over $Z$ has an integral isotropic vector is the dimension is at least $5$ and this bound is tight. Indeed, in dimension $4$ there are indefinite ...
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54
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Syntomic f-cohomology for open varieties
Syntomic cohomology $H^{i+j}_{\mathrm{syn}}(X,n)$ of a proper variety $X$ with good reduction over a $p$-adic field $K$ is computed via a spectral sequence in terms of $H^i_{\mathrm{f}}(G_K;H^j_{\...
- 15.5k
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50
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Diophantine approximation for polynomials in a single variable
Let $a_0, a_1, \cdots, a_n$ be non-zero algebraic numbers. Consider the polynomial
$$\displaystyle f(x) = a_n x^n + \cdots + a_1 x + a_0.$$
For a real number $\alpha$, put $\lVert \alpha \rVert$ for ...
- 25.9k
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46
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Proving we can minimize the number of crossings by having a planar embedding of $K_{2,2}$ encircle another out of any 2 such embeddings
Say that we draw a graph in the following way: we first draw $n$ planar embeddings of $K_{2,2}$ (that is, we first draw $n$ quadrilaterals) such there are no edges which cross. Then for each of the $...
- 1
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147
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Possible implications of the bound $\sum_{n\leq x}\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)=O\left(x\right)$
Let $\lambda(n)$ be the Liouville function and consider the Goldbach-type problem $\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)$. Assume the Riemann hypothesis, the ...
CommunityBot
- 1
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Tamari lattice and bicategory coherence
Reference links:
Tamari lattice (Wikipedia): https://en.wikipedia.org/wiki/Tamari_lattice
Associahedra: https://en.wikipedia.org/wiki/Tamari_lattice#/media/File:Tamari_lattice.svg
The Tamari lattice ...
- 62.3k
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51
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Relation between the field and $\mathbb{Z}$-algebra generated by eigenvalues of modular form
Cross-posted from MSE: https://math.stackexchange.com/questions/4944262/relation-between-the-field-and-mathbbz-algebra-generated-by-eigenvalues-of
Let $f$ be a cusp form of weight $k\in\mathbb{Z}$ for ...
- 337
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0
answers
82
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Evaluating the difference of weighted binomial coefficients
I encountered the following type of sum:
$$
\begin{align}
\left[
\sum_{k=1}^{t}\binom{k+i-2}{i-1}\binom{t-k+l_1-i}{l_1-i}\sum_{s=k}^{t}\binom{t-s+l_2-j+1}{l_2-j+1}\binom{s+j-3}{j-2}
\right] \tag{1} \\
...
- 11