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### Regularizing the divergent sum $1^k + 2^k + \cdots$

EDIT: Under this "regularization", the harmonic series can be interpreted as $s_{-1}$ and assigned a value $$s_{-1} = \lim_{k \rightarrow 1} \zeta(k) (2-2^k) = -2 \log 2$$ I was looking at ...
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### How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in François Brunault's answer to the question Freeness of a Z[x] module depicted below? -- Choosing colors according to the highest power of 2 which ...
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### If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?

One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
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### Composition of two formal series

There are two formal semi-infinite Laurent series $$f_+(z)=z+\sum_{k=2}^{\infty} a_k z^k$$ and $$f_-(z)=z+\sum_{k=0}^{\infty} b_k z^{-k}$$ Their composition (we assume that this composition ...
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### About the first decimal of $\sqrt {n!}$

Do we have : $$\sup\{\sqrt {n!} - E(\sqrt {n!}); n\in I\!\!N\}=1?$$ Where $E(\cdot)$ is the integer part function, and $n!=1\times 2...\times n$.
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### Analytic expression for the Tsirelson bound of the I3322 inequality?

Finding Tsirelson bounds for Bell inequalities is a well-loved problem in quantum information theory. A famous case where it is still open is for the I3322 inequality. In this paper Pál and Vértesi ...
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### Asymptotic behavior of a sequence of functions

For $n\in\mathbb{N}$ and $q\in(0,1)$, define $$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$ ...
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### Graphs with graphic imbalance sequences

Let $G$ be simple undirected graph and $e=uv\in E(G)$. The imbalance of the edge $e$ is the value $imb(e)=|d(u)-d(v)|$. Let $M_{G}$ denotes the imbalance sequence (or more correctly, multiset of ...
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None of the Harmonic numbers $H_n = \sum_{k=1}^n 1/k$ are integers for $n>1$ (e.g., this MSE question and answer). Q. Define the $r$-th root Harmonic number $H_n^{1/r} = \sum_{k=1}^n 1/{k^{1/r}}... 0answers 105 views ### On the comparison of Egyptian fractions of two kinds I posted the question on MSE here but it did not get any answer. Consider $$S(n)=\left\{(a_1 ,a_2,a_3, \dots, a_n)\mid a_1\le a_2\le\cdots\le a_n, \; \sum_{r=1}^{n}\frac{1}{a_r} = 1\right\} \subset \... 0answers 134 views ### “Shifted” Vandermonde determinant is nonzero? I have already posted this question at MSE here, but as it received a few upvotes, but no comments or answers I choose to cross-post it here. Let P be a degree-two polynomial, with roots \alpha,\... 0answers 116 views ### Prove that when converge, the following expansions are equal Prove f_1(x)=f_2(x)=f_3(x) when converge.$$f_1(x)=\sum_{m=0}^{\infty} \binom {x}m \sum_{k=0}^m\binom mk(-1)^{m-k}f(k)f_2(x)=\lim_{n\to\infty}\binom xn\sum_{k=0}^n\frac{x-n}{x-k}\binom nk(-1)^... 0answers 138 views ### A second polylogarithm ladder for the tribonacci and n-nacci constants In "A seventeenth-order polylogarithm ladder", on page 6 (eq 25), Bailey et al give a dilogarithmic ladder that begins with, $$0 = \operatorname{Li_2(\alpha_1^{-630})}-2\operatorname{Li_2(\alpha_1^{-... 0answers 152 views ### Shift-invariant submultiplicative seminorms of \ell^{\infty} Question: Is there a shift-invariant submultiplicative seminorm ||\cdot|| of \ell^\infty which satisfies the following property? If f:\mathbb{N}\rightarrow\mathbb{N} is an increasing function ... 0answers 272 views ### Convolution inverse of recursively defined sequence is alternating Consider the double sequence A(n,k) which is recursively defined by$$A(n,n)=1 \text{ for } n=0,1,2,\dots \text{ and }A(n,k)=2\sum_{l=1}^{k+1} \binom{2n+1}{2l} A(n-l,k+1-l) \text{ for }0\leq k &... 0answers 238 views ### Alternating sums of powers of the lngamma ($\small f_p(x) = \sum \log(k!)^p x^k $at$\small x=-1$) I'm still fiddling with this recent question and come to a detail, whether I can find closed forms for the sums of the$\small lngamma() $function. Precisely $$\small f_p(x) = \sum_{k=0}^{\infty} \... 0answers 765 views ### Method for variable substitution in multiple summation I want to ask: is there any general method for variable substitution in multiple summation? For example in the following equation a new variable \lambda=n+m-2\mu is introduced to transform the LHS ... 0answers 333 views ### Finch's sequence over \mathbb{F}_3 In http://algo.inria.fr/csolve/seqmod3.pdf -- "Periodicity in sequences mod 3" Steven Finch (also cited in Sloane's OEIS A112683) defines the following sequences in \mathbb{F}_3: For each positive ... 0answers 74 views ### Relating face polytopes of permutohedra to integer partitions The OEIS entries A019538, A049019, and A133314, relate a refinement of the face polynomials of the permutohedra (A049019) to partition polynomials (A133314) defined by multiplicative inversion of an ... 0answers 78 views ### How to estimate \prod_{t=1}^{N}\frac{1}{2-z^t} for large N? Based on the top answer to How to estimate of \prod_{k=a}^N \frac{1}{e^{k\kappa}-1} for large N? Can anyone find an approximate closed form for$$ \frac{\mathrm{d}^k}{\mathrm{d}z^k}\prod_{t=1}^{N}... 0answers 103 views ### Enumerating the number of degree d curves tangent to a planar conic This question is based on a special case of the Coparaso Harris formula, as described in Counting curves on rational surfaces - R. Vakil. Let$E$be a non-singular planar conic. Then every degree$d$... 0answers 62 views ### Proving convergence is impossible for a sum of hyperbolic cosines Suppose that$z$is some complex value. Is it possible to prove that $$\lim_{n \rightarrow \infty} \sum_{j = 1}^n {\sqrt{n \over j}} \cdot \cosh(z \log {n \over j}-\operatorname{ Arccoth} (2z))$$ ... 0answers 109 views ### Oscillation aspects of two-way infinite alternating series (a followup from the MO-question “functions that eventually oscillate”) In the recent question on "eventually oscillating function" I had a heuristic for the function$d(x)$that its amplitude is constant, but could not further describe that function. I just found a ... 0answers 204 views ### Minimizing$\{0,1\}$-sequence permutations Explanation: For a given bit sequence$f$, reposition the bits as to minimize$G$which can be thought of as a measure of how poorly proportional$f$is to each of its subsequences. Let$p \in \...
Is there any closed form expression for the infinite sum $\sum_{n \geq 0}q^{n(n+1)/2}(1+q)(1+q^2)\cdots(1+q^n)u^n$ where both $q$ and $n$ are variables and $n \in N \cup {0}$?
Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and $Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything on this sequence has been proved -- not even that $Q(n)$ is well-...