Stack Exchange Network

Stack Exchange network consists of 174 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

The tag has no usage guidance.

2
votes
0answers
84 views

Calculating the Taylor series, given a functional equation

I have two functions, whose Taylor series about infinity are given by $$ f(z) = \frac{1}{z} + \sum_{n=2}^{\infty} \frac{A_k}{z^k}, \quad g(z)=\frac{1}{z} + \sum_{n=2}^{\infty} \frac{B_k}{z^k} $$ and ...
5
votes
2answers
293 views

Partitions, $q$-polynomials and generating functions

Recall the integer partition function $P(n)$ with generating function $$\sum_{n\geq0}P(n)x^n=\prod_{k=1}^{\infty}\frac1{1-x^k}.$$ Let $[n]_q=\frac{1-q^n}{1-q}$ denote the $q$-analogue of the integer $...
5
votes
2answers
597 views

Searching for a proof for a series identity

The below identity I have found experimentally. Question. Is this true? If so, may you provide a "slick" (or any) proof. $$6\sum_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum_{k=1}^{\infty}...
7
votes
1answer
226 views

Descartes' rule of signs for infinite series

Consider the function given by $$f(x)=1-a_1x-a_2x^2-a_3x^3-\cdots$$ where each $a_k\geq0$ and some $a_j>0$. If $f(x)$ is a polynomial then Descartes' Rule of signs tells us there is exactly one ...
4
votes
0answers
62 views

Cohen structure theorem with explicit equations

By Cohen structure theorem, a complete regular equicharacteristic Noetherian local ring is isomorphic to a power series. In particular, this should hold for finite extensions of power series $k[[t]][\...
-1
votes
2answers
301 views

Are there integral solutions for $(2a-1)(2^{(b+c)}-3^c )=2^b-1$?

Can anyone prove this assertion? Or at least suggest a method of attack? It has come up in my research. There do not exist $a,b$ and $c$ such that$$ (2a-1)(2^{(b+c)}-3^c )=2^b-1 $$where $a>0,b&...
16
votes
1answer
412 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 ...
4
votes
0answers
141 views

power series and roots of unity

Let $p$ be an odd prime and $X$ and $Y$ be subsets of $p^{th}$ roots of unity, $|X|=|Y|=n,X\neq Y.$ Let $f(t)=\sum_{x\in X}x^{t}-\sum_{y\in Y}y^{t}$. If $f(t)=at^k+o(t^k)$ is the power series ...
4
votes
0answers
57 views

Rationality of power series whose coefficients are the ranks of a sequence of matrices

Recently, I stumbled several times about the problem to decide whether a certain formal power series $$ f = \sum_{n=0}^\infty d_n T^n \in \mathbb{Q}[\![T]\!]$$ is actually a rational function, where ...
1
vote
0answers
61 views

Relation between coefficients of expansions

Related to Relations between coefficients of expansions of a rational function at 0 and infinity I commented at the linked question that the question seemed less about what happened "at infinity", ...
10
votes
1answer
441 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}...
7
votes
2answers
236 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 ...
1
vote
1answer
60 views

Notions of convergence over extensions of finite fields

Let $\displaystyle Q_p[x] = \left\{\frac{p(x)}{q(x)} \mid \, p(x),q(x) \in \mathbb{F}_p[x], \, q(x) \neq 0 \right\}$ denote the field of fractions extending $\mathbb{F}_p[x]$. If we consider the ...
0
votes
0answers
54 views

Upper bound on the modulus of a power series and concentration inequalities for empirical processes

This is a research question I encountered when I as studying solutions of Lebesgue-Stieltjes integral equations. It is related to a new statistical method I am developing (which I cannot expose now) ...
1
vote
1answer
340 views

References on Power Sums

Consider a recent arXiv preprint 1805.11445. The author of 1805.11445 has done an overview of classical problem of simplifying of power sum $$\sum_{1\leq k\leq n}k^m, \ (n,m)\geq 0, \ m=\mathrm{const}...
2
votes
1answer
142 views

On an isomorphism between $p$-adic power series and an inverse limit

Let $K$ be an extension field of $\mathbb{Q}_p$, let $O$ be the ring of integers of $K$, and let $P$ be the maximal ideal of $O$. If $K$ is a finite extension of $\mathbb{Q}_p$, there is the well-...
13
votes
0answers
189 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 ...
1
vote
2answers
62 views

Confluent Heun Equation

Does anyone know any source in which I could find a recurrence relation for the coefficients of the series solution of the Confluent Heun Equation $$y''+\left( {\gamma\over z}+{\delta\over z-1}+\...
3
votes
1answer
134 views

Cancellation problem for Laurent polynomial rings and power series rings

Throughout, let $k$ be an algebraically closed field. For two $k$-algebras $A,B$ let us write $A \cong_k B$ to mean that $A,B$ are isomorphic as $k$-algebras. It is known that if $A$ is an integral ...
1
vote
0answers
148 views

Should the power series solution to $y' = y, y(0) = 1$ be obvious? [closed]

My Understanding: I would derive the Poisson random variable as follows: I consider an experiment which consists of a continuum of trials on an interval $[0,t)$. The result of the experiment takes ...
6
votes
0answers
85 views

Deciding when certain elements of $L[[x]]$, coming from recursions, are algebraic over $L(x)$

Let $L$ be a finite field of characteristic $2$. Suppose that for some $k > 0$ we are given elements $A(0),\, A(1), \dots, \, A(k-1)$ and $c(0),\, c(1), \dots,\, c(k-1)$ of $L[t]$. Define $A(n)$ ...
20
votes
1answer
563 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 ...
0
votes
1answer
108 views

Non-recursive expression for coefficients of the derivative of the logarithm of a power series

Let $f :(-1,1) \to \mathbb{R};\ \ f(x)=\sum_{n=0}^\infty a_n x^n$ be an analytical function expressible as a power series. Also, let $$g : (-1,1) \to \mathbb{R}; \ \ g(x)=\frac{d}{dx} \log{f(x)} = \...
2
votes
2answers
121 views

Boundary behavior of power series vs. boundedness of partial sums

Let $f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be a power series with $0$s and $1$s as its coefficients ($a_{n}\in\left\{0,1\right\}$ for all $n$) with a radius of convergence of $1$. I call such ...
39
votes
3answers
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 ...
1
vote
0answers
211 views

On the coherence of $K[[X_1,X_2,…]]$

Recall that a commutative ring is coherent if every finitely generated ideal is finitely presented, or equivalently if every submodule of every finitely generated module is finitely presented. Let $A ...
2
votes
0answers
290 views

Solution to algebraic equations over $\mathbb{C}$ and $\mathbb{C}[x]$

$t^n=a$, we get one solution to the equation: $$t=e^{\frac{1}{n}\int^a_1 \frac{1}{x}}$$ generalizing this result by replacing the exponential with an elliptic modular function and the integral with ...
-4
votes
1answer
215 views

Does $\int_{0}^{\infty}e^{-xz}\sum_{n=0}^{\infty}a_{n}\frac{x^n}{n!}dx$ converge for $z>0$ with $a_{n} > n! $, for $ n>1$? [closed]

Let $g$ be exponential generating function such that $g(x)= \sum_{n=0}^{+\infty}a_{n}\frac{x^n}{n!}$ extended by analytic continuation along $\mathbb{R+}$ and has a positive radius of convergence. We ...
0
votes
1answer
144 views

Is this a Borel summable $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ with $ a_k$ alternating sequence?

let $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ a divergent series such that $b_k=(-1)^k (k!)a_k >0 $ for $k>1$ , and $b_k$ signed this from $k=1$ to $20$ ,The asymptotic of the titled series ...
1
vote
1answer
137 views

What is a sufficient condition for summability of formel power series? [closed]

There are several kind of summability , i accrossed differents conditions for applying for example Borel summation or laplace transform which let me mixed and confused , really i don't know if i have ...
2
votes
1answer
124 views

What can we know about “the half” of the generating series of Bessel function

I am interested in the series $$\sum_{n\geq 1}I_n(x)\lambda^n$$ which is not the full generating series of the modified Bessel function of the first kind because it starts from $n=1$ and not at $-\...
2
votes
1answer
118 views

Functions on a field representable by Hahn series?

It is well known (see here for example) that a function over $\mathbb{R}$ is representable by a power series iff its analytic continuation to $\mathbb{C}$ is holomorphic on some open subset of $\...
1
vote
0answers
86 views

Differentiation and endpoints of power series [closed]

It is known that the power series $\sum_{n=0}^\infty a_n x^n$ and $\sum_{n=0}^\infty n a_n x^{n-1}$ have the same radius of convergence $r$. Is it true that if $r<\infty$ and $\sum_{n=0}^\infty ...
4
votes
0answers
148 views

Formal multidimensional Taylor series expansion over commutative rings

If $F:V\to W$ is a smooth at $a\in V$ function between finite-dimensional vector spaces over $\mathbb{R}$, then we have $$ F(x) = \sum_{k=0}^N\frac{1}{k!}(D^kF)(a)[(x-a)^{\otimes k}]+\text{remainder}, ...
6
votes
1answer
271 views

What's the summation of formal series $\sum_{n\geq0}\binom{n\delta}{n}x^n$?

$\delta$ is a positive number. Is this Taylor expansion of some function?
4
votes
1answer
156 views

Asymptotic growth of the of Taylor coefficients of the inverse of a function

Let $f(x)=\sum_{n\geq 1} c_n\cdot x^n$ be a function given by a power series. Further there is some $\alpha >1$ such that for all $n$, $c_n = \Theta(1/n^{\alpha})$. What can one say about the ...
0
votes
1answer
76 views

Bounds for the coefficients of the even entire function with positive coefficients

Suppose that the function $f$ is defined by $f(z) = \sum_{j=0}^\infty a_{2j} z^{2j}$ where $a_{2j} \ge 0, z \in \mathbb{C}$. My questions are the following: First I want to check this point: if we ...
3
votes
0answers
300 views

Is there any closed form expression for $\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$?

Is there any closed form expression for the following serie? $$\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$$ Or at least a proof that it is an irrational number. The ...
1
vote
0answers
68 views

Getting the singularities of a function defined by a series

I am trying to locate the singularities that a linear transformation creates. I will not try to motivate this question, since it is already quite far from its starting point. So, the question is the ...
2
votes
1answer
135 views

Yet another question about unrestricted partitions

I posed a question called "A Product Related to Unrestricted Partitions". As it stands it is too hard. Here's another variation which is easier to search for and hopefully might shed some light on ...
5
votes
1answer
246 views

An elementary question about a sequence of numbers

Let $\lambda_n$ be an increasing and unbounded sequence of positive real numbers and $a_n$ be a sequence of real numbers such that $$\sum_{n=1}^\infty a_n \lambda_n^k=0 \ \ \text{ for all }\ \ k\geq ...
4
votes
1answer
236 views

A strange (possible) fact about the Hecke operator T_3 in level 13 and characteristic 2

delta(z) + delta (13z) is a weight 12 modular form of level Gamma_0 (13). Let A in Z/2[[q]] be the mod 2 reduction of the Fourier expansion of this form. (The exponents appearing in A are the odd ...
4
votes
1answer
133 views

An analogue of rational functions for Hahn series

For any field $k$, we have both the field $k(t)$ of rational functions (formal quotients of polynomials, i.e. the field of fractions of $k[t]$) and the field $k((t))$ of formal Laurent series (which ...
1
vote
1answer
145 views

A constrained double summation

This is a question that I asked on Math StackExchange (see here), but I believe it is better to ask if any number theorist has encountered it before. Consider two positive integer $(k,l)$ and they ...
2
votes
0answers
129 views

Series representation of multiplication of two modified Bessel function

Series representation of multiplication of two Bessel function $J_{\mu}(az) J_{\nu}(bz)$ is in terms of sum of hypergeometric functions $_2F_1$, it given in book Treatise on Theory of Bessel Functions ...
3
votes
1answer
188 views

what is about the corresponding power series?

According to the papers The absolutely continuous spectrum of Jacobi matrices and these lecture notes: periodicity ~ potential well or lattice (order) lack of absolutely continued spectrum ~ Anderson ...
6
votes
3answers
442 views

Alternating power series $\sum_{k\geq 0}(-1)^k z^{(2k+1)^2}$

Suppose that $f(x):=\sum_{k\geq 0}(-1)^ke^{-(2k+1)^2x}$ has a holomorphic continuation to a neighborhood of $0$, that is, $f(x)=\sum_{n\geq 0}a_n x^n$ for $x> 0$ small. I want to know the value of $...
0
votes
2answers
137 views

Closed form of $\sum_{i=k}^\infty i h {i \choose {k-1}} h^{k-1} (1-h)^{i - (k-1)}$?

Is there a closed form solution to the expression below? Or, if there is no closed form solution but the series converges, is there some upper bound on this expression? $$\mathbb E_{i \sim Q}[i] = \...
3
votes
1answer
214 views

Multivariate Generating Function Related to Lambert $W$ Function and Counting Trees with a Certain Property

First, define a sequence $F_0,F_1,\dots$ of functions by $$F_0(x,z) = z,$$ $$F_k(x,z)=x\exp\left(F_{k-1}(x,z)\right) \quad \text{for }k\geq1.$$ So, for example, $$F_1(x,z) = x e^z, \quad F_2(x,z)=xe^{...
1
vote
1answer
96 views

Given 𝛾 ∈ (0, 1), why is 𝛾ˡ negligible for l ≫ 1/(1-𝛾)? [closed]

This comes from the paragraph following equation (27) on page 6 of this paper. It's not crucial to the argument — any such bound will do — but it's not clear to me why this particular ...