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6
votes
2answers
229 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
122 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
117 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
59 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”. ...
1
vote
0answers
64 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
168 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
251 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
103 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
435 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
96 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
202 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
284 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
182 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
98 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
129 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
362 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
291 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 ...
1
vote
2answers
415 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
230 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
151 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
28 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
124 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
287 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
268 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 ...
4
votes
3answers
192 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
240 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 $pR)$ $?$
27
votes
1answer
440 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) = ...
0
votes
1answer
160 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
112 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
92 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
240 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
174 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
137 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 ...
1
vote
1answer
168 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
133 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
181 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
285 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
168 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 ...
9
votes
1answer
220 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 ...
1
vote
0answers
55 views

Is it obvious that the defining conditions to obtain a particular singularity are well defined on the quotient space?

Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function vanishing at the origin, with the following properties: $$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 ...
6
votes
1answer
148 views

Is there an algorithm to find out the number of small solutions to a polynomial equation, when we vary all the coefficients?

Let $\Phi (z,t)$ be a polynomial given by $$ \Phi(z,t) := z^n + A_{n-1}(t) z^{n-1} + \ldots + A_1(t) z + A_0(t).$$ Assume that $\Phi(0,0) =0$. It is a fact that a solution $z(t)$ of the equation $$ ...
15
votes
1answer
683 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 ...
0
votes
1answer
445 views

construct a power series with infinitely many zeros in the complex plane, bounded coefficients???

Hi all. I want to construct a power series $F(z)=\sum_{n=0}^\infty c_nz^n$ centered at zero and with finite radius of convergence in the complex plane, and which has infinitely many zeros (in its ...
4
votes
0answers
153 views

Does this space of mod 2 modular forms admit a (Z/8)* degree decomposition?

Fix an odd N>0. Let M consist of all odd elements of Z/2[[x]] that are the mod 2 reductions of elements of Z[[x]] arising as the Fourier expansions of modular forms for (Gamma_0)(N); it's easy to see ...
3
votes
2answers
214 views

Sign of coefficients

Let $a_0,a_1,\dots$ be the sequence satisfying $$ \left(\sum_{n=0}^\infty a_n x^n\right)\left(\sum_{n=0}^\infty \frac{x^n}{n+1}\right)=1. $$ This means that $a_0=1$ and ...
0
votes
1answer
247 views

Uniform convergence of $g(x) = \sum_{n=1}^\infty (1-x) a_n x^n$

Let's assume that $ f(x) = \sum_{n=1}^\infty a_n x^n $ has a radius of convergence $1$ and that $ \lim_{x\to 1^-} f(x) = +\infty. $ Does it imply that power series $ g(x) = \sum_{n=1}^\infty (1-x) ...
0
votes
1answer
358 views

A simple question from mathematical analysis (assumption changed) [closed]

Let $\forall n=0,1,2,\dots$, $\alpha_{n}(x)$ are POLYNOMIALS in $x$. Next, let for all $x\neq0$ the power series $$\sum_{n=0}^{\infty}\alpha_{n}(x)t^{n}$$ has positive radius of convergence. Can one ...
4
votes
0answers
105 views

A subring of the Serre Swinnerton -Dyer ring of level N modular power series

Suppose ell is prime and (N,ell)=1. Consider those power series over Z that are expansions at infinity of modular forms for gamma_0 (N) of weight a multiple of ell-1. I'll say that an element of ...