Results for query "units of R`x` the formal power series. 3 Prove that"

Given a formal power series ,decide whether there exists a polynomial the series satisfies and if it exists,how to write it down?

Given a formal power series $$y(x)=\sum_{i=0}^{\infty} a_i x^i$$ Is there an algorithm that decides whether there exists a polynomial$$ P(x,y)=p_n(x)y^n+p_{n-1}(x)y^{n-1}+\cdots+p_0(x)=0,p_j(x)\in F[x]$$the series satisfies and if it exists,how to write it down?This is relevant because if an element of $\mathbf{Z}[[X]]$ is algebraic over
$\mathbf{Q}(X)$, then its reduction modulo any prime $p$ will be algebraic
over $\mathbf{F}_p(X)$.

Prove a family of series having integer coefficients

I encountered a certain family of infinite series in some work, which is given by
$$F_r(x)=\frac1{2^r}\sum_{k=0}^r\binom{r}k\frac1{1+x(2k-r)^2}.For each $r\in\mathbb{N}$, the Taylor series for $F_r(x)$ has integer coefficients.

Flat Quotients of Power Series Rings

I read the following statement in some algebraic topology notes and I want to know if it is true and, if so, why.Let $R$ be a commutative ring and $f(x)$ a power series in $R[[x]]$.

Generating function of the Thue-Morse sequence

Let $T$ be the generating function of the Thue-Morse sequence; thus,
$T(x)=x+x^2+x^4+x^7+\dotsb$.It is known that $T$ satisfies the nice
congruence
$$ (1+x)^3 T^2(x) + (1+x)^2 T(x) + x \equiv 0 \pmod 2 $$
(the congruence is actually modulo the principal ideal generated by $2$ in
the ring of formal power series ${\mathbb Z}[[x]]$).

Any closed form for series like $F(x)=\Sigma_{i=p}^{\infty}x^p$,p is prime$?

Any closed form for series like $$F(x)=\Sigma_{i=2}^{\infty}x^p,\text{p is prime}$$ or $$F(x)=\Sigma_{i=0}^{\infty}x^{i!More generally,we can obtain a power series from decimal expansion of a number r(0< r<1 ) by replacing $$(\frac{1}{10})^i$$ with $$x^i$$ like $$\frac{1}{3}=3(\frac{1}{10})^1+3(\frac{1}{10})^2+\cdots 3(\frac{1}{10})^i+\cdots$$,
we obtain :
$$f(x)==\Sigma_{i=1}^{\infty}3x^i$$

when f(x) is convergent,what restriction do we have to put on r(if r is c.

Irrationality measure of formal power series

I'm looking for an analogue of irrationality measure for formal power series with integer coefficient, the elements of $\mathbb{Z}[[x]]$.Note that $m_g(f)$ is finite for all $g$ if and only if $f$ is irrational, otherwise $m_g(f) = +\infty$ for $g$ sufficently large.