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-1
votes
0answers
12 views

Rigorious formulation of approximation of integral as area of a square and its radius of convergence [migrated]

We know that the taylor expansion of $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = a(t)\Delta t + \frac{1}{2}\frac{da}{dt}(t) \Delta t^2 + \frac{1}{3!}\frac{d^2a}{dt^2}(t) \Delta t^3 + \frac{1}{4!}\...
2
votes
1answer
135 views

Ellipsoidal harmonics - A Series expansion for Lame functions of the second kind

$\underline{Intro \;to \;skip}$ In the theory of ellipsoidal harmonics, Lame functions of the second kind $F_n$ arise as the second linearly independent solution (the first being Lame functions of ...
5
votes
1answer
98 views

Is there an exponential map on (Hahn) ordered fields?

If $F$ is an ordered field and $G$ is an ordered abelian group, one can define the Hahn product $F \boxtimes G$ to be the set of formal Laurent series with coefficients in $F$ and exponents in $G$. It ...
4
votes
2answers
112 views

Bieberbach-type bound for bounded univalent functions

Suppose $f: \mathbb{D}\to \mathbb{C}$ is a univalent function with $$f(z)=z+a_2z^2+a_3z^3+\cdots.$$ The Bieberbach conjecture/de Branges' theorem asserts that $|a_n|\leq n$ with equality for the Koebe ...
1
vote
0answers
113 views

Can an algebraic function be zero both at $z=0$ and at its leading singularity?

Apologies for asking possibly strange questions, but I am just a poor computer scientist trying to understand a mathematical paper on singularity analysis of algebraic functions that is apparently not ...
12
votes
2answers
319 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 ...
5
votes
0answers
82 views

Result of Larsen and Lunts on rationality of power series with coefficients in a free abelian group

Let $G$ be a free abelian group (not necessarily finitely-generated) and $F$ be the fraction field of the group ring of $G$. Let $\Theta$ be the set of power series in $F[[t]]$ such that each nonzero ...
1
vote
1answer
55 views

Can I apply Lagrange inversion theorem? [closed]

I want to invert the equation $$\eta = g(x)\sqrt{1+g'(x)^2}$$ to get $x$ as a function of $\eta$. Assume $g(0)=0$, $g'(0)=0$ and $g'(x)>0$ for $x>0$ (Think $g(x) = x^p$ for $p\geq 2$ integer). ...
1
vote
0answers
26 views

Need explicit formula for reversion of a Chern-character-like series

On the first sight this looks like homotopy theory question but actually came from need to simplify some expressions related to the Rasch model from the Item Response Theory. Let $$ s=\sum_i\frac{...
2
votes
0answers
66 views

Is the positive part of an algebraic bilateral p-adic convergent power series algebraic?

Let $\mathbb{Z}_p \{ X\}$ and $\mathbb{Z}_p \{ X , X^{-1}\}$ be the henselizations of $\mathbb{Z}_p [X]$ and $\mathbb{Z}_p [ X , X^{-1}]$ with respect to the ideals $p\mathbb{Z}_p [X]$ and $p\mathbb{Z}...
3
votes
1answer
131 views

Growth comparision between an entire function and a related function

Let $p$ be a prime number, $\mathbb C_p$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the valution $v(x)=-\deg(x)$. Let $\sum_{n\ge0}a_nz^n$ be a ...
3
votes
2answers
166 views

Lower Bounds for the Roots of Polynomials

I'm interested in the "size" of the roots of a sequence of Taylor Polynomials of an entire function. For example, consider $\mathrm f(z) = \mathrm e^z$. The Taylor Polynomials, or $k$-jets, are $$\...
2
votes
0answers
69 views

Inverses of probability generating functions: positivity of derivatives

Let $\mathcal{G}$ be the set of probability generating functions of random variables taking positive integer values, considered as functions on $[0,1]$. So $G\in\mathcal{G}$ can be written $G(x)=\...
5
votes
2answers
188 views

When does the radius of convergence of the product of two $p$-adic power series increase?

Let $p$ be a prime number and denote by $R(f)$ the radius of convergence of a power series $f(x) \in \mathbb{C}_p[[x]]$, where $\mathbb{C}_p$ is the completion of the algebraic closure of $\mathbb{Q}...
8
votes
4answers
389 views

Analytic continuation of the double sum $\sum_{n,m\ge0}x^ny^mt^{nm}$

Define function $f(x,y,t)$ as the analytic continuation of the series $$f(x,y,t)=\sum_{n,m\ge0}x^ny^mt^{nm}$$ This series definitely converges when all the arguments are small enough. I would like to ...
3
votes
1answer
59 views

Redundancy in transseries representation of functions?

"Transseries" are a kind of generalized power series that allow things like fractional exponents and exponentials (with another transseries as the exponent). I know very little about them but I have ...
6
votes
0answers
218 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 ...
0
votes
0answers
75 views

Orbits of some action of SL2 on Pontryagin dual of the field of formal Laurent series

Let $K=\mathbb{F}_2((t))$ be the field of formal Laurent series over the finite field $\mathbb{F}_2$. Now consider $K^3$ as an additive group and its dual group $\hat{K^3}$, which consists of all ...
1
vote
1answer
79 views

Inverse error function in Hardy space?

Let $\mathrm{erf}(x) := \frac{2}{\sqrt{\pi}} \int_{-\infty}^x \exp(-t^2) \, dt$ be the error function $\mathrm{erf}: \mathbb{R} \to (-1,1)$. It is monotonously increasing and therefore has an inverse $...
4
votes
2answers
165 views

Extraction of Coefficients in the Exponential Function of a Series

Question: Let $f(x) \in x\mathbb{C}[[x]]$. What is the (asymptotically) fastest algorithm for calculating the coefficient of $x^n$ in $e^{f(x)}$? Naive Solution 1: Using fast polynomial ...
2
votes
0answers
56 views

Hadamard Product of specific type of power series

I am consider the power series of the form $$F_n(t):=\frac{1}{\prod_{i=1}^n(1-t^i)}.$$ Given two power serires $A(t)=\sum_{i\ge 0}{a_it^i}$ and $B(t)=\sum_{i\ge0}{b_it^i}$, their Hadamard product is ...
18
votes
1answer
302 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 ...
0
votes
1answer
82 views

When can one infer degrees of generators of a ring from its hilbert series

I know that for a noetherian ring, it's hilbert series can be written as $$HS(t)=\frac{P(t)}{\prod_{i=1}^d{(1-t^{d_i})}}$$ where $P(t)$ is polynomial, and there are $d$ generators of degrees $d_1,\...
7
votes
1answer
104 views

When is the diagonal of a rational bivariate power series again rational

Given a rational bivariate power series $F(x,y)=\sum{a_{n,m}x^ny^m}$, the diagonal function $G(t):=\sum{a_{n,n}t^n}$ is known to be algebraic, although not rational in general. I was wondering if ...
0
votes
1answer
102 views

Frobenius method for multiple singular points

As we know, if the equation $$a(x)y''+b(x)y'+c(x)=0 \ \ \ \ \ \ \ \ \ (1)$$ has a regular singular point at $x=x_0$ then we seek solution of the equation as $$y(x)=\sum_{n=0}^{\infty}\beta_n (x-x_0)...
9
votes
2answers
452 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: $$\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$$ ...
3
votes
0answers
127 views

Involutions on $[0,1]$ given by power series (related to probability generating functions)

Let $A$ be a function from $[0,1]$ to $[0,1]$. $A$ is an involution if $A(A(x))=x$ for all $x\in[0,1]$. Which involutions $A$ exist such that $A(x)=\sum_{k=0}^\infty a_k x^k$ with $a_0=1$ and $...
3
votes
0answers
85 views

Crossed homomorphisms between power series groups

Consider the group $\mathbb{C}[[z]]_1$ of the power series of the form $a_1 z + a_2 z^2 + \cdots$, with $a_1\neq 0$, under the operation of composition, and $\mathbb{C}[[z]]$ as a $\mathbb{C}[[z]]_1$-...
1
vote
1answer
161 views

How do powers affect asymptotics in generating functions?

Let $a_n$ be a sequence of non-negative real numbers, and $A(x) = \sum_{n=0}^{\infty} a_n \frac{x^n}{n!}$ its exponential generating function. Also, suppose $B(x) = \sum_{n=0}^{\infty} b_n \frac{x^n}{...
1
vote
1answer
155 views

Formal Power series decomposition

Let $G$ be a linear algebraic group over $\mathbb C$ (say $SL_r$) consider a formal power series $$g(t)\in G(\mathbb C((t)))$$ My question is: Is it possible to decompose $g$ as $$g=ha$$ with $h\in G(...
1
vote
0answers
128 views

Power series with matrix coefficients

Let $A(t)\in SL_r(\mathbb C((t)))$ be a formal power series with matrix coefficients, and let $B(t)\in SL_r(\mathbb C[t])$ and $C(t)\in SL_r(\mathbb C[[t]])$ such that : $$A(t)=B(1/t) \;( ^tA(-t)^{-1})...
7
votes
1answer
200 views

Conjectured equivalent conditions on certain power-series

Let $P(x)=1+a_1x+a_2x^2+a_3x^3+...$ be a series such that every $a_i$ is an integer, $a_1<0$, and $a_i\ge 0$ for every $i\ge 2$. Are the following statements equivalent ? $P(y)=0$ for some $y&...
3
votes
0answers
93 views

Is there a name for the operation that stretches out an invertible series by a factor of $m$?

The question is whether there is an established word for the transformation that starts with an invertible formal power series over a field $k$, $u(x)=xg(x)=x(1+a_1x+a_2x^2+\cdots)$ and delivers the ...
15
votes
7answers
2k 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 ...
7
votes
2answers
306 views

Radial limit does not exist almost everywhere

Problem 4 in Chapter 4 of Stein's book "Real Analysis" says $\sum_{n\geqslant 0}z^{2^n}$ doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy ...
2
votes
0answers
63 views

Discrete “difference” equations that involve changes in both shift and scale

A standard use of the Z-transform ($F(z) = \sum_n (f[n] \cdot z^{-n} )$) is to understand the effect of a difference equation on a signal. For instance: $y[n] = x[n] + y[n-1]$ $Y(z) = X(z) + Y(z) \...
4
votes
0answers
164 views

Mod 2 modular forms in levels 5 and 25--how to account for this Hecke isomorphism?

The space $P1$ of my earlier question 203755 "Two spaces attached to mod 2 level 9 modular forms...", is essentially the space of mod 2 level 3 modular forms. That such a space should appear inside ...
1
vote
0answers
48 views

Simplifying closed form for Meta Operator

I was consider the set of linear operators: $$O_{a,k} = \frac{f(ax^k) - f(x)}{ax^k - x} $$' Particularly I am looking for the closed forms of the eigenfunctions of this operator, that is the ...
2
votes
2answers
484 views

Local coordinates on (infinite dimensional) Lie groups, factorization of Riemann zeta functions

Given a (finite dimensional) Lie group $G$ (real $k=\mathbb{R}$ or complex $k=\mathbb{C}$) and its Lie algebra $\mathfrak{g}$, one can prove (a basis $B=(b_i)_{1\leq i\leq n}$ of $\mathfrak{g}$ being ...
1
vote
0answers
166 views

Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism

MOTIVATION Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on ...
4
votes
1answer
185 views

Raising coefficients of a power series to some power

Suppose you are given a power series $P=\sum_{i=0}^\infty{a_nt^n}$. I am primarily concerned with those power series coming from rational functions of the form $$ \frac{1}{\prod_{i=1}^k{(1-t^{\...
2
votes
1answer
92 views

A question about decomposing mod 2 modular forms of level p^2

Fix an odd prime $p$. Each $f \in \mathbb{Z}/2[[x]]$ can be written as $f_{+} + f_{-} + f_0$ where each exponent k of $x$ appearing in $f_{+}$ (resp. $f_{-}$, $f_0$) has Legendre symbol $(k/p)$ equal ...
24
votes
0answers
450 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 ...
7
votes
0answers
173 views

Singularities of an analytic function over a non-archimedean field

What do we know about the types of singularities that a convergent power series over a non-archimedean field can have? More specifically: i) What types of essential singularities can occur? ii) Are ...
1
vote
1answer
69 views

Generating function for products of laguerre polynomials?

In a quantum physics context, I would like to evaluate $S=\sum_{n=0}^\infty z^n\cos(\pi L_n(x))$ for $z<1$. I found generating functions for squares of Laguerre polynomials but not for any higher ...
4
votes
1answer
740 views

How to prove this identity on double summation series?

I suspect the following identity is valid, but I can not prove it. I just calculate it numerically. $\sum_{m=0}^\infty\left[\sum_{n=0}^\infty\frac{(-1)^{n+m}}{(n+1)(n+m+1)}\right]=\sum_{m=0}^\infty\...
1
vote
0answers
78 views

Relative nonarchimedean disks and annuli

Let $A$ be a Huber (i.e. f-adic) ring, meaning a topological ring with an open subring $A_0$ which is adic and has finitely generated ideal of definition. Is there a good notion of closed disk of ...
30
votes
3answers
994 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\...
3
votes
3answers
579 views

Find an integrable, positive, unbounded, analytic function

Is there a standard example of a function $f \in L^1( \mathbb R)$ which is analytic, positive, integrable but not bounded? An example which comes immediately to mind is to take the series of narrower ...
2
votes
1answer
305 views

The rigid-analytic open disk

Let $K$ be a local field and $D_K$ the open unit disk, considered as a rigid space or adic space over $K$. What is the algebra of analytic functions on $D_K$? Proposition 1.1 of this article describes ...