Questions tagged [special-functions]
Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.
831
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Two-variable generating functions for Laguerre polynomials
Where can I find generating functions for orthogonal polynomials in two variables?
Lebedev's book (Special Functions and their Applications, Dover, 1972) gives a closed form for
$$
\sum_{n=0}^\infty \...
- 151
9
votes
2
answers
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How to do integrals involving two Bessel functions and another function?
I often encounter the integrals in the following form:
$\int_0^\infty{\rm Bessel}(ax)\cdot{\rm Bessel}(bx)\cdot f(cx)dx$,
where Bessel can be $J$, $N$, $H^{(1)}$, $H^{(2)}$, $I$, or $K$; and $f(x)$ ...
- 385
3
votes
0
answers
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Bessel functions in wave propagation and scattering
Is there a way to scale $J_n(\cdot)$ (Bessel of first kind) and $H_n(\cdot)$ (Bessel of third kind or Hankel)? I am having computer problems with higher orders (higher values of n) and small arguments....
- 41
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votes
3
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Asymptotic bounds for a confluent hypergeometric function $F_{1}[;1;x]$
I know that for infinite series and $|z|<1$ there exists a confluent hypergeometric expression
$
\sum_{k=0}^{\infty} \frac{z^k}{k!k!} = F_{1}[;1;z]
$
This is not very helpful though, and I 'd ...
- 335
1
vote
3
answers
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How to isolate $f(x)$ in $f(x+a)=f(x)+a\times g(x)$?
$a \in \mathbb{R}$
$f:\mathbb{R} \rightarrow \mathbb{R}$
$g:\mathbb{R} \rightarrow \mathbb{R}$
For generic functions $f$ and $g$, how isolate $f(x)$ in the equation below?
$f(x+a)=f(x)+a\times g(...
- 175
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votes
2
answers
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Sum of trigonometric functions
Do somebody know the closed form of the following sum (m is an integer)
$$f(\beta)=\sum _{k=1}^{2m+1} \sin^{2 m+1}\left[\frac{-\beta+k \pi
}{1+2 m}\right]$$
If instead of $n=1+2m$ we put $n=2m$, ...
- 161
1
vote
0
answers
526
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An infinte series involving the Modified Bessel Function of the second kind
The following series has had me held up for the past one week:
$$
\sum_{n=0}^\infty\frac{(2m)_n m^n}{(2m+1/2)_n n!}A^{3n/2} t^n K_{2m+n-1/2}(2\sqrt{A}t)~~~~ A>0, ~t>0, ~m\geq1/2
$$
where $K_{...
- 93
1
vote
1
answer
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Legendre Polynomial Identity
I have encountered the following sum involving Legendre polynomials, which I hope to reduce to something involving a $\delta$-function:
$$
\frac{d^2}{dx^2} \sum_{\ell=0}^{\infty} \frac{2 \ell + 1}{2 \...
- 11
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2
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On a polynomial related to the Legendre function of the second kind
The Legendre function of the second kind, $Q_n(z)$, along with the usual Legendre polynomial $P_n(z)$, are the two linearly independent solutions of the Legendre differential equation.
$Q_n(z)$ can ...
3
votes
3
answers
544
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Differential equation with some constraints
I posted this to stackexchange, and after some hours got a comment that was so pessimistic about finding some neat orderly solution, that I'm posting it here too. (In case anyone cares, this is ...
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0
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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
$$
\int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} |...
- 1,619
1
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2
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High dimensional beta integral (a typo in Stein's book "singular integrals")
Hello,
When I read Stein's book of Singular Integrals, at p. 118, there is an obvious mistake:
$$
\int_{R^n} |x-y|^{-n+\alpha} |y|^{-n+\beta}=\frac{\gamma(\alpha)\gamma(\beta)}{\gamma(\alpha+\beta)},...
- 1,619
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Geometric meaning of a trigonometric identity
It follows from the law of cosines that if $a,b,c$ are the lengths of the sides of a triangle with respective opposite angles $\alpha,\beta,\gamma$, then
$$
a^2+b^2+c^2 = 2ab\cos\gamma + 2ac\cos\beta +...
- 11.9k
1
vote
3
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Pairing function monotonic respect to product of arguments [closed]
Has anyone ever created a "pairing function" (possibly non-injective)
with the property to be nondecreasing wrt to product of arguments, integers n>=2, m>=2. (We can also assume that n and m are ...
- 31
2
votes
1
answer
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looking for monotonically increasing functions with range in [0,1] [closed]
Ideally, the function f(x) would tend toward zero as x tends towards negative infinity, and f(x) would tend towards 1 as x tends towards infinity, all the while being monotonically increasing.
for ...
- 121
7
votes
2
answers
456
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Asymptotics of the $q$-harmonic series as $q\to1$
The following (very simply looking!) problem occurs in regularization
of the harmonic series
which can be formally thought of as the limit as $q\to1$, $|q|<1$, of
$$
h(q):=(1-q)\sum_{n=1}^\infty\...
- 13.4k
0
votes
0
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Non-standard addition theorem for Legendre function of the first kind
There is known link text addition theorem for the Legendre functions of the first kind $P_{\nu}^m(x)$, which establishes the connection between $P_{\nu}^m(x)P_{\nu}^{m}(y)$ and $P_{\nu}(F(x,y))$ for ...
- 103k
3
votes
0
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Mathematica package for obtaining hypergeometric function
In my current research in electromagnetics I am encountering integrals of the form $$ \int_0^\infty dt J_0( r t) \frac{\exp(-h \sqrt{t^2 - a^2})}{\sqrt{t^2 - b^2}} t . $$ $a$ and $b$ are complex ...
- 1,077
2
votes
1
answer
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antiderivative involving modified bessel function
This little integral has been holding me up for weeks. Has anyone come across a similar integral in their work.
$\int {\frac{d x}{c-I_0(x)}} $
- 21
1
vote
1
answer
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Can you control the amplitudes of a finite collection of sine curves just by controlling the amplitude of their superposition?
Dear all,
I would like to know whether the following claim is true. In particular, if it is true, then I would like to know if there is some textbook that contains the statement and maybe even the ...
- 678
4
votes
0
answers
435
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Why is Mellin-inverse of Gamma periodic?
Specific Case
The periodicity is obvious from computation:
$$\cal{M}^{-1}\{\Gamma\}(x) := \frac{1}{2\pi i}\int_{c}\Gamma(s)x^{-s}d s=e^{-x}$$
However, is there a way to see directly from the integral ...
- 1,243
11
votes
1
answer
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A 2F1 Hypergeometric identity from a Feynman integral
Using two different approaches to evaluating the dimensionally regularized ($d=4-2\epsilon$ dimensional Euclidean space), equal mass ($x=m^2$), 2-loop vacuum Feynman diagram
$$
\begin{align}
I(x) &...
- 461
1
vote
1
answer
387
views
Maximum of a series of integrals of Hermite functions
Given the function $$f(A) := \sum_{n=1}^{\infty}\left( \int_A \varphi_0\varphi_n \right)^2,$$ where $A$ is any measurable subset of $\mathbb{R}$, and $\varphi_n$ is the $n$th Hermite function, I want ...
- 692
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votes
2
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Functions defined as infinite products
Are there standard references on infinite products of rational functions and their convergence properties? I'd appreciate information on finite products too!
The original motivation for this is the (...
- 1,801
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1
answer
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Asymptotic form of $L^1$-norm of Hermite functions
Background
Working on a quantum mechanics problem, I've stumbled on the problem of maximizing the functional
$$\int_{A} \varphi_m \varphi_n$$
in the limit of large $m$ and $n$, given that $n \gg m$. ...
- 692
0
votes
2
answers
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Cosine of a Partial Sum
Does anyone know of a closed formula for $\displaystyle \cos\left(\sum_{n=0}^m a_n\right)$? I've seen formulas for $\displaystyle \cos\left(\sum_{n=0}^\infty a_n\right)$ and $ \displaystyle \tan\left(\...
- 519
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What's the difference between a Riemann theta and a Siegel theta function?
One of the things I'm working on has required me to look into the literature of multidimensional theta functions, and I've gotten a bit confused on a few naming details.
A look at the DLMF says that "...
7
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Geometric meaning of trigonometric relations
According to a paper by Zhiqin Lu in the Mathematical Gazette (the British publication, not the Boston-area newsletter, if that still exists (or even if it doesn't)) in 2007(?), if $u+v+w=\pi$ and $a,...
- 11.9k
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Integral identity for Legendre polynomials
How does one prove the following integral identity, where $P_n(x)$ is the $n$th Legendre polynomial?
$$
\int_0^1 P_n(2t^2-1) dt = \frac{(-1)^n}{2n+1}
$$
Notes & Background
A variant of this ...
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1
answer
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Legendre Function "Types"
Legendre functions can be of first and second kinds; $P$, $Q$.
They can have order $\mu$ and degree $\nu$; $P^\mu_\nu$ $Q^\mu_\nu$
But do they also have Types? Some of the numerical software I am ...
- 31
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votes
4
answers
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Determining the asymptotic behavior of a series
I am trying to determine the behavior of the following series as $n\to\infty$. Let $0<\mu<1$ be fixed and for every positive integer $n\geq 1$, consider the function $f_n(t)$ of a real variable $...
7
votes
4
answers
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Estimating the probability that one Poisson RV is larger than another
Let $X$ and $Y$ be Poisson random variables with means $\lambda$ and $1$, respectively. The difference of $X$ and $Y$ is a Skellam random variable, with probability density function
$$\mathbb P(X - Y ...
- 8,392
5
votes
3
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Integral over error function and normal distribution
Help me understand why
$\int_{-\infty}^{\infty}\frac{1}{2}[1+\operatorname{erf}(\frac{\theta-x}{\sqrt{2q^2}})]\frac{1}{\sqrt{2\pi\sigma^2}}{\exp(-\frac{(x-\mu)^2}{2\sigma^2})}dx \approx \frac{1}{2}[...
- 153
3
votes
2
answers
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Product of hypergeometric functions/Jacobi Polynomials
Are there any theorems related to the product of Jacobi/Legendre Polynomials and/or Hypergeometric functions? Specifically, I'm interested in the product of ${}_{2}F_{1}[-n,-n+1;2;x]$ and ${}_{2}F_{1}[...
- 335
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votes
6
answers
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Why are hypergeometric series important and do they have a geometric or heuristic motivation?
Apart from telling that the hypergeometric functions (or series) are the solutions to the (essentially unique?) fuchsian equation on the Riemann sphere with 3 "regular singular points", the wikipedia ...
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Groups, quantum groups and (fill in the blank)
In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...
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votes
3
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About Turan`s problem(inequality) in multivariable
Hi. I have a question related to Turan`s problem, that is
Find a sequence of polynomial $P_n(x)$ satisfying $P_{n+1}(x)P_{n-1}(x) < P_{n}^2(x)$.
I am considering the generalized question for ...
- 21
8
votes
1
answer
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Positivity of a rational function
A rational function is called positive if all its Taylor coefficients are positive.
Friedrichs-Lewy conjecture states the positivity of the rational function
\begin{eqnarray*}\frac{1}{
(1-x)(1- y)+(...
- 1,858
2
votes
1
answer
421
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monotonicity of functions related to modified Bessel function
Dear colleagues,
I recently met some problems related to the modified Bessel funtions of the first kind and the second kind. I want to know if there exist some results on the monotonicity of
$\frac{...
- 35
2
votes
1
answer
345
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Estimating associated Legendre function
For past months I've been trying to estimate associated Legendre function $P_{-\frac{1}{2}+it}^m (\cosh r)$ in order to study Laplacian eigenfunctions on a hyperbolic surface. I found reasonably sharp ...
0
votes
1
answer
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Results on derivatives with respect to the parameter of Modified Bessel Function [duplicate]
Possible Duplicate:
Derivate Bessel Function with respect to order
Dear colleagues,
I have a question about the modified Bessel function of the second kind, $I_\nu(x)$ and $K_\nu(x)$. I want to ...
- 35
5
votes
2
answers
652
views
generalization of (Rogers) dilogarithm
Let $C$ and $S$ be abbreviations for $\cosh$ and $\sinh$, and consider the following
function:
$$f(x,y) = \int_{-y\le r+l \le y} \frac{ (C(x)S(l)C(r) - C(l)S(r))(C(x)C(l)S(r)-S(l)C(r)) }
{(C(x)C(l)C(...
- 2,562
4
votes
2
answers
506
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Analytic continuation of $_4F_3(1)$
The Gauss theorem
$${_2F_1}(a,b;c;1)=\frac{\Gamma(c-a)\Gamma(c-b)}{\Gamma(c)\Gamma(c-a-b)}$$
allows to compute the analytic continuation of ${_2F_1}(a,b;c;1)$ for $a+b>c$ when the series ...
- 41
9
votes
2
answers
3k
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Sums of arctangents
$$
\begin{align}
\arctan(x) = {} & \arctan(1) + \arctan\left(\frac{x-1} 2 \right) \\
& {} - \arctan\left(\frac{(x-1)^2} 4 \right) + \arctan\left(\frac{(x-1)^3} 8 \right) - \cdots
\end{align}
$$...
- 11.9k
4
votes
1
answer
449
views
Summations in $\tan^2$
Hey all,
I was just wondering if anyone had come across the following identities, valid for $m\in\mathbb{N}$. I've used Abramowitz and Stegun, Maple, Mathematica etc but can't find them anywhere. I ...
- 211
2
votes
1
answer
2k
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invert complete elliptic integral of first kind K(k)
I am not a mathematician although I use it in hydrodynamics research. I have a question regarding elliptic integrals for my research in wave theory
Given the value of the complete elliptic integral of ...
- 21
5
votes
2
answers
578
views
A problem on sums of arctangents of rationals
Let $S$ be a set of rational numbers.
For "special" sets $S$, we can ask if $\pi$ can be written as a $\mathbb{Q}$-linear or $\mathbb{Z}$-linear combination of elements from '$\{\tan^{-1}(x):...
- 341
8
votes
3
answers
2k
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Conformal Mappings for hyperbolic polygon
I am searching for a conformal mapping from the upper halfplane onto a hyperbolic polygon, i.e. the sides of the polygon have to be geodesics.
The classical Schwarz Christoffel theorem does the job ...
9
votes
1
answer
1k
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Is this sequence of polynomials well-known?
While working on a problem in p-adic Hodge theory, and needing to write down a solution to a certain equation involving p-adic power series, I stumbled across a certain sequence of polynomials. Define ...
- 36.2k
8
votes
0
answers
864
views
On Stark's conjecture for imaginary quadratic fields
In the famous paper "L-Functions at s = 1. IV. First Derivatives at s = 0" of Stark from 1980, it is shown that in the case of an imaginary quadratic field $K$ certain numbers of the form $$exp(-\frac{...
- 2,009