Questions tagged [power-series]
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395
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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 ...
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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 ...
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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 ...
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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 ...
user119110
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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 ...
user119110
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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 $-\...
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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 $\...
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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 ...
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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
$$
F(x) = \sum_{k=0}^N\frac{1}{k!}(D^kF)(a)[(x-a)^{\otimes k}]+\text{remainder},
...
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What's the summation of formal series $\sum_{n\geq0}\binom{n\delta}{n}x^n$？
$\delta$ is a positive number. Is this Taylor expansion of some function?
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Asymptotic growth of the of Taylor coefficients of the inverse of a function
Let $f(x)=\sum_{n\geq 1} c_n\cdot x^n$ be a function given by a power series. Further there is some $\alpha >1$ such that for all $n$, $c_n = \Theta(1/n^{\alpha})$. What can one say about the ...
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Bounds for the coefficients of the even entire function with positive coefficients
Suppose that the function $f$ is defined by
$f(z) = \sum_{j=0}^\infty a_{2j} z^{2j}$ where $a_{2j} \ge 0, z \in \mathbb{C}$. My questions are the following:
First I want to check this point: if we ...
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Is there any closed form expression for $\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$?
Is there any closed form expression for the following serie?
$$\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$$
Or at least a proof that it is an irrational number. The ...
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Getting the singularities of a function defined by a series
I am trying to locate the singularities that a linear transformation creates. I will not try to motivate this question, since it is already quite far from its starting point.
So, the question is the ...
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Yet another question about unrestricted partitions
I posed a question called "A Product Related to Unrestricted Partitions". As it stands it is too hard. Here's another variation which is easier to search for and hopefully might shed some light on ...
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An elementary question about a sequence of numbers
Let $\lambda_n$ be an increasing and unbounded sequence of positive real numbers and $a_n$ be a sequence of real numbers such that
$$\sum_{n=1}^\infty a_n \lambda_n^k=0 \ \ \text{ for all }\ \ k\geq ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $...
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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?
$$\mathbb E_{i \sim Q}[i] = \...
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Multivariate Generating Function Related to Lambert $W$ Function and Counting Trees with a Certain Property
First, define a sequence $F_0,F_1,\dots$ of functions by
$$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^{...
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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 ...
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Linearizing a power series by conjugation
Let $\mathfrak{I}:=\big\{ \, f:=\sum_{k=0}^\infty f_k z^k \in\mathbb{C}[[z]]\; : \text{s.t. }\; f_0=0 \;\text{ and }\; f_1=1\big\}$. A most basic result about linearization states that, for any $f\...
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Formal Cauchy-Riemann equations for formal power series without complex analysis
Consider the ring $\mathbb{C}[[X,Y]]$ and its subring $\mathbb{C}[[X+iY]]$, where $i=\sqrt{-1}$. One can show that $f(X,Y):=u(X,Y)+iv(X,Y)\in \mathbb{C}[[X,Y]]$ lies in $\mathbb{C}[[X+iY]]$ iff $u$ ...
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how to pass from algebraic power series to the analytic ones
Fix a field of zero characteristic, $k$, e.g. $\Bbb{R}$ or $\Bbb{C}$. Suppose $k$ is normed (and complete for its norm). Consider the ring extensions: $k[x_1,..,x_n]\subset \ k<x_1,..,x_n> \ \...
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Transformation converting power series to Bernoulli polynomial series
I wonder, can anyone describe an expression or formula of a transform that converts
$$\sum_{k=0}^\infty \frac{a_k x^k}{k!}$$
into
$$\sum_{k=0}^\infty \frac{a_k B_k(x)}{k!}$$
where $B_k(x)$ are ...
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Does the divergent solution of this equation :$f'=e^{f^{-1}}$ of Gevrey type and could be Borel summation applied for it?
This question was asked here in MO by someone seeking for the solution of the functional -differential:$f'=e^{f^{-1}}$ not exactly an O.D.E, and again here seeking for the growth rate of it solution ...
user99666
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Question about the limit of a series [closed]
What's the exact value of $\lim\limits_{n\rightarrow \infty}\frac{e^n}{\sum\limits_{i=0}^{i=n}\frac{n^i}{i!}}$?
p.s. I suppose it may be 2, but I cannot prove it.
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Finite group action on germs of holomorphic functions
Let $f(x,y) \in \mathcal{O}^\ast_{\mathbb{C}^2,0}$, a germ of holomorphic function at the origin of $\mathbb{C}^2$ with $f(0,0)=1$. Let $$\varphi(x,y)=(ax+by,cx+dy)$$ be a linear germ of ...
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Double series problems
How to calculate$$\sum_{n=-\infty}^{\infty}{\sum_{m=-\infty}^{\infty}{\frac{\left(-1\right)^n}{\left(6m\right)^2+\left(6n+1\right)^2}}}.$$Follow this,we first get $$\sum\limits_{k = - \infty }^\infty ...
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A solution to the differential equation $Y'' + M(x^2 Y)' - x^2 Y = 0$
In a problem I'm working on relating to plasma instabilities, the following boundary value problem showed up
\begin{equation}
\frac{d^2Y}{dx^2} + M\frac{d}{dx}\left[x^2Y(x)\right] - x^2 Y(x) = 0\;\;\;\...
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asymptotic estimate for log-tan sum
I am finding the following first order estimate.
Question. As $y\rightarrow\infty$,
$$\sum_{n=1}^{\infty}\frac{\log n}n\,\arctan\frac{y}n\,\,
\sim\,\,\frac{\pi}4\log^2y.$$
Is it true?
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Puiseux decomposition over a field with positive characteristic
Let $K$ be an algebraically closed field with characteristic $p>0$, and let $f(t,x)\in K[t,x]$ be a polynomial separable in $x$. Denote:
\begin{equation} \Lambda = \bigcup_{i\in \mathbb{N}} K((t^\...
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A conjecture for q series that is similiar Jacobi Triple Identity
During my research on extension of the Jacobi Triple Product. , I finally got an interesting conjecture that is similar and extension of of the Jacobi Triple Product for higher terms . I asked a ...
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Closed expression for hypergeometric sum
I am trying to simplify an expression and find a closed form for
$$\sum_{m=0}^l \binom{s-m}{s-l} \binom{s-1+m}{s-1}x^m$$
How could I get rid of this summation?
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Sufficient condition on coefficients for a complex power series to be bounded
Let $f(z)$ be an entire function (on $\mathbb{C}$). Assume it has a power series of the form
$$\displaystyle \sum_{n=0}^\infty (-1)^nc_{2n}z^{2n},$$
where $c_{2n}\geq 0$ for all $n$.
Is there a ...
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Continuity (and possibly smoothness) of a multivariable powerseries with positive coefficients bounded on a curve
Consider a multivariable power series with positive coefficients such that it is known to converge on a $C^\infty$ (bounded) curve of $\mathbb{R}^n$, where $n$ is the number of variables. In addition, ...
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Principal ideal subrings of formal power series rings
In the formal power series ring $\mathbb{F}[[x]]$ over a field $\mathbb{F}$ of characteristic $p>0$, consider an element of the form $f=\sum_{i=0}^\infty a_ix^{p^i}$. Let $R$ denote the unitary ...
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Convolution in Hardy spaces
Question Are there non-trivial restrictions on the coefficients of functions in Hardy spaces ($H_p(\mathbb{D})$, $p<1$) that make a subspace that is closed under convolution?
Definition The Hardy ...
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A problem involving power series
We define an entire function on $\mathbb{C}^m$ by
$$
f(z_1,\cdots,z_m)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}t^{2n}(z_1^2+\cdots+z_m^2)^n,
$$
here $t$ is some (positive) real number. Of course, $f(x)=...
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$q$-Eulerian type B enjoy symmetry
Let $(q;q)_n=(1-q)(1-q^2)\cdots(1-q^n)$ with $(q;q)_0:=1$. Define a $q$-exponential by
$$e(z;q)=\sum_{n\geq0}\frac{z^n}{(q;q)_n}.$$
There is a notion of $q$-Eulerian polynomials, see the reference. I ...
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How to work with this power series? [closed]
Let's say we have a sequence $a_n$ that is defined for all $n\in\mathbb{Z}$
and i want to work with its GF $$A(z)=\sum_{n\in\mathbb{Z}}a_nz^n$$
But there are some problems with convergence. For ...
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Linking formulas by Euler, Pólya, Nekrasov-Okounkov
Consider the formal product
$$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,$$
on the ...
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On a particular case of Dirichlet series [closed]
I've this series:
$$ \sum_{\ell = 1}^{+ \infty} e^{-t \ \ell^2} \sin{(k\ell)} = f(k, t) $$
where $ t \in [0,\infty]$ , $ k \in [0,2\pi] $.
I need the limit of series like an analytic function of $...
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Formal theory of (some) generating functions in $t$ and $t^{-1}$?
I am interested in using series of the form $\sum_{n=-\infty}^{\infty} a_nt^n$ (where $a_n\in\mathbb C$) as generating functions. In general, multiplication of such series goes against the "formal ...
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Positivity of a finite sum involving Stirling numbers
In my research in theoretical physics, I have arrived at some coefficients $a_{n,m}$ depending on two integers, $n\geq 1$ and $0\leq m\leq n$:
$$
a_{n,m}=\sum_{j=0}^{n-1} {2j \choose j+m} \left(\frac{...
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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 $...
user97565