Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [formal-power-series]

The tag has no usage guidance.

1
vote
1answer
39 views

Is this a maximal ideal of the ring of formal power series?

Let $ k $ be an algebraically closed field, and $ k[[T]] $ be the ring of formal power series in variables $ (T_{1},\dots,T_{n}) = T. $ Let $ \mathfrak{m}^{l} $ be the ideal of $ k[[T]] $ consisting ...
0
votes
1answer
19 views

Find the radius of convergence of summesion (n=0 to ∞)(an*x^n), where an=[sin(n!)/n!] and ao=0? [duplicate]

found a question like this and there are four options R>=1 R>=2*pi R<=4*pi R<=pi, where R denotes radius of convergence. I've tried hard but cant get it how to show in equality. Also there is ...
-1
votes
3answers
27 views

Power series: convergence on right-open interval

Given the power series $f(x) = \sum_{k\geq 1} a_k x^k$ where $a_k \geq 0$, is the following statement true? Let $f(x) = \sum_{k\geq 1} a_k x^k$ with $a_k \geq 0$. If $f(x) < \infty$ for some ...
1
vote
0answers
24 views

Involutions in $\mathbb{F}_p[[x]]$

Question: For a prime $p$, is every involution in $\mathbb{F}_p[[x]]$ a reduction modulo $p$ of some involution in $\mathbb{Z}[[x]]$? Obviously, a reduction mod $p$ of an involution in $\mathbb{Z}[[x]...
2
votes
0answers
75 views

What is the difference between $K[X_1, \ldots, X_{m - 1}]\color{blue}{[X]}$ and $K[X_1, \ldots, X_{m - 1}]\color{blue}{[X_{m}]}$?

In my textbook Analysis I by by Amann/Escher, there are definitions as follows Let $R$ be a nontrivial (not necessarily commutative) ring with unity. The formal power series ring over $R$ is the ...
1
vote
1answer
33 views

Associativity of formal group law in elliptic curves

In Silverman's AEC there is the following paragraph. Firstly, $F(z_1,z_2)$ lies in $\mathbb{Z}[a_1,...,a_6][[z_1,z_2]]$. Should I treat the $a_i$s as indeterminates or should I treat them lying in ...
5
votes
1answer
79 views

Using “formal” formulas to get non-formal results

So-called "formal" operations--like "formal differentiation", "formal integration", etc.--have always made me a bit uneasy, because it seems to be used sometimes as a snake-oil solution for dealing ...
2
votes
1answer
36 views

Formal power series rings and p-adic solenoids

For any prime $p$, the ring of $p$-adic integers can be generated as the quotient of the formal power series ring $\Bbb Z[[x]]/(x-p)$. My questions: If we instead use $\Bbb Q[[x]]/(x-p)$, do we get ...
4
votes
0answers
105 views

Relationship between $p$-adic numbers and analytic continuation of $1+x+x^2+x^3+…$

The infinite sums $1 + 2 + 4 + 8 + ...$ $1 + 3 + 9 + 27 + ...$ $1 + 5 + 25 + 125 + ...$ $1 + 7 + 49 + 343 + ...$ ... of powers of primes do not converge in the usual sense. However, by analytically ...
3
votes
1answer
92 views

Is $A[[x_1,x_2,\dots]]$ flat over $A[x_1,x_2,\dots]$? ($A$ noetherian and commutative)

Let $A$ be a noetherian commutative ring with one and $x_1,x_2,\dots$ indeterminates. Question. Is $A[[x_1,x_2,\dots]]$ flat over $A[x_1,x_2,\dots]\ ?$ Recall that $A[[x_1,x_2,\dots]]$ is the set ...
0
votes
1answer
24 views

Generalizing an implicit function theorem for formal power series

This exercise is from Greuel & Lossen & Shustin's Introduction to Singularities and Deformations, Exercise 1.2.5. Let $f\in\mathbf{C}\langle \mathbf{x},y\rangle$, where $\mathbf{C}\langle \...
0
votes
1answer
50 views

Proving that the given element is irreducible but not prime

This question is from Introduction to Singularities and Deformations, Greuel et al. Let $R=\mathbf{C}\langle x,y,z\rangle/\langle x^2-yz\rangle,$ where $\mathbf{C}\langle x,y,z\rangle$ is the ring ...
0
votes
1answer
80 views

Spectrum of the ring of formal power series over integers [closed]

Let $\mathbb{Z}[[X]]$ be the formal power series ring over $\mathbb{Z}$. I want to understand the set of prime idelas $\rm{Spec}(\mathbb{Z}[[X]])$, maximal ideals $\rm{Spm}(\mathbb{Z}[[X]])$ and ...
0
votes
1answer
33 views

Transferring big Witt vectors from “power series form” to “vector form”

I am currently attempting to read Hazelwinkel's paper "Witt Vectors 1", in which the ring of big Witt vectors $W(R)$ for $R$ a commutative unital ring is defined to be the set $R^\mathbb{N}$ with ring ...
3
votes
0answers
56 views

On $A$ algebra homomorphisms $A[[X_1,…,X_n]]\to Q(A)$, where $A$ is a complete DVR

Let $(A,\mathfrak m)$ be a complete Discrete Valuation Ring (complete w.r.t. the $\mathfrak m $-adic topology) with fraction field $K$. Let $\phi : A[[X_1,...,X_n]]\to K$ be an $A$-algebra ...
1
vote
0answers
29 views

On integral closedness of formal power series ring over an integrally closed domain satisfying Krull intersection principle

Let $R$ be a normal domain (i.e. an integral domain integrally closed in its fraction field) such that for every non-unit $t\in R$, $\cap_{n\ge 1} (t^n)=(0)$ ; then is it true that $R[[X]]$ is normal (...
1
vote
1answer
43 views

Determining solutions for convergence of formal power series

Consider the following power series $$ S = \sum_{n \geq 0} a_n x^n $$ where $(a_0, a_1, \dotsc)$ is a sequence of complex numbers and $x$ is a complex variable. If, say, for all $n$ the coefficients $...
1
vote
0answers
55 views

Show that $e^{\frac{-\hbar}{2}\partial_{a}\partial_{b}}e^{\frac{-ab}{\hbar}}=2e^{\frac{-2ab}{\hbar}}$

I think this question could be answered by only using mathematics (it relates to physics). Where, $\partial_{x}f$ is denoted as partial derivative of $f$ w.r.t $x$, and first exponential term behaves ...
0
votes
0answers
26 views

How do I prove the following statement to do with formal power series?

Let f(t) be a formal power series with ord(f) = 1 and let g(t) be the inverse of f. Then b0 = 0 and bn = (Res(1/(f(t))^n))/n where n>0 I am completely lost in understanding how I should show that ...
0
votes
0answers
20 views

How do I solve this question to do with residue of a formal power series?

If f(t) is a formal power series with ord(f) = 1, then it follows that Res(f'(t)/f^n(t)) equals 0 when n is not equal to 1, but equals 1 when n is equal to 1. I know that residue is defined for ...
4
votes
2answers
145 views

How to show that $\gcd(a_1,a_2,\cdots,a_k) = 1$ implies that there exist a non-negative solution to $\sum_{i=1}^{n}a_ix_i = n$ for large $n.$

I was reading about the Coin-problem and I am unable to fully understand the following argument: On the other hand, whenever the GCD equals 1, the set of integers that cannot be expressed as a ...
-2
votes
1answer
34 views

Any identity (involving addition, multiplication, substitution) between real/complex power series is an identity in the ring of formal power series.

I need help to prove the following principle- Any identity between real or complex power series, involving addition, multiplication (possibly infinite sums and products), and substitution, is an ...
0
votes
0answers
31 views

show that we can write $ [B_{\frac{1}{N}}(x)]^N=(1+x)$, similar to Binomial expansion

We know binomial expansion $$B_{a}(x)=(1+x)^a=\sum_{n=0}^{\infty} \frac{a(a-1)(a-2) \cdots (a-n+1)}{n!}x^n$$ For any $a \in \mathbb{R} $ or $\mathbb{C}$, this series converges in $\mathbb{R}$ or $\...
0
votes
0answers
13 views

How can one prove that for a formal power series f with ord(f) = 1, then ones has the following…?

Let f(t) be a formal power series and ord(f) = 1, then prove that: Res(f'(t)/f^n(t)) = 0, if n!= 1 = 1, if n=1 Maybe I don't understand what the order of a formal power series is ...
2
votes
0answers
55 views

Preimage of a prime ideal in a ring of formal power series.

Let $k = \overline{k}$ be a field. We have the inclusion $\iota:k[x,y] \to k[\![x,y]\!]$ and the prime ideal $\mathfrak{p} = (y - \sum_{n\geq 1} x^n/n!)$. This ideal is prime, because it is the kernel ...
1
vote
1answer
53 views

Exponentiate generating functions as formal power series

In my discrete math class, we are studying generating functions. We learned that $$ e^x = \sum_{i = 0}^{\infty} \frac{x^i}{i!}, $$ which is certainly an identity in calculus. However, in the ring of ...
0
votes
0answers
12 views

special partition function

Let following be the partition function in infinitely many variables $x_i$ the linear coordinates on the vector space $V$. $$ \mathcal{Z}=exp\Big(\sum_{\substack{g\geq 0\\n\geq 1}}\frac{h^{g-1}}{n!}\...
3
votes
1answer
76 views

Why $K[[X]]$ is PID and what's the form of the ring's ideals?

Let $K$ be a field. We write $K[[X]]$ for the ring of all formal power series with coefficients from the field $K$. Then, we will try to prove the next theorem. Theorem. If $K$ is a field then: ...
1
vote
1answer
64 views

Uniqueness in Weierstraß p-adic preparation theorem

I have given the Weierstraß p-adic preparation theorem. It is stated as follows: Let $f=a_0 + a_1T + ... \in \mathbb{Z}_p[[T]]$ for a prime $p$ such that $p \mid a_0,...,a_{n-1}$ and $p\not\mid a_n$. ...
4
votes
5answers
126 views

Find the sum of $1-\frac17+\frac19-\frac1{15}+\frac1{17}-\frac1{23}+\frac1{25}-\dots$

Find the sum of $$1-\frac17+\frac19-\frac1{15}+\frac1{17}-\frac1{23}+\frac1{25}-\dots$$ a) $\dfrac{\pi}8(\sqrt2-1)$ b) $\dfrac{\pi}4(\sqrt2-1)$ c) $\dfrac{\pi}8(\sqrt2+1)$ d) $\dfrac{\...
5
votes
0answers
65 views

What is a common framework for these divergent sums?

If you expand $2^x$ using a finite difference series you end up with the formula $$ 1 + x + \frac{1}{2!}x(x-1) + \frac{1}{3!}x(x-1)(x-2) ... = \sum_{n=0}^{\infty} \frac{(x)_n}{n!} $$ Now these ...
3
votes
2answers
144 views

Computing quotient by dividing formally in $p$-adic number system

From Gouvea's p-adic Numbers: To pass from positive integers to positive rationals, we simply do exactly as in the other case, that is we expand both numerator and denominator in powers of p, ...
0
votes
0answers
50 views

Formal power series with uncountable infinitely many zeros

Let $K$ be an arbitrary field of characteristic zero (may assume it is algebraically closed if necessary), and let $$\Phi(y_1,\dots, y_n) \in K[[y_1^{\pm 1},\dots, y_n^{\pm 1}]]$$ be a (Laurent) ...
0
votes
2answers
30 views

Multiplication of Power Series [Complex]

I know that given two power series, $A$ and $B$, such that $A= \sum_{n=0}^{\infty} a_n x^n$ and $B= \sum_{n=0}^{\infty} b_n x^n$ then, $AB= \sum_{n=0}^{\infty} (\sum_{i=0}^{n}a_i*b_{n-i})x^n$ But ...
2
votes
1answer
37 views

Degree of a formal power series involving Mobius function

I am reading Enumerative Combinatorics by Richard Stanley, and I came across the following expression: $(1-x^n)^{\frac{-\mu(n)}{n}}$, where $\mu(n)$ is the usual Mobius function from number theory. ...
3
votes
2answers
119 views

What does it mean for a formal power series to be “well-defined”?

I see the term "well-defined" used in Stanley's Enumerative Combinatorics, but I'm not sure what it means. Is it equivalent to saying that the formal power series converges to a certain function, or ...
0
votes
0answers
24 views

I need some advice on how to organize some drafts and what to with a new formula I found

Found a new formula to get the sum of k to the power of any positive integer , k from 1 to n .I know bernoulli already found another formula in which he created his bernoulli numbers. This new fomula ...
3
votes
2answers
211 views

Formal Laurent series and field of fractions of Taylor series

Let us define the field of formal Laurent series of $n$ variables as $K=k((x_1))((x_2))...((x_n))$. As a subring it contains the ring of formal Taylor series $R=k[[x_1]][[x_2]]...[[x_n]]$. And the ...
2
votes
0answers
43 views

When should we suspect an integral polynomial/power series to have coefficients with meaning behind them?

When one has a question regarding the coefficients of polynomial or power series with integer coefficients, especially if they are positive, it seems like a good strategy to realise the coefficients ...
1
vote
1answer
34 views

What does it mean for a member of formal power series over a field to be algebraic over polynomial ring of that field?

What does it mean for a member of formal power series over a field to be algebraic over polynomial ring of that field? For example what does it mean for a $f$ in $k[[t_1 ,...,t_n ]]$ which is ...
-1
votes
2answers
200 views

Cauchy product of multivariate formal power series

Cauchy product of two univariate formal power series is pretty straight forward. If $$A=\sum_{i=0}^\infty a_i x^i \, ,$$ and $$B=\sum_{j=0}^\infty b_j x^j \, ,$$ then $$A \times B = \sum_{k=0}^\...
0
votes
1answer
79 views

Rings of formal power series is finitely generated as module.

The question rises when I was reading Kemper's "A course in Commutative Algebra." Let $K$ be a field and consider the ring of formal power series $K[[x]]$. In an exercise I proved $K[[x]]$ is not ...
0
votes
1answer
83 views

Taylor expansion for formal power series

Let $P = p_1x + p_2x^2 + \dots$, $Y = y_1x + y_2x^2 + \dots$, and $V = v_1x + v_2x^2 + \dots$ all be formal power series with indeterminate $x$ and coefficients in some field $\mathbb{F}$, satisfying $...
1
vote
0answers
61 views

Does $\sum_{n\ge1}x_n^n$ converge $\mathfrak m$-adically in $K[[x_1,x_2,\dots]]\ ?$

Let $K$ be a field, let $x_1,x_2,\dots$ be indeterminates, and form the $K$-algebra $A:=K[[x_1,x_2,\dots]]$. Recall that $A$ can be defined as the set of expressions of the form $\sum_ua_uu$, where ...
0
votes
1answer
64 views

absolute value and valuation on the field of formal power series

I'm currently working on (non-Archimedean) valuations and absolute values. My text states that one example, where I have both a non-Archimedean valuation and a corresponding absolute value is $\mathbb{...
1
vote
0answers
48 views

When do we know a quotient of the polynomial ring over a local ring is torsion free

Let $R$ be a local ring (e.g. the discrete valuation ring $\mathbb C[[T]]$) and $\mathfrak{m}$ its maximal ideal. Consider the polynomial ring $R[X_1,\dots, X_n]$ in $n$ variables and a finitely ...
0
votes
0answers
34 views

Holonomic functions and degree bounds

Let $A(x)$ is a generating function annihilated by the following differential equation of order $r$ and degree $d$. The degree of the differential equation is given by the maximal degree of the ...
0
votes
1answer
30 views

identity of a formal power series

Recently I encounter an identity $$z_{0}^{-1}\delta(\frac{z_{1}-z_{2}}{z_{0}})=z_{1}^{-1}\delta(\frac{z_{0}+z_{2}}{z_{1}})$$ where $\delta(x)=\Sigma_{n\in \mathbb{N}}x^{n}$. I tried to expand both ...
0
votes
1answer
27 views

Formal power series: $F_i(x)$ converges if and only if $\lim_{i\to\infty}deg(F_{i+1}(x)- F_i(x))=\infty.$

The following notations and definitions are taken from Richard Stanley's book Enumerative Combinatorics Volume $1,$ second edition. Recall that a formal power series $F(x)$ is of the form $$\sum_{n\...
1
vote
1answer
52 views

Power rule of exponents for formal power series

I am trying to prove the following basic fact about exponentiation for a version of binomial coefficients for formal power series, based on R.P. Stanley's definition in "Enumerative Combinatorics ...