For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.
2
votes
2answers
113 views
Closed form for $\prod_{1 \leq i < j \leq k} (j - i)$?
Is there a closed form for $\prod_{1 \leq i < j \leq k} (j - i)$? It looks like something like a determinant of a Vandermonde matrix, but I can't seem to get it to fit.
0
votes
0answers
102 views
What is the difference between multiplying two functions and sum their product over $x$?
I am studying wavelets and I found this equation in the DWT section:
$$W\varphi(j_0,k)= \frac{1}{M}\sum_x \left(f(x) \varphi(j_0,k)(x)\right)$$
I am wondering which is the difference with this:
...
4
votes
2answers
169 views
Closed form expression for a product.
A simple method for evaluating a product is term cancellation. For example, the product
$$\begin{align*}
\prod_{k=2}^{n}\left(1-\frac{1}{k}\right)&=\prod_{k=2}^{n}\left(\frac{k-1}{k}\right)\\
...
3
votes
1answer
161 views
Is $\prod_{\mathbb{R}}\mathbb{R} = \mathbb{R}^\mathbb{R}$?
(If the title is unclear, I'm looking at infinite cartesian product of $\mathbb{R}$ indexed by $\mathbb{R}$.)
I thought that I had reasoned this rather well, as follows:
$\mathbb{R}^\mathbb{R} = ...
1
vote
3answers
765 views
Rules for algebraically manipulating pi-notation?
I'm a bit of a novice at maths and want to learn more about algebraically manipulating likelihoods in statistics.
There are a lot of equations that involve taking the product of a set of values given ...
2
votes
1answer
104 views
Reference about product of elliptic curves
I am wondering if there is some accessible reference to learn about product of elliptic curves and their 'properties'. For dimension 1, there is plenty to find. I think the dimension 2 case is done as ...
5
votes
1answer
99 views
Modulus of infinite product of complex functions
We know that for complex (entire) functions $f,g$ we have $|f(z)g(z)|=|f(z)||g(z)|$, where $|.|$ means complex modulus.
What about if we have an infinite product? Is it true that
$$\bigg| ...
1
vote
1answer
179 views
Gluing together mathematical structures, how?
By structure, I mean that which is defined here:
http://en.wikipedia.org/wiki/Structure_%28mathematical_logic%29
What I'm looking for is a way of gluing together structures so that each structure ...
2
votes
1answer
93 views
Infinite product of recursive sequence
Let $a_{n+1}=\sqrt {(a_n+a_{n-1})/2}$ and $a_0=a_1=2$, how to prove convergence of the product $a_0 a_1 a_2 a_3...a_\infty$, and possibly find its value?
-1
votes
1answer
124 views
product of an orthogonal and a diagonal matrix
Been a while since my first degree, and I can't seem to solve this kiddy-level question.
Please indulge me:
A an orthogonal matrix, D a diagonal matrix.
is it true that
A^DA = D
?
(where ^ is the ...
3
votes
5answers
593 views
The limit of infinite product
Is there any elementary proof that the limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0?
3
votes
2answers
120 views
Simple properties of a direct product
I am working on some homework for modern algebra class. The problem I just finished seems relatively easy, but I have learned to be wary of that feeling when it comes to this material. Below are the ...
14
votes
3answers
721 views
A “fast” way for computing $ \prod \limits_{i=1}^{45}(1+\tan i^\circ) $?
Which is the fastest paper-pencil approach to compute the product $$ \prod
\limits_{i=1}^{45}(1+\tan i^\circ) $$
-1
votes
1answer
96 views
Kronecker Product
Is this right
$$\mathbf{A}\left(\mathbf{B}\otimes\mathbf{C}\right)\mathbf{D}=\left(\mathbf{A}\mathbf{B}\mathbf{D}\otimes\mathbf{C}\right)$$
Thanks in advance for your help.
1
vote
2answers
93 views
Product and Square Root Proof
Let $a_1$ and $a_2$ be positive integers and let $m = a_1 a_2$.
Prove that at least one of $a_1$ or $a_2$ is at least $\sqrt m$.
Disclosure:
This is for a homework question, though the question is ...
7
votes
2answers
310 views
Proving an infinite product formula
I have found this formula and I am trying to prove it , but I have not any idea how to deal with it:
$$e^{ax}-e^{bx} = ...
3
votes
1answer
116 views
Dyson series and T product (II)
After reading the previous posts related to the Dyson series, I have decided to open a new thread because there is something that I am still not understanding. It concerns the expression:
$$
...
5
votes
1answer
247 views
Showing $\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n b_i \right)^\frac1{n}\le\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n a_i \right)^\frac1{n}$?
If $a_1\ge a_2 \ge a_3 \ldots $ and if $b_1,b_2,b_3\ldots$ is any rearrangement of the sequence $a_1,a_2,a_3\ldots$ then for each $N=1,2,3\ldots$ one has
$$\sum^N_{n=1}\left(\prod_{i=1}^n b_i ...
2
votes
2answers
473 views
The derivative of a product of more than two functions
I'm trying to generalize the product rule to more than the product of two functions using the fact that I can treat the product of $n$-1 functions as a single one. Here is an example of what I mean:
...
4
votes
1answer
286 views
When is $\displaystyle \prod_i \prod_j a_{i} a_{j} = \Bigl(\prod_i a_i\Bigr)^2$
In statistical mechanics, I used to use the procedure that if $a_{ij}=a_i a_j$ $$\prod_i\; \prod_j a_{i}a_{j} = \biggl(\prod_i a_i\biggr)\vphantom{\Bigr)}^2$$
However, today I noticed, $$\prod_i\; ...
13
votes
1answer
273 views
A question about $\prod_{x\in \mathbb{R}^{*}}{x}$
When I was an elementary school student, I encountered some puzzles like "What's product of numbers of hair of each people in the world?", which answer was zero, because there's people with no ...
2
votes
3answers
250 views
About the factors of the product of prime numbers
If a number is a product of unique prime numbers, are the factors of this number the used unique prime numbers ONLY? Example: 6 = 2 x 3, 15 = 3 x 5. But I don't know for large numbers. I will be using ...
4
votes
1answer
285 views
Dyson series and T product
One of the most important tool in quantum mechanics is the Dyson series because it is the basis of the perturbative theory. There is a step in the derivation that I can't understand.
$\{H(t_i)\}$ are ...
6
votes
1answer
177 views
Which is the Abel's theorem invoked in the context of convergence of this infinite product?
Motivation: As I wrote in this answer the following product is evaluated in Proposition 5 of Jean-Paul Allouche and Jeffrey Shallit's paper The
ubiquitous Prouhet-Thue-Morse sequence
...
6
votes
1answer
269 views
Generalization of the series for $\frac{\pi^2}{6}$? Is there a more elementary proof?
In the same vein as:
$ \frac{\pi ^2}{6} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{25} \cdots $
Starting with:
$ \displaystyle \prod_{n=1}^{\infty} \left( 1 -\frac{q^2}{n^2} \right) = ...
4
votes
1answer
240 views
What is $\prod_{k=1}^n (1-x^k)$?
I'd like to know what
$$\prod_{k=1}^n (1-x^k)$$
evaluates to (assuming there is a simple closed form) and what it "is" in the context of commutative algebra (of which I knew little and recall ...
2
votes
2answers
518 views
Formula for Geometric Progression
Can someone help me understand the idea behind constructing a formula for the following:
For $n\in\mathbb{N}$, $n\geq 2$, find and prove a formula for:
$$\prod_{i=2}^n \left(1 - ...
1
vote
1answer
160 views
Efficient calculation of polynomial product
I have 2 polynomials $p_1(x_1,\ldots,x_n)$ and $p_2(x_1,\ldots,x_n)$, of which I have to compute the product, with a special property: The exponent of each variable is always either $0$ or $1$, where ...
3
votes
2answers
55 views
interval for a product to infinity
I was wondering - how would I specify the interval (the amount that n increases each time) between terms? Is that possible? What if I want it to increase by, say, ...
20
votes
4answers
820 views
What is to geometric mean as integration is to arithmetic mean?
The arithmetic mean of $y_i ... y_n$ is: $$\frac{1}{n}\sum_{i=1}^n~y_i $$
For a smooth function $f(x)$, we can find the arithmetic mean of $f(x)$ from $x_0$ to $x_1$ by taking $n$ samples and using ...
15
votes
4answers
1k views
Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$
While trying some problems along with my friends we had difficulty in this question.
True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...
