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19
votes
0answers
194 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 ...
6
votes
0answers
113 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 ...
0
votes
0answers
35 views

t-linked extension

Let $A\subseteq B$ be an extension of commutative integral domains. the extension is t-linked if it satisfies the following property: If P is a finitely generated ideal of A such that $P^{-1}=A$ than ...
1
vote
1answer
39 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
433 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. ...
1
vote
0answers
49 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
842 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 ...
3
votes
3answers
329 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
130 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 ...
2
votes
1answer
93 views

Matching power series to infinity

As pointed out by Makoto, on this question about power series rings and the axiom of choice, an idea I had needed the axiom of dependent choice to work. However, the construction raises another ...
7
votes
3answers
324 views

properties of formal delta functions

The formal delta function is $\,\,\displaystyle\delta(x):=\sum_{n\in\mathbb Z}x^n. $ If we agree that expressions $(x+y)^n$ for $n\in\mathbb Z$ are always expanded in non-negative powers of the second ...
4
votes
1answer
127 views

Estimate of the sum Taylor's coefficients

Let $f(x) = \begin{cases}\ln\frac{x}{e^x-1}, \quad x > 0; \\ 0, \quad\qquad x=0; \\ \ln\frac{x}{e^x-1}, \quad x < 0. \end{cases}$ Power series in 0: $f(x) = \sum_{n=1}^{\infty} a_n x^n = ...
0
votes
1answer
215 views

Is there any algorithm to decide whether a series with integral coefficiens is a algebraic function? [closed]

Given a series with integral coefficiens as following: $$F(x)=\sum_0^i a_i x^i,\text{where }a_i\in \mathbb{N}\bigcup 0 $$$$\text{and there is a computable function $\psi$ such that } \forall i ...
1
vote
1answer
130 views

Ordinary or Rational Generating Function for Associated Stirling Numbers $b(n,k)$

I am trying to identify or find the ordinary or rational generating function (not the exponential generating function) for the Associated Stirling numbers of the Second kind, denoted ...
0
votes
0answers
61 views

For which recurrence relations is it decidable whether a formal power series has a maximal zero coefficient?

In this MSE question, I asked whether one can prove that a generating function has infinitely many coefficients equal to zero. The answer given (and accepted) to that rather broad question was “No”. ...
2
votes
0answers
66 views

Orders of certain quotients of power series rings

Let $\Lambda_d := \mathbb{Z}_p[[T_1, \ldots, T_d]]$ denote the ring of formal power series in $d$ variables over the ring of $p$-adic integers. Suppose that $g \in \Lambda_d$ is an irreducible ...
1
vote
0answers
190 views

integral curves and differential equations on arcs

I am trying to prove a statement that is obivious in analytic setting, but makes me feel at a loss in formal algebraic setting. Let $M$ be a smooth curve over an algebraically closed field $k$. Let ...
8
votes
2answers
263 views

How do I find coefficients of a product expansion

Any power series $f(t) = 1 + t \mathbb{Z}[[t]]$ can be uniquely expanded in the following two ways: $$1 + \sum_{i=1}^\infty f_i t^i = \prod_{i=1}^\infty (1-t^i)^{-n_i}$$ Here, the $f_i$ and $n_i$ ...
0
votes
0answers
150 views

Examples of functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition

there are examples of lacunary functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition.I want to know more examples of those functions,the more the ...
9
votes
3answers
455 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 ...
0
votes
1answer
97 views

A pole of function in the article of Springer

I read the article Springer, T.A. On the invariant theory of $SU_2$, Indag. Math. 42, 339-345 (1980). Author considers $\mathbb{C}$-linear map at page $340.$ If $n$ is a positive integer, then write ...
1
vote
0answers
209 views

R[[X]] flat as a R[X]-module?

I assume $R[X]\rightarrow R[[X]]$ is not flat in general, but I was wondering if any conditions on a commutative ring $R$ are known such that $R[[X]]$ is flat as a $R[X]$-module. Would $R$ noetherian ...
5
votes
1answer
288 views

Laurent expansion of a principal value integral

Let $f(t)$ be a nice Hölder continuous function. Also, suppose that $f$ is even. I'm interested in evaluating integrals of the form: $$\oint (1-z)^{k+1}\int_0^1 \frac{f(t)}{(1-zt)^{n+1}}dtdz,$$ ...
2
votes
0answers
214 views

closed form expression for an infinite series

Is there any closed form expression for the infinite sum $\sum_{n \geq 0}q^{n(n+1)/2}(1+q)(1+q^2)\cdots(1+q^n)u^n$ where both $q$ and $n$ are variables and $n \in N \cup {0}$?
-1
votes
1answer
105 views

derivatives and uniformly convergence [closed]

Let $f$ be a function of a real variable expandable in power series on $\mathbb R$: there exists a sequence $(a_n)_{n\in\mathbb N}$ of reals such that for all $x\in\mathbb R$, one has ...
-3
votes
1answer
131 views

Randomness about coefficients of series

$B\subset \mathbb{N}\bigcup \{0\}$ is finite and not empty, infinite series:$$f(x)=\sum_{i=1}^{\infty}a_i x^i,a_i \in B$$ Now $f(x)$ is rational or has a natural boundary. Now,the question :if ...
3
votes
2answers
377 views

Sum of series $a^{i^2}$

Is there any closed form known for the expression $\sum_{i=1}^\infty a^{i^2}$ where $|a|<1$? Thanks!
3
votes
3answers
300 views

An apparently simple question (behaviour at infinity of a power series)

Let $(a_n)$ be a sequence of real numbers, and suppose that the real power series (function) $S(x):=\sum_{n=0}^{\infty} a_n x^n$ converges for every $x\in\mathbb{R}$. $\mathbf{Question}$: Suppose ...
2
votes
2answers
420 views

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 ...
6
votes
3answers
232 views

On formal solutions to differential equations

Let $k$ be a field of characteristic zero. Put $K=k[\![ t]\!]$ and $W=k\langle t,\partial\rangle / ([\partial,t]=1)$. Then $W$ operates on $K$ in the obvious way ($\partial f = \frac{d f}{dt}$), and ...
3
votes
0answers
157 views

derivatives of composite function [closed]

There's a formula for the $n$th derivative of a composite function $f(g(x))$ - it's called Faa di Bruno's formula - but I'm not really interested in the formula but in the proof given in the book of ...
1
vote
0answers
30 views

Saturation of a subalgebra over the Tate-algebra inside the power series ring

Let $A$ be a discrete valuation ring and $\pi$ a uniformizer. Over $A$ we consider the Tate-algebra $$A\langle t \rangle =\{ f=\sum_{n=0}^\infty a_nt^n \mid a_n\in A, \lim_{n\to \infty} \lvert ...
7
votes
1answer
136 views

Positivity of coefficients of a power series

How does one check for the positivity of coefficients of a rational function,say, for example $\frac{p_1(x,t)}{(1-xt)(1-x^2t)(1-x^3t)}$ where $p_1(x,t) = 1 + tx + 2t^2x^2 - 3x^3t^2 -x^5t^3 - ...
7
votes
0answers
294 views

name for a degree-like invariant of a power series over a commutative ring

Let $R$ be a commutative ring, and let $f \in R[\![X]\!]$ be a formal power series. Sometimes (and for example, this will always be possible if $R$ is Noetherian), one may write $f$ in the form $$ f ...
3
votes
2answers
281 views

existence and uniqueness of solutions for ODEs in formal power series?

I came across this question and it looked like something that is likely to have been looked into, but I couldn't find a reference. Let $k$ be some (algebraically closed, if needed) field. There is a ...
5
votes
4answers
254 views

Exponential of a specific hypergeometric series

This is motivated by this question. Let $f$ be the hypergeometric series $ f(x) = 2 x \, _{4}F_3([1, 1, 4/3, 5/3], [2, 2, 2], 27 x) $ which is explictly given by $ f(x) = \sum_{n \geq 1} ...
2
votes
1answer
243 views

Localisation of $\mathbb{Z}_p[[X]]$ at ideal $(p)$

Let $R=\mathbb{Z}_p[[X]]$ where $\mathbb{Z}_p$ denotes the $p$-adic integers and $p$ is a prime. Then what is $R_{(p)}$ $(R$ localised at the ideal $pR)$ $?$
30
votes
1answer
567 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) = ...
-1
votes
1answer
172 views

An extension of the real semiring with multiple degrees of infinity

Is it possible to define an extension of the probability semiring $(\mathbb{R}^+, +, \times, 0, 1)$ such that Closure $a^* = 1 + a + a^2 + \ldots$ is defined for every element of the semiring, not ...
1
vote
0answers
116 views

Automorphism on F_2[[X,S]]

Let us define the automorphism $\sigma$ on ${\Bbb F}_2[[X,S]]$ such that $\sigma \colon S \mapsto S + S^2 + S^3$ $\sigma \colon X \mapsto X + S$. It is easy to see that the ideal $(S)$ is stable ...
2
votes
1answer
94 views

Convolution-type operator for series

Suppose $f(z)=\lambda(a_1z+a_2z^2+\cdots)$ is holomorphic in $\{|z|<1\}$ with $\lambda>0$. For each $d\geq 1$ , I am trying to define an operation, $\star_d$ , so that $f(z)\star_d ...
11
votes
1answer
244 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 ...
3
votes
1answer
176 views

Are these powers of a characteristic 3 power series annihilated by certain Hecke operators?

Let D in Z/3[[x]] be sum ((a_n)(x^n)) where the sum runs over all n prime to 6 and a_n is the mod 3 reduction of the number of ideals of norm n in the ring of integers of Q(root(-3)). (So ...
1
vote
1answer
179 views

What's the asymptotic behavior of this function at large distance? [closed]

This question is based on some Physics motivation. Define a distance function $f(\mathbf{r})=\int_{\Omega }d^2k\int_{\Omega }d^2q \cos[(\mathbf{k}-\mathbf{q})\cdot\mathbf{r}]$, where ...
4
votes
0answers
140 views

Are these subspaces of $\mathbb{Z}/3[[x]]$ stable under the shallow Hecke algebra?

This is a characteristic $3$ analog of part of my earlier question, "Are these two subspaces of $\mathbb{Z}/2[[x]]$ the same?" Notation Fix a prime $N$ other than $3$. Let $F,G \in ...
2
votes
1answer
172 views

elementary question on a completion of a ring

Let $k$ a field, and $k[\epsilon]=k[X]/(X^{2})$ , what is the completion of the ring $k[\epsilon][t]$ with respect to the ideal $(t^{2}+\epsilon)$?
2
votes
0answers
146 views

Solving ODE with negative expansion power series

I'm moving this here, as suggested from physics.stackexchange. The original is here. So, I need to solve a system of ODE, using negative power expansion. I will give all the necessary equations and ...
5
votes
1answer
187 views

Is it decidable whether the support of a rational $\mathbb{Z}$-series is a regular language?

Let $S \in \mathbb{Z}\langle\langle A\rangle\rangle$ be a rational series in noncommutative variables. The support of $S$ is the set of all words $u \in A^*$ such that $(S, u) \not= 0$. It is ...
5
votes
0answers
290 views

Are these two subspaces of $\mathbb{Z}/2[[x]]$ the same?

The following questions arise from modular form theory. But this theory isn't needed to formulate or understand them, and I'm not using the modular-forms tag. NOTATION Fix an odd prime $N$. Let $$ ...
1
vote
0answers
184 views

Can the series $\sum\limits_{n=0}^\infty q^{F_n}$ be expressed in terms of theta functions?

Let $F_0=0,F_1=1,...$ be the Fibonacci numbers. Is there a known closed form for the sum $\sum\limits_{n=0}^\infty q^{F_n}$? By closed form, I mean in terms of well-known functions, the first ones to ...