All Questions
Tagged with ca.classical-analysis-and-odes special-functions
22 questions
4
votes
1
answer
387
views
$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$
The Fejer-Jackson inequality as follows:
$$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$
I conjecture that the inequality as follows holds:
$$\sum_{...
- 1,776
42
votes
7
answers
5k
views
How should an analytic number theorist look at Bessel functions?
(And a related question: Where should an analytic number theorist learn about Bessel functions?)
Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...
- 7,347
11
votes
1
answer
1k
views
Has anyone seen this series?
I come across the following infinite series.
$$
\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
...
- 1,649
40
votes
1
answer
2k
views
Curious $q$-analogues
Consider the Fibonacci polynomials
$$F_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \right\rfloor }\binom{n-j}{j} x^{n - 2j} $$
and the Lucas polynomials
$$L_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \...
- 5,622
21
votes
1
answer
2k
views
Trigonometry related to Rogers–Ramanujan identities
For integers $n\ge2$ and $k\ge2$, fix the notation
$$
[m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad
[m]!=[1][2]\dots[m], \qquad m\in\mathbb Z_{>0}.
$$
Consider the following trigonometric numbers:...
- 13.5k
12
votes
1
answer
352
views
A problem involving the Error Function
I am looking at the following function on the domain $x\geq 0$:
$$F(x)=(x+a)e^{x^2}(1-\mathrm{erf}(x))-\frac{b}{\sqrt\pi},$$
where $a>0$, $0<b<1$ are parameters. From plotting this function ...
- 389
8
votes
1
answer
607
views
Rotation invariance of an integral
Consider the integral depending on 2 parameters
$$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$
where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...
- 21.8k
6
votes
2
answers
1k
views
A (likely) positivity property of the Lerch zeta-function
The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where
$$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$
is the Lerch zeta-...
- 128k
6
votes
2
answers
1k
views
Generalized trigonometric functions $Cos(n) v$ and $Sin(n) v$.
I just discovered a paper from 1948, Eine Verallgemeinerung der Kreis-und Hyperbelfunktionen by R. Grammel which introduces functions he calls Cos(n) and Sin(n), representing a parameterization of the ...
5
votes
1
answer
1k
views
Asymptotic form of $L^1$-norm of Hermite functions
Background
Working on a quantum mechanics problem, I've stumbled on the problem of maximizing the functional
$$\int_{A} \varphi_m \varphi_n$$
in the limit of large $m$ and $n$, given that $n \gg m$. ...
- 702
4
votes
1
answer
211
views
Perform an integration over the unit interval of a two-parameter expression involving a Gauss hypergeometric function
In a quantum-information-theoretic context, I've encountered the problem
of integrating over $r \in [0,1]$, the function
\begin{equation}
r^{2 d-1} \, _2F_1\left(-\frac{d}{2},\frac{d}{2};\frac{d+2}{2}...
- 723
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
394
views
Closed form for the integral of a squared Legendre function
Is there a closed form for the integral $$\int_0^{\pi/2}(P_\nu^\mu(\cos\theta))^2\,\mathrm d\theta,\quad\mu>\nu\gt-\frac12$$ where $P_\nu^\mu(x)$ is the associated Legendre function of the first ...
3
votes
1
answer
386
views
A Bessel function integral identity involving $\int_0^\pi \frac{K_{j-1/2}(w)}{w^{j-1/2}}\sin^{2p-1}(\theta)\, d\theta$
Suppose that $w=\sqrt{R^2 + s^2 -2Rs\cos\theta}$ with $R\ge s>0$, that $p$ is a positive integer and that $j$ is an integer with $0\le j\le p$. Let $I$ and $K$ denote the modified Bessel functions ...
- 1,271
2
votes
1
answer
205
views
An extreme of Jacobi elliptic function on an interval
Consider the Jacobi elliptic function $sn(\cdot,k)$ restricted to the interval $(0,2K)$, where $K=K(k)$ is complete elliptic integral of the first kind. If $0<k<1$, then it is well known the ...
- 2,188
2
votes
1
answer
341
views
What functions do we need to solve linear second order differential equations with polynomial coeficients? [closed]
.
Final edit: The problem I had in mind is properly asked in THIS MO QUESTION, so I'll vote to close the present post e recommend anyone interested in the topic to visit that link.
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Below is ...
- 480
2
votes
1
answer
662
views
Clausen’s identity for associated Legendre polynomials
Clausen’s identity for Legendre polynomials has the form (see, for example,
A generating function of the squares of Legendre polynomials, by Wadim Zudilin: https://arxiv.org/abs/1210.2493)
$$P_n(\cos{\...
- 16.5k
1
vote
1
answer
507
views
Upper bound of the fraction of Gamma functions
Is there a simple upper bound of the following fraction of Gamma functions for any $a,b\geq1/2$:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}$$
An upper bound in the following form is ...
- 1,495
1
vote
0
answers
127
views
About writing solutions of linear ODE's: Is this statement correct?
A motivating example: Consider the Hypergeometric equation
$$z(1-z) \frac{d^2y}{dz^2}+(c-(a+b+1)z) \frac{dy}{dz}-aby=0,$$
it has a solution given by the Gauss's Hypergeometric function
$$_2F_1(a,b;c;z)...
- 480
0
votes
1
answer
125
views
A slight generalization of triconfluent Heun equation: what is known?
I have recently come across an ODE of the form
$$y''+(a+b x^2)y'+(c+dx+h/x^2)y=0 \hspace{30mm} (*)$$
where $y=y(x)$ and $a,b,c,d,h$ are arbitrary constants.
As far as I understand (please correct ...
- 312
0
votes
0
answers
298
views
High dimensional beta integral (question following the previous post)
Hello,
This post is a question following the previous post. In one dimensional case, we have
$$
\int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} |...
- 1,649
0
votes
1
answer
375
views
Bringing a Heun equation into canonical form
It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...
- 213