Questions tagged [closed-form-expressions]

For questions that specifically ask for determining a closed form of equations, integrals etc.

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2 votes
0 answers
74 views

Closed form for coefficients related to excedance set of permutation

Working on suitable closed form for A329369, I discovered very useful coefficients, which have the following recurrence relation: $$T(0,1)=T(0,2)=1$$ $$T(n,1)=1, n>0$$ $$T(0,k)=0, k>2$$ $$T(2n+1,...
2 votes
0 answers
90 views

Closed form for $a(2^m(2^n-2^p-1))$

Let $q(n)$ be A007814, i.e., the number of trailing zeros in the binary representation of $n$. Here $$q(2n+1)=0, q(2n)=q(n)+1$$ Let $a(n)$ be A329369. Here $$a(2n+1)=a(n), a(2n)=a(n)+a(n-2^{q(n)})+a(...
4 votes
1 answer
153 views

Closed-form examples of CMC surfaces

Besides the trivial cases of cylinders and spheres, are there any other known examples of non-zero constant mean curvature surfaces which can be represented explicitly in a closed form? I am ...
1 vote
1 answer
96 views

Number of steps required to get one ball in each box for $n=2^k$

Given $n$ balls, all of which are initially in the first of $n$ numbered boxes, $a(n)$ is the number of steps required to get one ball in each box when a step consists of moving to the next box every ...
3 votes
1 answer
425 views

Is there real or complex analytic function whose positive real zeros are the primes?

Related to this question Q1 Is there real or complex analytic function $f(x)$ such that its positive real zeros are the primes and it is given in closed form of compositions of already named ...
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8 votes
1 answer
704 views

Closed form for ₄F₃(n,n,n,2n;1+n,1+n,1+n;−1)

For positive integer $n$ the following value of a hypergeometric function $$_4F_3(n,n,n,2n,1+n,1+n,1+n,-1)$$ based on the first few terms looks like $$ R_1(n) + R_2(n) \pi^2$$ where $R_{1,2}(n)$ are ...
1 vote
0 answers
114 views

Solution that minimizes the sum of squared errors, with quadratic constraints

Given symmetric and positive definite $n \times n$ (real) matrices $A_1, \dots, A_m$ and $b_1, \dots, b_m \in {\Bbb R}^{n}$, I am trying to find the solution with the least sum of squared errors of ...
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1 vote
0 answers
49 views

How to extend this sum involving generalized harmonic numbers?

It is well-known since Euler that the Generalized harmonic numbers, defined for $n\in\mathbb N$ by $$H_n^{(r)}=\sum_{k=1}^n\frac1{k^r},$$ can be naturally extended for non integer $n$ in terms of ...
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1 vote
1 answer
115 views

How to solve for $a$ in $\sum_{j=i}^n (a -j) \binom{n}{j} y^j (1-y)^{n-j} = 0$

The Problem Given $i, n, y$, I am trying to find a closed form solution for $a$ in the equation $$ \sum_{j=i}^n (a -j) \binom{n}{j} y^j (1-y)^{n-j} = 0 $$ Do you have any recommendations on how to ...
3 votes
0 answers
119 views

Closed form for $\sum_{k=1}^\infty\frac{H_k^{(m)}}{k^n}$

Let's define $$\sigma(m,n)=\sum_{k=1}^\infty\frac{H_k^{(m)}}{k^n}$$ where $H_k^{(m)}=\sum_{n=1}^{k}\frac{1}{n^m}$ is the k-th generalized harmonic number of order $m$. In mathworld site Eq (20), I ...
1 vote
1 answer
257 views

Is this long closed form for pi trivial?

With the help of wolfram alpha we got very long closed form for $\pi$ in terms of algebraic numbers, logarithms of algebraic numbers and $cot^{-1},coth^{-1}$ which can be expressed as logarithms. From ...
  • 23.7k
8 votes
3 answers
472 views

Why mpmath computes $\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right)$

Working with precision 500 decimal digits, mpmath in sage computes: $$\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right).\tag{1}\label{1}$$ We believe the LHS of \eqref{1} ...
  • 23.7k
3 votes
3 answers
1k views

Surprisingly long closed form for simple series

For natural $A$ define $$ f(A)=\sum_{n=1}^\infty \frac{1}{A^n}\left(\frac{1}{An+1}- \frac{1}{An+A-1}\right)$$ $f(A)$ is BBP (Bailey-Borwein-Plouffe) formula and allows digit extraction in base $A$. ...
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0 votes
0 answers
29 views

How to find close form roots or at least good approximation of roots of such function?

I need to solve for $D$: $$ KD^{N-1} S + P = KD^N + \left(\frac{D}{N}\right)^N, $$ where $$ S = \sum_i x_i, \, P = \prod_i x_i, $$ $$ K_0 = P \left(\frac{N}{D}\right)^N, \, K = AK_0 \frac{\gamma^2}{(\...
3 votes
0 answers
77 views

Minimal eigenvalue of infinite dimensional matrix

Consider the following symmetric, positive-definite matrix $$ H_{nm}=-\frac{(4 m+4 n+1)}{(4 m-4 n-1) (4 m-4 n+1)}\sqrt{\frac{(4 m-1)!! (4 n-1)!!}{(4 m)!! (4 n)!!}} $$ where $n,m=0,1,2\dots$ (Here $!!$ ...
5 votes
1 answer
157 views

Asymptotics of error function integral with square root

I am interested in the asymptotics of the integral $$I(a):=\int_0^\infty \sqrt{x}\operatorname{Erfc}(x+a)\,\mathrm{d}x$$ for $a>0$. I think that $I(a)$ should be decaying exponentially as $I(a)\...
  • 563
4 votes
0 answers
132 views

Trigonometric sum and residues

I am interested in the sum $$ \sum_{n=1}^k 2\biggl[\sin\biggl(\frac{n\pi}{2k+2}\biggr)\biggr]^{-2g} $$ where $k$, $g$ are integers. It is not too hard to show that this can also be expressed as $$ -1-...
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2 votes
0 answers
154 views

Width of the peaks in a Dyck's path

I have a pretty simple question for which I was not able to find a so simple answer. Introduction I was playing around with some of the mathematical objects that can be enumerated by Catalan numbers. ...
0 votes
0 answers
57 views

Weak derivative of projection onto probabilist's simplex

Let $\Delta_n:=\{x\in [0,1]^n:\boldsymbol{1}^{\top}x=1\}$ denote the probabilist's $n$-simplex and let $P:\mathbb{R}^n\rightarrow\Delta_n$ denote the (Euclidean) metric projection onto this simplex ...
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1 vote
1 answer
75 views

Simplification of $\sum_{m=0}^\infty \text B_z(m+a,b-m)x^m,\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}$ in terms of Kampé de Fériet function

Here is the goal sum where the Pochhammer Symbol with the Incomplete Beta function series $$\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}=\sum_{m=0}^\infty z^{m+a}\sum_{n=0}^\infty\frac{(1-(b-m))...
3 votes
1 answer
177 views

Volumes of unit balls in $S^2 \times\mathbb R$ and $H^2 \times\mathbb R$

The following integrals are equal to the volume of a unit ball in $S^2 \times \mathbb R$ and $H^2 \times\mathbb R$, respectively: $$8\pi\int_0^1\sin^2 \frac{\sqrt{1-h^2}}2 \, dh$$ $$8\pi\int_0^1\sinh^...
1 vote
0 answers
110 views

Closed form for $\sum _{k=1} ^n \frac k 2 \,\operatorname{sgn} \left( \frac 1 {k^2} + \cos \frac {2\pi n} k-1 \right)$ [closed]

I am looking for the closed form of $$\sum _{k=1} ^n \frac k 2 \,\operatorname{sgn} \left( \frac 1 {k^2} + \cos \frac {2\pi n} k-1 \right) \ .$$ Wolfram Alpha cannot do this for me, so I am forced to ...
2 votes
0 answers
33 views

Reduction of the general Lauricella hypergeometric function $F_B$ for identical parameters and variables

The Lauricella function $F_B^{n}$ of $n$ variables is defined as $$F_B^{(n)}(a_1, \ldots, a_n, b_1, \ldots, b_n, c; x_1, \ldots x_n) = \sum_{k_1, \ldots, k_n = 0}^\infty \frac{1}{(c)_{k_1 + \ldots + ...
  • 121
0 votes
1 answer
191 views

Where is the source of the formula $\sum_{j=0}^\infty \bigl(j+\frac{1}{2}\bigr)^{n-1}\frac{2^{j+1/2}}{\binom{2j+1}{j+1/2}}$ for an integer sequence?

The infinite series representation \begin{equation} \frac1\pi\sum_{j=0}^\infty \biggl(j+\frac{1}{2}\biggr)^{n-1}\frac{2^{j+1/2}}{\binom{2j+1}{j+1/2}}, \quad n\ge0 \end{equation} for the positive ...
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2 votes
0 answers
215 views

Is there a closed form of $ \displaystyle \sum_{k=0}^{\infty}{\frac{\phi^{xk}}{k!_F}}$

where $\phi = \frac{1+\sqrt{5}}{2}$ and $k!_F$ is the fibonorial of $k$, or the product of the first $k$ Fibonacci numbers? My hunch is that, this can be represented as a function in terms of the ...
1 vote
0 answers
158 views

How to explicitly obtain an analytic function whose power series coefficients are sums over integer compositions?

Starting with the following differential equation, \begin{eqnarray} x \frac{\partial^3}{\partial x^3} P[h, x] - \frac{\partial^2}{\partial h^2} \left( h \frac{\partial}{\partial h} P [h, x] \right) ...
1 vote
0 answers
83 views

Any known relations to this doubly exponential constant?

Two years back I was working with lacunary series. In my explorations I had derived that the following series is periodic with period 1: $$ f(x)= \sum_{n=0}^{\infty} \left[ e^{2^{x+n}} \right] + x - \...
1 vote
0 answers
29 views

Solution to dynamic program-type recursion

I have the following dynamic programming principle-type problem. Suppose that we are given a sequence $\beta_1,\dots,\beta_n\in (0,\infty)$, some target $y\in (0,\infty)$ with $y>\sum_{t=1}^N \...
0 votes
0 answers
166 views

How to determine the closed form of this Fourier series?

Consider the Series $$ S(z) \equiv \sum_{n \in \mathbb{Z}, n \ne 0} \frac{ 1 }{ \sin n\pi \tau \sin 2n \pi \tau } e^{2\pi i n z} \ , \quad \operatorname{Im}\tau > 0 $$ I am trying to find its ...
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1 vote
1 answer
216 views

Generalizing closed form representations related to conjectured analytic formulas for $f_a(x)=\sum\limits_{n=1}^x a(n)$

Consider the summatory function $f_a(x)$ defined in formula (1) below where the related Dirichlet series $F_a(s)$ defined in formula (2) below converges for $\Re(s)\ge 2$. $$f_a(x)=\sum\limits_{n=1}^...
0 votes
0 answers
89 views

Requesting proof of closed form of sum involving Fibonacci and Lucas numbers

$$ \sum_{n=0}^{k+1}\frac{3F_{n+1}-L_{n+1}}{2n!}\frac{(k+1)!}{(k-n+1)!}x^{k-n+1}=(\varphi+x)^k\left(\frac{\sqrt{5}}{5}-\frac{\sqrt{5}-5}{10}x\right)+(\psi+x)^k\left(\frac{\sqrt{5}+5}{10}x-\frac{\sqrt{5}...
2 votes
1 answer
280 views

Closed form of $\prod_{k=1}^{n}\left(\cos(kx)-1\right)$

Is there any closed form of $$\prod_{k=1}^{n}\left(\cos(kx)-1\right)?$$ I failed to find references on this problem in the internet.
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3 votes
2 answers
359 views

About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions

NOTE: I post this question on math.stackexchange but nobody answered, so I try here. For a work we need to evaluate the following integral $$\int_{0}^{1}\frac{\log\left(x\right)}{\sqrt{1+x^{4}}}dx=\,-...
3 votes
0 answers
199 views

Spherical harmonic expansion of a power function

Let $f$ be an even continuous function on the sphere $S^{n-1}$. Find a relation of the spherical harmonic expansion between coefficients of $f^n$ and those of $f$.
  • 109
1 vote
0 answers
84 views

Expression for the single common root

Let $ \mathbb{F} $ be a field, consider the polynomial ring $ \mathbb{F} \left[ x\right] $ and suppose that the polynomials $f,g \in \mathbb{F} \left[ x\right]$ have degrees $2,2^n$, respectively, ...
3 votes
1 answer
76 views

General formula for the integral w.r.t to Marchenko-Pastur density, of the ratio of degree $\le 2$ polynomials

Question. Is there a closed-form formula (via standard objects like rational functions, radicals, special functions, special numbers like Catalan numbers, etc.) expressing integrals of rational ...
  • 6,094
0 votes
2 answers
255 views

Closed form for $\sum_{i=1}^n{a^{i^2}}$

Let $a$ be an element of some ring or field, possibly finite. Is there closed form for $\sum_{i=1}^n{a^{i^2}}$? sage and wolframalpha couldn't solve it. If $a$ is primitive n-th root of unity this is ...
  • 23.7k
6 votes
4 answers
395 views

Is this closed-form summation a special case of known Lerch zeta function formulas?

With some Poisson summation manipulations (credit: Michał Pacholski) I have convinced myself of a closed form expression for this conditionally convergent series: $$\sum_{n=-\infty}^\infty \frac{e^{in\...
7 votes
0 answers
260 views

Closed form of the sum $\sum_{r\ge2}\frac{\zeta(r)}{r^2}$

Note: This question has been brought here from MSE. I have been working on various sums involving the zeta function (which come up frequently in my research), and it turned out that many of them had ...
user avatar
0 votes
1 answer
96 views

Closed-form for recursive "geometric-like" recursion

I asked this question of MSE, but to no avail; alas, here I am. Let $k>0$, $C\geq 1$, $\alpha \in (0,1]$, and let $(x_n)_{n\geq 1}$, be a sequence of real numbers given by the recursion $$ x_{n+1} =...
  • 4,971
1 vote
1 answer
119 views

An ambitiouser binomial coefficients sum

I asked how to calculate $$\sum_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$ and got amazing answers. A bit later, however, I figured I needed something rather more complicated: I need to find ...
  • 369
-1 votes
3 answers
272 views

Binomial Coefficients sum [closed]

Any idea on whether or not $$\sum_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$ has a closed formula on $a$ and $b$ (and on what it is, in case it does)? It is supposed that $b \le a$.
  • 369
5 votes
2 answers
407 views

Closed form for $\sum_{j=0}^{\infty}{\alpha \choose j} {\beta \choose j}x^j$

As stated, I wonder if there is a closed form for the generating function $F_{\alpha,\beta}(x):=\sum_{j=0}^{\infty}{\alpha \choose j} {\beta \choose j}x^j$ where $\alpha,\beta \in\mathbb{N}$. Calling ...
  • 2,217
2 votes
0 answers
104 views

How to solve a QCQP where constraints are balls?

I want to solve the following optimization problem in variables $\theta_1, \theta_2, \dots, \theta_K$ \begin{equation} \begin{aligned} & \underset{\theta}{\text{minimize}} & & \...
8 votes
3 answers
391 views

Invertibility of specific function

This is my first post. I'm not a mathematician, just an electronics engineer who loves mathematics. In one of my projects, I arrived at the following function: $$V\left(\varphi\right)=\frac{A\sqrt{\pi-...
6 votes
0 answers
129 views

Fourier transformation of a distribution

We have no idea how to tackle the following Fourier transformation of a distribution: $$ \lim_{\epsilon\to0^+}\int_{-\infty}^{\infty}\mathrm{d} t\int_{\mathbb{R}^{d-1}} \mathrm{d}^{d-1}\vec{r} e^{-\...
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1 vote
1 answer
191 views

Analytically controlling sizes in modular arithmetic to demonstrate Dirichlet pigeonhole application

Given $a,b,c,d\in\mathbb Z$ with $ad+bc=p$ a prime there is an $m\in\mathbb Z$ with $-\lceil1+\sqrt{p}\rceil< r_1,r_2<\lceil1+\sqrt{p}\rceil$ and $$r_1\equiv mac\bmod p$$ $$r_2\equiv mbd\bmod p$$...
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4 votes
0 answers
165 views

Can $ x \sum_{k=1}^{\infty} \frac{1}{k} \Big{(}- \gamma - \psi \big{(}1-\frac{x}{k} \big{)} \Big{)} $ be simplified?

I'm interested in sums of the form $$f_{p} (x) = \sum_{k=2}^{\infty} \zeta(k)^{p} x^{k} .$$ For $p=1$, the following result is known: $$f_{1} (x) = -x \big{(}\psi(1-x) + \gamma \big{)} .$$ (That is, ...
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-2 votes
1 answer
210 views

Is it possible to express $\int_{0+\epsilon}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ in elementary functions?

let $\epsilon >0$, I tried to evaluate $\int_{0+\epsilon}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ , using the fact $x= \cos t$ yield to have integrand using $\sin $ ...
1 vote
0 answers
90 views

Extension of closed-form solvability of polynomial equations of x and exp to rational expressions?

Is my conjecture below true? It's a new conjecture. It's an extension of Lin's theorem in [Lin 1983] which is cited in [Chow 1999] - see both references at the bottom of this page. It seems that Ferng-...
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