All Questions
Tagged with ca.classical-analysis-and-odes power-series
19 questions
0
votes
1
answer
74
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Singularities at the circle of convergence: generalization of Cauchy-Hadamard theorem
Consider a series $\sum a_n z^n$ with finite radius of convergence $R$. Cauchy-Hadamard theorem gives $1/R = lim\ sup |a_n|^{1/n}$.
Q: Suppose for some reason (e.g. numerical) we know that there is ...
- 593
0
votes
0
answers
43
views
The reciprocal of the normalized tail of the Maclaurin power series expansion of the hyperbolic sinc function is a convex function
The classical Bernoulli numbers $B_j$ are generated by
\begin{equation}\label{Bernoulli-No-Generating}
\frac{x}{\operatorname{e}^x-1}=\sum_{j=0}^\infty B_j\frac{x^j}{j!}=1-\frac{x}2+\sum_{j=1}^\infty ...
- 1,091
7
votes
1
answer
488
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On a paper by Dimitrie Pompéiu and on one (in two parts) by Edmund Landau
To celebrate the new year and the future of mathematics (or the mathematics of future), I see no better way to ask a question stemming from my researches on power series.
The two papers the title ...
- 6,367
1
vote
0
answers
52
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A problem on monotonicity rule for the ratio of two Maclaurin power series
In the paper [1] below, a monotonicity rule for the ratio of two Maclaurin power series was presented as follow.
Let $a_k$ and $b_k$ for $k\in\{0\}\cup\mathbb{N}$ be real numbers and
the power series ...
- 1,091
4
votes
1
answer
155
views
Could the range of $\sum_{k\geq 1}r^{n(k)}$ for $r\in \big(\frac{1}{2}, 1\big)$ be continuous?
Let $\mathcal{F}_N$ be the set of all strictly increasing sequences of positive integers. For every two $F_1, F_2\in\mathcal{F}_N$, if we use $\delta(F_1,F_2)$ to denote the first $n$-th coordinate ...
- 307
1
vote
1
answer
447
views
Closed form series for reciprocal cubic function
consider a cubic of the form f(x)=$x^3-2x+z$
Is it possible to derive a power series of coefficients for the function $x^y/f(x)$, for some $y=0,1,2...$ that does not require the use of Faà di Bruno'...
- 367
3
votes
1
answer
109
views
Small power series "approximating" a Dirac
Does there exist a (sequence of) power series $\sum_{i\geq 0} a_{n,i} x^i$ that is $1$ at $0$ and $0$ at integers from $1$ to $n$, and such that $\sum_{i\geq 0} \vert a_{n,i}\vert n^i=O(n^p)$ for some ...
- 2,772
1
vote
2
answers
165
views
Convergence radius of double series with Pochhammer symbols
I would like to know the convergence radius of the following two double power series of $(x,y) \in \mathbb{C}^2$:
\begin{align}
\sum_{m,n=0}^\infty \frac{(d-a)_{n+m}(d+b)_{n+m}(d+a)_n(d-b)_n}{n!m!(2d-...
- 21
3
votes
2
answers
266
views
Integral expressions for Bessel-like power series
I'm interested in power series of form $$f(z)=\sum_{k=0}^\infty \frac{z^k}{(k!)^\alpha}.$$ When $\alpha=1$, this becomes $\exp(z)$. For $\alpha=2$ this is a Bessel function and for larger integer $\...
- 1,324
-3
votes
1
answer
148
views
A proposition about power series
Is this proposition established?
Suppose that $0<\nu<1$, $x\in[0,1]$ and absolutely converge power series
$$p(x)=\sum_{n=0}^\infty a_nx^n,$$
$$P(x)=\sum_{n=0}^\infty \frac{\Gamma(n+1)}{\Gamma(n+...
- 13
5
votes
1
answer
278
views
"One half of a theta-function" - is there something in the literature about it?
In an answer to my question Enumeration of lattice paths of a specific type (which was in turn caused by an older one, "Special" meanders) I encountered the series
$$
F(t,q):=\sum_{n=1}^\...
- 18.4k
1
vote
2
answers
134
views
Confluent Heun Equation
Does anyone know any source in which I could find a recurrence relation for the coefficients of the series solution of the Confluent Heun Equation
$$y''+\left( {\gamma\over z}+{\delta\over z-1}+\...
- 207
1
vote
0
answers
98
views
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 ...
- 893
4
votes
1
answer
118
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 ...
- 66.8k
7
votes
2
answers
437
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 ...
- 171
1
vote
1
answer
204
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 ...
- 11
3
votes
2
answers
462
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!
- 61
4
votes
3
answers
621
views
Power series with double zeros
How many power series of the form
$1+\sum_{k=1}^{\infty} a_{k}x^{k}$ with $a_{k}\in \{-1,0,1 \}$, that have a double zero $f(x)=f'(x)=0$ in $(0,1)$, are there. Ok, there are many ways to understand ...
- 1,648
57
votes
8
answers
10k
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Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
I have spent some time using gp-pari. There is, of course, a formal power series solution to
$ f(f(x)) = \sin x.$ It is displayed below, identified by the symbol $g$ because I am not entirely sure ...
- 25.7k