# Tagged Questions

For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.

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Suppose I have known polynomials $p_i, i=0\ldots k-1$. I have the following horrific looking recursive equation: $$p_{k}(x)=-\prod_{i=1}^{k}{(x+i)}^{\left\lfloor\frac{k}{i}\right\rfloor-1}\left[p_o(x)... 0answers 42 views ### How I can calculate this product How I can calculate this product:$$s=∏_{j=1}^{p-3}\sinh^2[2^{j-1}\cosh^{-1}( 2)]$$for a natural number p>3. 0answers 128 views ### Product of Summations for All Subsets We have a set X of n integers \{$$x_1$,$x_2$, .. ,$x_n$$\}, for which there are 2^n total subsets. The summation s of a subset X' is simply the sum of all integers present in X', ... 0answers 284 views ### Product of sines I am looking to evaluate$$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n}$$without using complex numbers. I can show the result if n is a power of 2, but if n is anything else I reach a point where I ... 0answers 183 views ### Infinity Product Equality. Let \{I_n\}_{n\in\mathbb{N}} be a sequence of intervals in the form$$ I_n = \Big [ \frac{q_n}{b_n}, \frac{q_n + 1}{b_n} \Big),$$where q_{n} is some integer, for all n\in\mathbb{N}. Define ... 0answers 31 views ### What is \prod _{j=1}^n \left(\sqrt{j}+1\right)? By the Fundamental Theorem of Algebra, it is easily seen that for a monic polynomial p(x) \in \mathbb{C}[x],$$\prod _{j=1}^n p(j) = \frac{\prod_{p(r)=0}\Gamma(1+n-r)}{\prod_{p(r)=0}\Gamma(1-r)},... 0answers 42 views ### Multiplication of polynomials of the same degree Consider polynomials of the form $$p(x)=x^{n-2r}\sum_{i=0}^ra_ix^{2i},$$ where \begin{align} r&=n/2, \quad n \quad \text{even},\\ r&=(n-1)/2, \quad n \quad \text{... 0answers 123 views ### What notation would I use to differentiate between a cartesian product and a cotangent bundle of surfaces? If the S^1 is defined by x^2 + y^2 = r^2 , T^2 = S^1 \times S^1 is defined by \left(\sqrt{x^2 + y^2} -R\right)^2 + z^2 = r^2 , T^3=S^1\times S^1\times S^1 is defined by \left(\sqrt{\left(... 0answers 42 views ### How can I find the elements generating a group in a special way? Suppose, a finite permutation group G is given. I want to find the minimal set x_1,...,x_n such that every element of G can be uniquely written in the formx_1^{j_1}...x_n^{j_n}$$with 0\le j_i\... 0answers 45 views ### Is (\omega \times \omega)^{\omega}\cong \omega \times \omega \times… \cong \omega^{\omega}? Where “\cong” means homeomorphic. I'm interested in the circumstances for when we can conclude that two ordinal spaces are homeomorphic by an examination of their written form. Specifically, I'm taking an ordinal space, say (\omega^... 0answers 104 views ### Is the product of two discrete \sigma-algebras necessarily discrete? I know that the answer to this question is negative, since proving the opposite is an exercise in Terrance Tao's Measure Theory book. However, it doesn't make sense to me. In another part of the same ... 0answers 64 views ### Cleaning Up Messy Product Notation Suppose I have the following: Let N_1<...<N_m. Let T_{N_k}(x)=\sum_{i=0}^{N_k}{\frac{x^i}{i!}},  t(i,j,x)=(T_{N_i}-T_{N_j})(x) I'm trying to define a polynomial p_{k,m}(x) like this:... 0answers 53 views ### How the second form of following equation is derived form first form (i.e. given first line, what are the steps involved in writing second line How the second form of following equation is derived form first form (i.e. what are the steps involved in writing second line) 0answers 131 views ### Evaluate this product n \times \frac{n-1}{2} \times \dots \times \frac{n-(2^k-1)}{2^k} For k = \lfloor \log_{2}(n+1) \rfloor - 1 evaluate n \times \frac{n-1}{2} \times\frac{n-3}{4} \times \frac{n-7}{8} \times \dots \times \frac{n-(2^{k}-1)}{2^k} So the product goes up to k and I ... 0answers 30 views ### Vectorial product analog operation in 4+ dimensions? I am thinking about a such operation. Which it need to have: It needs to be \mathbb{R}^n\times{\mathbb{R}^n}\rightarrow\mathbb{R}^n The result needs to be perpendicular to the arguments (thus, ... 0answers 32 views ### Fourier Series from product of to functions I have to calculate the Fourier Series of x\sin(x) beeing 2\pi periodic on [-\pi,\pi]and i did it the standard way. But then i wanted to solve the problem with multiplication of two fourier ... 0answers 201 views ### The logarithm of a product Let p be a prime number, C\in \mathbb{N} and C is not a square. Then define$$F=\prod_{|z| \leq \sqrt{\frac{x}{2}} \atop |y|\leq \sqrt{\frac{x}{2D}}}{|z^2-Cy^2|}.$$Note that we omit the term with ... 0answers 88 views ### Multiplicative group into ring operation My question is simple, though it proves to be much more difficult than it sounds. Suppose I want to find a binary operation to add extra structure to a multiplicative group (so it becomes a ring). ... 0answers 134 views ### “Product” bundle notation. Let \newcommand{\Spin}{\operatorname{Spin}}M and M' be two manifolds, equipped with a principal \Spin_n and \Spin_{n'} bundle called P and P', respectively. Then there is an induced ... 0answers 14 views ### nth product of sequential matrices \forall n \in \mathbb{N}, let:$$P_n = \left( \begin{matrix} a & 1-a \\ b_n & 1-b_n \end{matrix} \right). $$Whereby \{b_n\}_{n \in \mathbb{N}} is a monotonically increasing sequence of ... 0answers 19 views ### In which cases are the main diagonal elements of a product of positive definite matrices positive? Let A and B be symmetric positive definite (pd) n \times n matrices and C = A \cdot B. In which cases is then every c_{ii}, the i-th main diagonal elements of C, positive? When A ... 0answers 27 views ### which values of k satisfies special property to formulate L function Consider x*\prod_{a=1}^{n}(1-x^a)^k Famously for k=24 this product satisfies the condition to be an L-Function. More information can be found here My question is for what other values of k, such ... 0answers 15 views ### Where can I find methods to evaluate products? I found it was slightly difficult to find resources that discussed methods for evaluating products, like \Pi_{n=0}^ka_n Preferably, I want to start with the basics and move through some readings on ... 0answers 32 views ### Product of Several Functions Becomes Very Small: Scaling? I have the following ratio:$$\frac{\sum_{i = 1}^n Y_i \prod_{p = 1}^P \lambda_p^{z_{i,p}}}{\sum_{i = 1}^n \prod_{p = 1}^P \lambda_p^{z_{i,p}}}$$where \lambda_p \in (0,1] is a parameter, and z_{... 0answers 20 views ### Product of a matrix and a tensor I need to know how to compute the following product: M(x)\frac{\partial M(x)}{\partial x}M(x) \quad where x \in R^{n}. Assuming the dimensions of the matrices are compatible,how do we take ... 0answers 28 views ### Binomial square sum and product Given c,n\in\Bbb N what is the expression for$$S(n,c)=\binom{n}c^2+\binom{n-c}c^2+\dots+\binom{x}c^2$$and$$P(n,c)=\binom{n}c^2\cdot\binom{n-c}c^2\cdot\dots\cdot\binom{x}c^2$$where x-c<c\leq ... 0answers 33 views ### How to find \sup \Pi_{i=0}^{n} (\sin(i)^2 - \frac{25}{16})? Let \sup,\inf,{\rm dif} denote resp supremum , infimum and \rm dif = supremum - infimum. Does any of the 3 below have a closed form ? \sup \Pi_{i=0}^{n} (\sin(i)^2 - \frac{25}{16}) \inf \Pi_{... 0answers 12 views ### How can I prove the following inequality? Let be N_{n+1}(x)=\prod_{i=0}^n(x-x_i). Now I have to prove that$$||N_{n+1}(x)||_{\infty,[-5,5]}\leq n!\frac{h^{n+1}}{4},\qquad h:=\frac{5-(-5)}{n}=\frac{10}{n}.$$I've started with$$\|N_{n+1}(...
Let $B_{i,n}$ with $i=1,...,n$ be the triangular Bernoulli array defined as $$B_{i+1,n} = B_{i,n}\,R_{i+1,n}+\left(1-R_{i+1,n}\right)\,F_{i+1,n},$$ where $R_{i,n}$ and $F_{i,n}$ are iid Bernoulli ...