Articles of closed form

The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$

Let $n\in\mathbb N$ ($n>0$), and define the $n$th integral in the sequence $I$ to be $$I_n = \int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx.$$ Evaluating such integrals, especially for small $n$, is essentially a straightforward exercise in complex analysis (integrate on a semicircle in the UHP using the residue theorem, send the radius to $+\infty$ and show that the integral on […]

Closed Form Expression of sum with binomial coefficient

I have the following equation which is making me problems. $$A_{n} = \sum_{k=0}^{n} \binom{n-k}{k}(-1)^{k}$$ where $n\in\mathbb{N}$. The task is to find a closed form expression for $A_{n}$. I have two questions: What is a closed form expression? We never defined it in my combinatorics class and on the internet there are several different definitions. How […]

Integral involving a dilogarithm versus an Euler sum.

By using the integral representation of harmonic numbers and by using elementary integration we can fairly easily find the generating function of squares of harmonic numbers: \begin{equation} {\mathcal S}(x) := \sum\limits_{n=1}^\infty H_n^2 x^n = \frac{Li_2(1)- Li_2(1-x)}{1-x} + \frac{-\log(x) \log(1-x) + [\log(1-x)]^2}{1-x} \end{equation} Now let us take $q \ge 2$. By consecutively dividing both sides by […]

closed form expression of a hypergeometric sum

After playing around with transforms of a certain parametric integral, I am inclined to think that the linear combination $$f(n):=\dfrac1{n-2}\left({\,}_2F_1(\dfrac{n-2}{4n},\dfrac12;\dfrac{5n-2}{4n};-1)\right)+\dfrac1{n+2}\left({\,}_2F_1(\dfrac{n+2}{4n},\dfrac12;\dfrac{5n+2}{4n};-1)\right)$$ has a closed form for integer $n$. I know for example that $f(3)=\dfrac{1}{12^{3/4}}\dfrac{\Gamma(\frac14)^2}{\sqrt{\pi}}$. Any ideas? Edit: putting $a:=\frac14-\frac1{2n}$, we can define $g(a):= \frac{8}{1-4a}f(\frac{8}{1-4a})$ to get arguments closer to the “standard” notation used in formula collections. […]

Multi-index power series

What is closed-form expression for the summation $$ S(n,m)=\sum_{|\alpha|=m} p^{\alpha} = \sum_{\alpha_1 + \cdots + \alpha_n = m} \prod_{i=1}^n p_i^{\alpha_i} $$ as a function of $n$ and $m$? Here $\alpha=(\alpha_1,\cdots,\alpha_n)$ is a $n$-tuple of non-negative integers. For specific values of $n$ the sum has a closed form, e.g. $S(1,m)=p_1^m, S(2,m)=\frac{p_2^{m+1}-p_1^{m+1}}{p_2-p_1}$, etc. I wonder if there […]

Closed form for $\int \frac{1}{x^7 -1} dx$?

I want to calculate: $$\int \frac{1}{x^7 -1} dx$$ Since $\displaystyle \frac{1}{x^7 -1} = – \sum_{i=0}^\infty x^{7i} $, we have $\displaystyle(-x)\sum_{i=0}^\infty \frac{x^{7i}}{7i +1} $. Is there another solution? That is, can this integral be written in terms of elementary functions?

A closed form for $\int x^nf(x)\mathrm{d}x$

When trying to find a closed form for the expression $$\int x^nf(x)\mathrm{d}x$$ in terms of integrals of $f(x)$ I found that $$\int xf(x)\mathrm{d}x=x\int f(x)\mathrm{d}x-\iint f(x)(\mathrm{d}x)^2$$ $$\int x^2f(x)\mathrm{d}x=x^2\int f(x)\mathrm{d}x-2x\iint f(x)(\mathrm{d}x)^2+2\iiint f(x)(\mathrm{d}x)^3$$ $$\int x^3f(x)\mathrm{d}x=x^3\int f(x)\mathrm{d}x-3x^2\iint f(x)(\mathrm{d}x)^2+6x\iiint f(x)(\mathrm{d}x)^3-6\iiiint f(x)(\mathrm{d}x)^4$$ In general, it seems to be the case that $$\int x^nf(x)\mathrm{d}x=\sum_{k=0}^n(-1)^k\frac{\mathrm{d}^k}{(\mathrm{d}x)^k}x^n\int^{k+1}f(x)(\mathrm{d}x)^{k+1}$$ Using a formula for the repeated derivative of […]

How to derive a closed form of a simple recursion?

Consider: $$T(n) = 2 T(n-1) + 1$$ with $T(1)$ a positive integer constant $a$. I just stuck in finding a closed form for this simple recursion function. I would appreciate it, if someone gives me a hint.

How can I show that $\prod_{{n\geq1,\, n\neq k}} \left(1-\frac{k^{2}}{n^{2}}\right) = \frac{\left(-1\right)^{k-1}}{2}$?

Assume $k$ positive integer. How can I show that $$ \tag 1 \prod_{{n\geq1,\, n\neq k}} \left(1-\frac{k^{2}}{n^{2}}\right) = \frac{\left(-1\right)^{k-1}}{2}? $$ I know that $$ \tag 2 \underset{n\geq1}{\prod}\left(1-\frac{k^{2}}{n^{2}}\right)=\frac{\sin\left(\pi k\right)}{\pi k}$$ and so if $k$ is an integer the product is $0$ but how can I use these information?

Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$

This question already has an answer here: Evaluate $\sum_{n=1}^\infty nx^{n-1}$ 2 answers