# Tagged Questions

40 views

Consider the polynomial series defined by the following recursion formula: \begin{align} &\mathrm{P}_0 = 1 \\ &\mathrm{P}_1 = x-r \\ &\mathrm{P}_n = x\mathrm{P}_{n-1} - ... 1answer 21 views ### Cardinality of a set with a recurrence relation. Let A = \left\{ f\in \mathbb{N}\rightarrow \mathbb{C} \mid \forall n\in \mathbb{N}. f(n+3) + 3f(n+1) = f(n+2)+f(n) \right\} What is \left|A\right|? Well, I tried to treat f as a recurrence ... 2answers 72 views ### Zeroes of polynomialc_1,c_2 \text{ are polynomial's }g(x)=x^2+ax+b \text{ roots } \Leftrightarrow \begin{cases} g(c_1)=c_1^2+ac_1+b=0 \\ g(c_2)=c_2^2+ac_2+b=0 \end{cases}$$Prove that for every polynomial with integer ... 0answers 32 views ### Linear Independence of Powers of “roots vector” [duplicate] Let us be working over the field of complex numbers. Suppose f(x)= a_n x^n + \cdots +a_1 x + a_0 is a degree n polynomial with n distinct roots z_1,\ldots,z_n. Is the following matrix always ... 2answers 66 views ### number theory of coefficients in an infinite sequence of polynomials EDIT: equivalent formulation by Hurkyl in comments: if n is odd and p^\nu \parallel n and n > 2k, then$$ p^{(\nu + 2 + 2 k - n)} \; | \; \sum_j \left( \begin{array}{c} n \\ 2j \end{array} ...
A friend and I were examining polynomials of the form $p_n (x) = x (x+1) (x+2) \cdots (x+n -1)$ and we were trying to come up with some kind of closed form for the coefficients when the polynomial is ...