Questions tagged [power-series]

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54 votes
8 answers
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Does the formal power series solution to $f(f(x))= \sin( x) $ converge?

I have spent some time using gp-pari. There is, of course, a formal power series solution to $ f(f(x)) = \sin x.$ It is displayed below, identified by the symbol $g$ because I am not entirely sure ...
Will Jagy's user avatar
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50 votes
1 answer
2k views

Rearrangements of a power series at the boundary of convergence

Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series $$f_{\sigma}(z) = \sum_{n=0}^...
echinodermata's user avatar
44 votes
5 answers
3k views

An "analytic continuation" of power series coefficients

Cauchy residue theorem tells us that for a function $$f(z) = \sum_{k \in \mathbb{Z}} a(k) z^k,$$ the coefficient $a(k)$ can be extracted by an integral formula $$a(k) = \frac{1}{2\pi i}\oint f(z) z^{-...
MCH's user avatar
  • 1,304
40 votes
3 answers
1k views

Characterizing positivity of formal group laws

The formal group law associated with a generating function $f(x) = x + \sum_{n=2}^\infty a_n \frac{x^n}{n!}$ is $$f(f^{-1}(x) + f^{-1}(y)).$$ In my thesis, I found a large number of examples of ...
Jair Taylor's user avatar
31 votes
3 answers
1k views

Which power series have bounded integral coefficients and have an inverse given by a series having bounded integral coefficients

Let $A=1+\sum_{n=1}^\infty \alpha_nx^n\in\mathbb Z[[x]]$ and $B=\frac{1}{A}=1+\sum_{n=1}^\infty\beta_n x^n$ two mutually inverse power series having bounded integral coefficients (ie. $\vert \alpha_n\...
Roland Bacher's user avatar
31 votes
1 answer
2k views

Is this formal noncommutative power series identity known?

I recently discovered the following cute formal noncommutative power series identity: if $(x_i)_{i \in I}$ is some finite collection of noncommuting variables, then the formal power series $$ 1 + \...
Terry Tao's user avatar
  • 108k
29 votes
2 answers
16k views

power series of the reciprocal... does a recursive formula exist for the coefficients [closed]

Let $f(x)=\sum _{n=0}^{\infty } b_nx^n$ and $\frac{1}{f(x)}=\sum _{n=0}^{\infty } d_nx^n$. Then the coefficients of the reciprocal of $f(x)$ can be written down. The first few terms are: $d_0 = \frac{...
AUK1939's user avatar
  • 569
29 votes
0 answers
1k views

Linking formulas by Euler, Pólya, Nekrasov-Okounkov

Consider the formal product $$F(t,x,z):=\prod_{j=0}^{\infty}(1-tx^j)^{z-1}.$$ (a) If $z=2$ then on the one hand we get Euler's $$F(t,x,2)=\sum_{n\geq0}\frac{(-1)^nx^{\binom{n}2}}{(x;x)_n}t^n,$$ on the ...
T. Amdeberhan's user avatar
26 votes
1 answer
1k views

Identities for power series like $\sum_n z^{n^3}$

Probably, one of the first power series that every mathematician encounter is the geometric series $$\sum_{n=0}^\infty z^n = \frac1{1-z}, \quad z \in \mathbb{C},\; |z| < 1 .$$ Also, a particular ...
user avatar
24 votes
2 answers
3k views

A product approximation to the Taylor series of the exponential

I recently came across the following in something I'm working on, and I'd never seen it before. Consider \begin{align*} f_1(x) &= (1+x)^{1/1} \\\ f_2(x) &= (1+x)^{2/1} (1+2x)^{-1/2} \\\ f_3(...
Charles Rezk's user avatar
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21 votes
1 answer
722 views

Are $\mathbb C$ , $\mathbb C[X]$ definable in $\mathbb C[[X]]$?

Let $L$ be a first-order language and $M$ be an $L$-structure. Let $D \subseteq M^n$ . Let us say $D$ is definable in $M$ if for some finite set (possibly empty) $A=\{a_1,...,a_m\} \subseteq M$ and ...
user avatar
19 votes
7 answers
3k views

Is this a rational function?

Is $$\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$ In a slightly different vein, given a sequence of real numbers $\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient ...
Pablo's user avatar
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19 votes
3 answers
1k views

The sum of integers being a bijection

What are the pairs $(P,Q)$ of subsets of $\mathbb N$ for which the map \begin{eqnarray*} P\times Q & \rightarrow & {\mathbb N} \\\\ (p,q) & \mapsto & p+q \end{eqnarray*} is a bijection ...
Denis Serre's user avatar
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19 votes
2 answers
744 views

Is $K[[x_1,x_2,\dots]]$ an $\mathfrak m$-adically complete ring?

I asked this question on Mathematics Stackexchange (link), but got no answer. 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 ...
Pierre-Yves Gaillard's user avatar
19 votes
0 answers
1k views

Does this variant of a theorem of Hasse (really due to Gauss) have an "elementary" proof?

BACKGROUND Here are 3 theorems of varying difficulty. Let $M$ be the $Z/2$ subspace of $Z/2[[x]]$ spanned by $f^k$, with the $k>0$ and odd, and $f=x+x^9+x^{25}+x^{49}+\cdots$. For $g$ in $M$, let $...
paul Monsky's user avatar
  • 5,412
18 votes
3 answers
3k views

When are roots of power series algebraic?

Let $K$ be a field and consider a power series $f(T) \in K[[T]]$. Under what conditions (on $K$ and/or on $f$) can we conclude that if $\alpha$ is a root of $f(T)$ then $\alpha$ is in fact algebraic ...
Jeff H's user avatar
  • 1,412
17 votes
3 answers
2k views

If a formal power series over the complex numbers satisfies a polynomial identity, does it imply that the power series has a radius of convergence?

Let $ P(z) $ be a $\textit{formal}$ power series in $z$ that a priori may not have a non zero radius of convergence. Assume that $P(0) =0$. Let $\Phi(w,z)$ be a polynomial in two variables, that ...
Ritwik's user avatar
  • 3,235
17 votes
1 answer
566 views

Positivity of coefficients of the inverse of a certain power series

Consider the unique formal power series $g(z)$ with $g(0)=0$ and $g'(0)=1$ satisfying the equation $$ g(z)-g(z)^8+g(z)^{15}=z, $$ that is the inverse of $$ z-z^8+z^{15} $$ in the group of formal ...
Vladimir Dotsenko's user avatar
17 votes
2 answers
1k views

Higher level analogs of Nicolas-Serre theory

NICOLAS-SERRE THEORY Let $F \in Z/2[[x]]$ be $x+x^9+x^{25}+...$, the exponents being the odd squares, and $V$ be the space spanned by the $F^k$ with $k$ odd. Nicolas and Serre define formal Hecke ...
paul Monsky's user avatar
  • 5,412
16 votes
3 answers
7k views

What are the prime ideals of k[[x,y]]?

Let $k$ be a field. Then $k[[x,y]]$ is a complete local noetherian regular domain of dimension $2$. What are the prime ideals? I've browsed through the paper "Prime ideals in power series rings" (...
Martin Brandenburg's user avatar
16 votes
0 answers
452 views

A Product Related to Unrestricted Partitions

Start with the product for unrestricted partitions: $(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...$ Now replace some of the plus signs with minus signs and expand the product into a series. Is it ...
David S. Newman's user avatar
15 votes
5 answers
1k views

Comparing two power-series

I have been wrestling with the following problem for some time now. If possible, I would prefer a hint rather than a full solution, because I would like to "solve" it myself. Let $f(z)$ be a ...
Arkadij's user avatar
  • 914
15 votes
1 answer
714 views

Positivity of a finite sum involving Stirling numbers

In my research in theoretical physics, I have arrived at some coefficients $a_{n,m}$ depending on two integers, $n\geq 1$ and $0\leq m\leq n$: $$ a_{n,m}=\sum_{j=0}^{n-1} {2j \choose j+m} \left(\frac{...
Tomeu Fiol's user avatar
15 votes
1 answer
493 views

Partial sums of $\sum_0^\infty z^n$

Let $z$ be a complex number with $|z|<1$. For every subset $A\subset\mathbb N$, the series $\sum_{m\in A}z^m$ is convergent. Denote $S(A)\in\mathbb{C}$ its sum and $\Sigma_z$ the set of all numbers ...
Denis Serre's user avatar
  • 51.5k
14 votes
2 answers
1k views

When are growth series rational?

For a finitely generated group $G$ with generating set $S$, one can define the word metric. Using this, we are able to define the sequence $\sigma(n)$ by $ \sigma(n) = |\mathcal{S}(e,n)| $, which is ...
Tom Ultramelonman's user avatar
14 votes
1 answer
676 views

Power series $x^n/n$ with one plus, then two minuses, then three plusses, and so on

Which function is represented by the powerseries $$1-\frac{x}{2}-\frac{x^2}{3}+\frac{x^3}{4}+\frac{x^4}{5}+\frac{x^5}{6}-\frac{x^6}{7}-\frac{x^7}{8}-\frac{x^8}{9}-\frac{x^9}{10} + \dots$$ (one plus, ...
user115748's user avatar
14 votes
4 answers
824 views

Combinatorial interpretation of the power of a series

I am trying to understand a result involving the power of a series that occurs in Gradstein and Ryzhik's Table of Integrals, Series, and Products. Result 0.314 (p.17, 7th ed.) is: $$\left(\sum_{k=0}^\...
Jiahao Chen's user avatar
  • 1,870
13 votes
1 answer
751 views

Can formal power series become polynomial often, when composed with polynomials?

Let $F$ be a finite field. Let $F[X]$ and $F[[X]]$ denote the ring of polynomials and power series over $F$, respectively. I'm trying to show a statement like the following: Fix a $d > 0$. Let $g\...
Henry Yuen's user avatar
  • 1,899
13 votes
2 answers
643 views

Generating functions, Tutte polynomials, and the bivariate series $\sum_n x^n y^{n^2} / n!$.

A few years ago I computed the Tutte polynomials of the matroids given by the classical Coxeter groups, and found that their generating functions are all simple variations of the series $\sum_n \frac{...
Federico Ardila's user avatar
13 votes
0 answers
369 views

Integral element over p-adic power series

Let $p$ be a prime number. and $R[[X]]$ be the ring of formal series with coefficients in a $p$-adic field $R$. Let $\Lambda=\mathbb{Z}_p[[X]]$. Question 1) Does there exist an explicit description ...
Sharpsel's user avatar
  • 173
12 votes
1 answer
1k views

Singularities of power series

The power series $\sum_{n=1}^\infty \ln(n)z^n$ has radius of convergence $1$ and $z=1$ is a singular point. Is $z=1$ an isolated singularity? If yes, what kind of isolated singularity? I am only able ...
Bogdan Ion's user avatar
12 votes
1 answer
643 views

Relations between coefficients of expansions of a rational function at 0 and infinity

This question goes in the bucket of "this must be well known, but I don't see it and am not sure where to look it up." Given two Laurent power series $A(t)=\sum_{k>N}a_kt^k$ and $B(t)=\sum_{k>M}...
Ben Webster's user avatar
  • 43.9k
12 votes
2 answers
902 views

An analogue of the Bass-Quillen conjecture with power or Laurent series

The famous Quillen-Suslin theorem (formerly known as Serre's problem/conjecture) states that every projective module over $k[x_1,\dots, x_n]$ is free for $k$ a field. Replacing $k$ by a more general ...
Lennart Meier's user avatar
12 votes
1 answer
976 views

Divergent series summation beyond natural boundaries

I'm hoping to investigate the effects of divergent summation methods on series which cannot be analytically continued due to a dense set of singularities. At least a priori, it doesn't seem that a ...
Caleb Briggs's user avatar
  • 1,662
12 votes
3 answers
3k views

What is known about the polynomial factorization of power series?

Some power series factorize; $1+\sum_{n=1}^\infty x^n=\prod_{n=1}^\infty (1+x^{2^n})$ and $1+\sum_{n=1}^\infty x^{2n}/(2n+1)!=\prod_{x=1}^\infty (1+x^2/n^2\pi^2)$ for example; while others do not----...
John Bentin's user avatar
  • 2,427
12 votes
0 answers
260 views

Mixed characteristic analogue of algebraicity of the diagonal of two-variable power series?

Let $f=\sum_{n,m \geq 0}^{\infty}[a_{nm}]p^ny^m \in \mathbb Z_p[[y]]$, where $a_{nm} \in \mathbb F_p$ and $[\cdot]$ means the Teichmüller lifting. Define $I(f)=\sum_{n \geq 0}[a_{nn}]p^nt^n \in \...
sawdada's user avatar
  • 6,148
11 votes
1 answer
937 views

In the rational numbers, is every convergent power series a Taylor series for a rational function?

David Roberts wrote in the comment section of the blog post "Convergence of an infinite sum in the rationals" the following paragraph: Someone mentioned (I think on Twitter) that the Taylor ...
Madeleine Birchfield's user avatar
11 votes
1 answer
1k views

flatness of power series rings

It is known that $A[[X]]$ is flat if $A$ is noetherian (see for example Bourbaki, Algèbre commutative, Ch. III, §3, Cor. 3 p. 146). What happens if A is not noetherian? Is there an easy counter-...
Baptiste Calmès's user avatar
11 votes
1 answer
367 views

Derivative of an algebraic power series in positive characteristic

Let $K$ be a field. It is easy to see that if the characteristic of $K$ is $0$ and $f(T)=\sum_{n\ge0}a_nT^n$ is a power series algebraic over $K(T)$, then $f'$ belongs to $K(T)(f)$. Indeed let $P(X)=\...
joaopa's user avatar
  • 3,655
10 votes
2 answers
1k views

Closed form for $\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$

I am looking at a certain sequence, and consequently I am wondering if anyone knows if there happens to exist a closed form solution for this sum: $$\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$$ ...
Tdonut's user avatar
  • 219
10 votes
3 answers
845 views

Combinatorial interpretation of composition of power series?

This is a minor curiosity that came up in a joint project recently. Consider the sequence $a_n=3\frac {(2n)!}{(n+2)!(n-1)!}$ (A000245 in OEIS). It has multiple combinatorial descriptions. One can ...
Lev Borisov's user avatar
  • 5,166
10 votes
2 answers
577 views

Lagrange inversion for power-series with rational powers

One can use Lagrange inversion to find the power series $F(x)$, which solves $F(x) = x(1+F(x)^p)$, where $p$ is a positive integer. Now, what if $p$ is not an integer, but rather a positive rational ...
Per Alexandersson's user avatar
10 votes
2 answers
787 views

Principal ideal subrings of formal power series rings

In the formal power series ring $\mathbb{F}[[x]]$ over a field $\mathbb{F}$ of characteristic $p>0$, consider an element of the form $f=\sum_{i=0}^\infty a_ix^{p^i}$. Let $R$ denote the unitary ...
Rocky Smith's user avatar
10 votes
1 answer
791 views

Is $k(\!(x,y)\!)$ a topological field?

More generally, let $(R,m)$ be a Noetherian local domain with fraction field $K$. The $m$-adic topology turns $R$ into a topological ring. When $R$ is a discrete valuation ring, this topology extends ...
Laurent Moret-Bailly's user avatar
9 votes
2 answers
1k views

Faster computation of p-adic log

As I see it, $p$-adic integers work very similar to formal power series over $x$ (e g. with regards to Hensel lifting). When it comes to computing $\log P(x)$, one may use the formula $$ (\log P)' = \...
Oleksandr  Kulkov's user avatar
9 votes
1 answer
862 views

A series that is rational?

Let $k=\mathbb F_q(T)$. Can one prove (or disprove) that the series $\sum_{n\ge0}(1-TX^{q^n})Y^{q^n}\in k[[X,Y]]$ belongs to $k(X,Y)$? At first, it looked like it was simple. But in fact, I have no ...
joaopa's user avatar
  • 3,655
9 votes
2 answers
555 views

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

Question: For a prime $p$, is every involution in $\mathbb{F}_p[[x]]$ with a zero constant term a reduction modulo $p$ of some involution in $\mathbb{Z}[[x]]$? Here involution in $A[[x]]$ means $f\in ...
Alexander Burstein's user avatar
9 votes
4 answers
2k views

Formal power series is Taylor expansion of rational function iff Hankel determinants vanish?

Let $$ u(T)=\sum_{n = 0}^\infty a_nT^n$$ be a formal power series over a field $K$. Then why does $u(T)$ lie in $K(T)$ (i.e. is the Taylor expansion of a rational function) if and only if there is an $...
user avatar
9 votes
2 answers
719 views

Algebraic power series of finite order

Apologies if the question is too elementary/something well-known. I believe it is a well-known fact that the rational formal power series $F(z)=\frac{P(z)}{Q(z)}$ which have finite order under ...
Sam Hopkins's user avatar
  • 22.7k
9 votes
2 answers
324 views

Algebraic power series over $\mathbb{F}_2$ as roots of polynomials of special form

Let $F = \mathbb{F}_2$ be the field with two elements. I will denote the rings of polynomials and formal power series over $F$ as $F[t]$ and $F[[t]]$ respectively. Suppose that $x \in F[[t]]$ is ...
Kaban-5's user avatar
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