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Taylor series is a method to analyze functions as polynomials.

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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 ...
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111 views

Generating Function for x+x^2+x^4+x^8+

I am trying to find a closed form solution for the generating function $x+x^2+x^4+x^8+...=\sum_{i=1}^\infty x^{2^i}$. I began with the fact that the positive integers can be neatly partitioned into ...
2
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0answers
169 views

Generalization of Carleman coefficients to multivariable functions - Carleman tensor?

Recently I learned about a matrix called Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying. Carleman linearization is a technique used to embed a finite ...
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0answers
131 views

Functional Taylor expansion for differential entropy

Consider an continuous distribution $F$ with density $f$. The (differential) Shannon entropy of $f$ is $h(f)=-\int f(x)\log f(x) dx$. In the literature of differential entropy estimation, ...
6
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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?
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1answer
145 views

Mean value theorem in terms of Wirtinger calculus?

The mean value theorem for vector-valued function in the real domain $f: \mathcal{R}^n \rightarrow \mathcal{R}^d$ can be expressed as \begin{equation} f(x)-f(y)=\int_{0}^{1}\nabla f(x(\tau))d \tau \...
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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 ...
7
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1answer
307 views

Is there a bound for Lipschitz constant in terms of second differences?

It is easy to show that if $f\colon[0,1]\to\mathbb R$ and $|f|\leq A$ and $|f''|\leq B$ then~$|f'|\leq 4A+B$. Indeed, by Taylor formula with remainder $f(x)=f(c)+(x-c)f'(c)+\frac12(x-c)^2f''(d)$ where ...
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1answer
120 views

A polynomial and its reciprocal expansion [closed]

Suppose $f(x)=\prod_{k=1}^n(x-a_k)$ where all $a_k>0$. Expand the function $\frac1f$ at $\infty$ so that $$\frac1{f(x)}=\frac{b_n}{x^n}+\frac{b_{n+1}}{x^{n+1}}+\cdots.$$ Does it follow that each $...
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3answers
356 views

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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3answers
666 views

A curious series related to the asymptotic behavior of the tetration

The tetration is denoted $^n a$, where $a$ is called the base and $n$ is called the height, and is defined for $n\in\mathbb N\cup\{-1,\,0\}$ by the recurrence $$ {^{-1} a} = 0, \quad {^{n+1} a} = a^{\...
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1answer
83 views

Determining the rate of spread of geodesics when the sectional curvature is zero

I have posted this question in mathSE a few weeks ago (and proposed a bounty) but so far got no response. In the book Riemanian geometry (by do-Carmo), the following result is proved (Corollary 2.9 ...
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1answer
404 views

Taylor expansion of cumulant generating function

For the characteristic function $\mathbf E e^{i t X}$ of a random variable $X$ with $n+1$ finite moments, there is the well known and easy to prove bound on the remainder of the Taylor series $$\left\...
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1answer
163 views

A closed form of a summation or the taylor series expansion of some function with a closed form?

Let $Z_N = \displaystyle{\sum_{k+j\leq N}} \frac{N!N^{k+j}}{N^{N+1}}\frac{u^kv^j}{k!j!}\binom{N-j}{N-j-k}$ where $u$ and $v$ are two unknowns. My question is: Is there a closed-form for $Z_N$ or is $...
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0answers
125 views

Smoothness of coefficients of remainder term in Taylor expansion

Given a $C^{k}$ function $f:\mathbb{R}^d\to\mathbb{R},$ we can use Taylor's theorem to write it as $$f(x)=\sum_{|\alpha|\le k-1} c_\alpha x^\alpha + R(x),$$ where $R$ is $C^k$ and can be expressed ...
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3answers
206 views

Compute Inverse Series for Implicit Equation

In financial mathematics, the inverse Series of: $$b(x) = -\frac{\log(1-e^{-x})}{x}$$ is needed in order to perform fast calculation on swaptions for G2++ calibration model. (see this post for ...
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1answer
376 views

Taylor Series expansion for an implicitely defined family of functions

Can we find a Taylor Series expansion for $y(x)$ implicitly defined by: $$\sum _{i=1}^nA_ie^{a_ix+b_iy} = 1 ?$$ In financial mathematics, the two-additive-factors Model G2++ is commonly used for ...
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2answers
187 views

aproximate sum involving binomial coefficients

I have the problem for computing the j-derivative of a logarithm, with $j\gg1$ \begin{equation} c_j=\left.\frac{\partial^j}{\partial s^j}\log\left(1+Ae^s+Be^{2s}\right)\right|_{s=0}, \end{equation} ...
4
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2answers
284 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 $$\...
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1answer
154 views

Approximate $\mathbf{G}=(a\mathbf{H}+\mathbf{M})^+$ by Taylor expansion [closed]

Suppose we have a complex matrix $\mathbf{M}$. Let $\mathbf{M}^+=(\mathbf{M}^*\mathbf{M})^{-1}\mathbf{M}^*$ be the pseudo-inverse of $\mathbf{M}$, where $^*$ denotes the conjugate transpose. Let $\...
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2answers
126 views

Zeros of partial sums of $(1+z)/(1-z)$ near $z=-1$

Let $p_n(z)$ be the $n^\text{th}$ partial sum of the Maclaurin series for $f(z) = (1+z)/(1-z)$. For large $n$ the zeros of $p_n$ appear to avoid the point $z=-1$: Figure: Zeros of $p_{40}$ and the ...
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163 views

Asymptotics to Taylor expansions?

I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments. Maybe you guys can help. https://math.stackexchange.com/questions/1440931/...
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205 views

Generating a series representation for the inverse of the operator $f(f)$

I am considering the following problem: Suppose you are given a function $u: C \rightarrow C$, find a function $g$ such that $g(g) = u$ (Let's assume that such a function exists). And by "find", I ...
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1answer
404 views

A function with positive decreasing Taylor coefficients?

The function $\frac{x}{\ln(1-x)}$ has a Taylor series $-1+c_1x+c_2x^2+\cdots$ and I want to show $c_1>c_2>\cdots>0$. More generally, is there a result about how a function has positive Taylor ...
3
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0answers
118 views

why can't taylor series capture memory effects? [closed]

I am trying to understand when to use Volterra series.I found this on wikipedia 'The Volterra series is a model for non-linear behavior similar to the Taylor series. It differs from the Taylor series ...
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1answer
215 views

Discrete Taylor's Formula in n dimensions [closed]

I am searching for discrete form of Taylor's formula in n dimensions. Please share the appropriate resources.
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107 views

Properties and name of some polynomials

I have encountered in a problem some polynomials given by $P_k(x) = \prod_{j=0}^{k-2} (kx-j)$. I need to understand if these polynomials are known, and if they have certain special properties, as ...
5
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2answers
373 views

An interesting calculation of derivative

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: $G(s) = e^{a(s-1)^2}=\sum s^np(n)$ I need first to do Maclaurin expansion of the exponential and ...
4
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0answers
141 views

Szegő curve for partial sum of Taylor series of Riemann $\Xi(z)$ function

I am sorry that this is long post. But it might be of interest to you. This post is related to zeros of partial sum of Taylor series of $e^x-1$. Entire functions $e^z$, $\cos(z)$, and $\sin(z)$ can ...
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116 views

Positive, Uni-modal, Log-concave Combinatorics

We define a sequence, $\{a_n\}_{n=0}^\infty$, to be a uni-modal sequence if for some $m$, $$a_0<a_1<\cdots<a_m,\ \ \ \ a_m>a_{m+1}>a_{m+2}>\cdots.$$ We define a sequence, $\{a_n\}_{...
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3answers
371 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 ...
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1answer
76 views

An algebraic equation question [closed]

My question is this: If $\frac{\sqrt[n]{\prod_{i=1}^n(p_i + 1)}}{\sqrt[n]{\prod_{i=1}^n(m_i + 1)}} = e ^\beta$ can I find an expression (either exact or approximate) for $\frac{\sqrt[n]{\prod_{i=1}^...
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1answer
173 views

Approximate the square root of (1-X) efficiently through (nested) products

Currently, I encountered a problem of approximating the following series: $$ (I-X)^{-\frac{1}{2}}=I+\frac{1}{2}X+\frac{1\cdot3}{2\cdot4}X^{2}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6}X^{3}+\ldots $$ where ...
3
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0answers
217 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 ...
5
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3answers
601 views

Taylor series coefficients

This question arose in connection with A hard integral identity on MATH.SE. Let $$f(x)=\arctan{\left (\frac{S(x)}{\pi+S(x)}\right)}$$ with $S(x)=\operatorname{arctanh} x -\arctan x$, and let $$f(x)=\...
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1answer
149 views

A nonlinear initial-boundary value problems with Taylor expansion of parameter [closed]

Let $u(x,t; \epsilon)$ satisfy the nonlinear initial boundary value problem $$ u_{tt} = (u_{x} + u_{x}^3)_{x} + u_{xxt}, \space 0 \lt x \lt 1 $$ $$ u(0,t) = 0 \\ u(1,t) = 0 \\ u(x,0) = \epsilon f(x) \...
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1answer
265 views

Laurent expansion of inverse of vandermonde determinant

I wish to find the coefficients of the Laurent expansion of the inverse of the Vandermonde determinant, that is, the Laurent expansion at 0 of $$\prod_{1\leq i<j \leq n}(x_j-x_i)^{-1}.$$ We can ...
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votes
1answer
795 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 ...
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1answer
133 views

Estimate the scale of the power series with Poisson pdf/pmf-like terms

I would like to have an estimate for the series $$P(t) = \sum\limits_{k = 0}^\infty (e^{-t}\frac{t^k}{k!})^m,$$ where $e$ is the base of natural logarithm, $k!$ is the factorial of the integer $k$, $t$...
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1answer
894 views

References on Taylor series expansion of Riemann xi function

I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$. $$ \xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$ where $$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/...
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0answers
141 views

Series expansion with remaining $log n$

Hi, I'm studying the asymptotic behavior $(n \rightarrow \infty)$ of the following formula, where $k$ is a given constant. $$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$ I'm trying to do a ...
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1answer
224 views

Inequality of Partial Taylor Series

Hi, For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds: $$\sum_{k=0}^{N} \frac{x^k}...
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votes
1answer
243 views

A question about approximation of Real analytic functions

Define $B$ to be the set of functions $f:[0,1]\rightarrow \mathbb{R}$ for which there exists a dense set $C\subset [0,1]$ of computables numbers and an algorithm $F$ such that for any $x_0\in C,$ in ...
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0answers
653 views

Calculating entropy of adjacency matrix using eigenvalue decomposition?

How to calculate entropy using the eigenvalues when the eigenvalues are negative? Is there a simple relation between the entropy of a matrix and its characteristic polynomial?
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2answers
220 views

Taylor expansion convergence relation to power-spectrum

Is there some connection between the power-spectrum of a real function $f:\mathbb{R}\to\mathbb{R}$ (that is, its Fourier transform) and the convergence radius of its Taylor expansion around arbitraty $...
3
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1answer
413 views

nonlinear delay differential equation

Consider the delay differential equation: $ y_x(x) = \sqrt{y(x-\bar{x})} $ where $y$ is the unknown function of $x$, and where $\bar{x}$ is a fixed parameter. This equation does not seem to have a ...
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2answers
4k views

Numerical Computation of arcsin and arctan for real numbers [closed]

I'm coding some numerical methods and I do not know what the correct analysis would be for choosing the implementation for $arcsin$ and $arctan$ for real numbers. Here's what I know: Both functions ...
3
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1answer
496 views

Converse of the taylor's theorem in Banach Spaces

I would like to known if the following converse of the taylor's theorem is true: Let $E$, $F$ Banach spaces, and $f:E\rightarrow F$ continuous. Suppose there are $k$ continuous functions $T_i: E \...
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1answer
425 views

Approximation:- Algorithmic considerations

Hello I want to approximate a function $f$ on $(a,b)$. The function is singular at the points $a$ and $b$, however I have asymptotic expansions at these points. I can also construct Taylor ...
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417 views

Applying the ideas of power series to certain convolutions - which identities transfer?

Let's suppose I'm working with some set of functions $f_k(n)$. $f_1(n)$ is essentially the root of my functions, and could be nearly anything, and then $f_k(n) = (f_1(n) * f_{k-1}(n))$ for some ...