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.
829
questions
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Analytic continuation of Dixon's identity
Many well-known combinatorial identities has an analytic version. For example, the following identities
$$
2^n = \sum_{k=0}^n \binom{n}{k}
$$
$$
\binom{2n}{n} = \sum_{k=1}^n \binom{n}{k}^2
$$
can be ...
7
votes
0
answers
436
views
Is there any literature on $\sum_{i=1}^{k} \left[ {k \atop i} \right] \zeta(i+1) $?
As per these questions, I'm trying to evaluate $$\sum_{n=2}^{\infty} \big{(} \zeta(n)^{2}-1 \big{)} = 1+ \sum_{m=2}^{\infty} \frac{H_{-\frac{1}{m}}}{m}. $$
Here, $H_{x}$ is a generalized Harmonic ...
7
votes
0
answers
263
views
Degree of automorphic forms, SL(3,Z), and the elliptic Gamma function
In this article, the authors interpret a certain special function, the elliptic Gamma function, defined as
$$
\Gamma(z,\tau,\sigma)=\prod_{j,k=0}^\infty\frac{1-e^{2\pi i((j+1)\tau+(k+1)\sigma-z)}}{1-...
7
votes
0
answers
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views
Why does a polynomial with discriminant $d=-163$ appear in this sequence?
Fact 1: Given the eta quotient,
$$x_d= e^{2\pi\rm{i}/48}\,\frac{\eta(\tau)}{\eta(2\tau)}$$
where $\tau=\frac{1+\sqrt{-d}}2$, then $x_d$ for $d=11,19,43,67,163$ are the roots of the simple cubics,
$$...
7
votes
0
answers
288
views
An integral for the tribonacci constant and the general case
When I asked for integrals involving the tribonacci constant $T$, user @nospoon gave the nice answer,
$$\int_0^{\infty} \eta( i \, x)\,\eta(i \,11 x) \,dx = \frac{ \ln T}{\sqrt{11}} $$
However, the ...
7
votes
1
answer
338
views
Level sets of weakly differentiable funtions
Let $C$ be a $C^1$ hypersurface in $R^n$ and let $u \in C^1(R^n)$. Suppose
$$\nabla u(x) \cdot \eta(x)=|\nabla u| \ \ \forall x\in C$$
where $\eta(x)$ is the normal vector to $C$ at $x$ ($\nabla u$ ...
7
votes
0
answers
201
views
Are these identities Newton series?
Newton series is the following expansion of a function:
$$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$
Now ...
7
votes
0
answers
716
views
Inequality between incomplete beta and gamma functions; or when is binomial distribution function above/below its limiting Poisson
Please note: this question was posted first (September 4) in math.stackeschange.com and then (September 16) in stats.stackeschange.com. It got no answers in neither of those sites.
Let the ...
6
votes
4
answers
955
views
An identity for the cosine function
Let $x = \pi/(2k+1)$, for $k>0$.
Prove that
$$
\cos(x)\cos(2x)\cos(3x)\dots\cos(kx) = \frac{1}{2^k}
$$
I've confirmed this numerically for $n$ from $1$ to $30$.
I'm finding it surprisingly ...
6
votes
1
answer
503
views
A surprising identity: $\det[\cos\pi\frac{jk}n]_{1\le j,k\le n}=(-1)^{\lfloor\frac{n+1}2\rfloor}(n/2)^{(n-1)/2}$
On the basis of my computation, here I pose my following conjecture involving the cosine function.
Conjecture. For any positive integer $n$, we have the identity
$$\frac1{2n}\det\left[\cos\pi\frac{jk}...
6
votes
2
answers
498
views
Any reference for the series expansion of $\Bigr[-\log(1-t)\Bigr]^x$?
Any reference that we can find the following $$\Bigr[-\log(1-t)\Bigr]^x = t^x + x t^x \sum_{k=0}^\infty \psi_k(x+k)\,t^{k+1}; \quad \mbox{for all} \, x\in \mathbb R, \, |t|<1$$
where $\psi_k(.)$ ...
6
votes
2
answers
1k
views
Generalized trigonometric functions $Cos(n) v$ and $Sin(n) v$.
I just discovered a paper from 1948, Eine Verallgemeinerung der Kreis-und Hyperbelfunktionen by R. Grammel which introduces functions he calls Cos(n) and Sin(n), representing a parameterization of the ...
6
votes
2
answers
433
views
Prove $\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1$
The question is to prove:
$$
\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1.
$$
Numerically it seems to hold true. So I have made some attempts to ...
6
votes
2
answers
721
views
Strange pattern in rounding errors?
This will look at first like a posting about trigonometry, then maybe about statistics, then finally about peculiarities of either
a certain random process; or
the pseudorandom number generator that ...
6
votes
2
answers
259
views
Asymptotics of error function integral with square root
I am interested in the asymptotics of the integral
$$I(a):=\int_0^\infty \sqrt{x}\operatorname{Erfc}(x+a)\,\mathrm{d}x$$
for $a>0$. I think that $I(a)$ should be decaying exponentially as $I(a)\...
6
votes
1
answer
422
views
Diagonalizing the ‘restricted’ Hilbert transform on $L^2(0,1)$, $f(z_1) \mapsto \mathrm{p.v.} \int_0^1 \frac{i}{z_1-z_2}f(z_2) dz_2$
Consider the following operator on functions $\mathcal{T}: L^2(0,1) \to L^2(0,1)$ over the complex numbers.
\begin{equation}
(\mathcal{T} f)(z_1) = \mathrm{p.v.} \int_0^1 \frac{i}{z_1-z_2}f(z_2) dz_2
...
6
votes
2
answers
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A (likely) positivity property of the Lerch zeta-function
The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where
$$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$
is the Lerch zeta-...
6
votes
3
answers
983
views
Asymptotic bounds for a confluent hypergeometric function $F_{1}[;1;x]$
I know that for infinite series and $|z|<1$ there exists a confluent hypergeometric expression
$
\sum_{k=0}^{\infty} \frac{z^k}{k!k!} = F_{1}[;1;z]
$
This is not very helpful though, and I 'd ...
6
votes
1
answer
367
views
When are the chirp signals orthogonal?
Assume that we have two bounded-time chirp signals,
\begin{align}
x(t)&=\exp\Big(j\pi(\alpha t^2+\beta t+\gamma)\Big),\quad 0\leq t\leq T,\\
y(t)&=\exp\Big(j\pi(\alpha' t^2+\beta' t+\gamma')\...
6
votes
1
answer
379
views
Calculate Ramanujan's class invariant by using modular equation of degree $5$
Let $$K(k):=\int_{0}^{\frac{\pi}{2}}\frac{d\phi}{\sqrt{1-k^2\sin^2\phi}}=\frac{\pi}{2}{ _2F_1\bigg(\frac{1}{2},\frac{1}{2},1;k^2 \bigg)}$$
where $0<k<1$.
Let $K, K′, L$ and $L′$ denote the ...
6
votes
1
answer
528
views
Asymptotic Expansion of Bessel Function Integral
I have an integral:
$$I(y) = \int_0^\infty \frac{xJ_1(yx)^2}{\sinh(x)^2}\ dx $$
and would like to asymptotically expand it as a series in $1/y$. Does anyone know how to do this? By numerically ...
6
votes
1
answer
341
views
Asymptotic behaviour of an integral
For $k\in\mathbb{N}_{0}$ and $x\in\mathbb{R}$, define
$$I_{k}(x):=\int_{0}^{\pi/2}\cos(xg(\theta))\sin^{2k}\theta\,\mathrm{d}\theta$$
where
$$g(\theta)=\int_{\sin\theta}^{1}\frac{\mathrm{d}t}{\sqrt{(1-...
6
votes
1
answer
157
views
About the high-order derivatives of Lambert function
In the mid seventies, in my former research group, we found that the $n^{\text{th}}$ derivative of $W_0(x)$ could write
$$\frac {d^n\,W_0(x)}{dx^n}=(-1)^{n+1}\,\,\frac{\,P_n(w)}{ e^{nw}\,(1+w)^{2n-1}}\...
6
votes
1
answer
401
views
Interesting behaviour of binomial coefficients
Let $\binom{n}{k}:=\frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(1-k+n)}$ be the generalized binomial coefficient then I noticed by playing around with Mathematica that the function $f:[0,n/2] \rightarrow \...
6
votes
2
answers
402
views
Cardinality of a special set of continuous functions
Let $C$ be a set of continuous functions with a domain $[0,1]$ and for every input $x$ in a domain there is a set $S(x)$ that contains all values that functions in $C$ will output given that input.
...
6
votes
1
answer
388
views
Spherical Bessel functions. Sum of squares
In (1) there is a property of spherical Bessel functions, which's derivation I can not find in the literature.
${\mathsf{j}_{n}^{2}}\left(z\right)+{\mathsf{y}_{n}^{2}}\left(z\right)=\sum_{k=%
0}^{n}\...
6
votes
1
answer
195
views
Modern comprehensive account of the Barnes G-Function
I previously put this forward on the math.stackexchange community with little luck:
I am looking for a comprehensive account of the properties and applications of the Barnes G-Function. Everything ...
6
votes
2
answers
2k
views
Simultaneously computing a complete elliptic integral and its complement
The complete elliptic integral of the first kind
$K(m)=\int_0^{\pi/2}\frac{\mathrm{d}t}{\sqrt{1-m\sin^2t}}$
is easily computed via the arithmetic-geometric mean iteration; to wit,
$K(m)=\frac{\pi}{...
6
votes
1
answer
344
views
Double Series involving Gamma function
Does anyone have any ideas on howto verify $$\sum_{n,m=0}^\infty \frac{\Gamma(n+m+3x)}{\Gamma(n+1+x)\Gamma(m+1+x)}\cdot \frac{1}{3^{n+m+3x-1}} = \Gamma(x)$$ for $x>0$?
I posted this question also ...
6
votes
2
answers
299
views
Choice of branch cuts in logarithmic integral
According to 8.111 from Lewin's book "Polylogarithms and associated functions", it is expected that
$$\int\limits_0^2\frac{\ln{(1-x)}\ln{(1+x)}}{x}\,dx=Li_3(-3)+\zeta(3)-2Li_3(3)+$$ $$\ln{3}\left[Li_2(...
6
votes
3
answers
2k
views
Sharp upper bounds on hypergeometric function ${}_2F_1[a,b,c;z]$ when $|z|\geq1$
Generally, hypergeometric function ${}_2F_1[a,b,c;z]$ is defined using Gauss series ${}_2F_1[a,b,c;z]=\sum_{n=0}^\infty\frac{(a)_n(b)_n}{(c)_nn!}z^n$ with $|z|<1$, and there seems to be a lot of ...
6
votes
1
answer
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views
Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions?
I am trying to somewhat simplify a series, whose coefficients feature generalised hypergeometric functions ${}_1F_2(1;a,a+1;z)$. I was unable to find useful functional relations for this specific ...
6
votes
1
answer
991
views
Contour Integral with Gamma functions and 2F1
Given the following contour integral
$$\frac{1}{2\pi j}\int^{c+j\infty}_{c-j\infty} \frac{\Gamma(-1+a+s)\Gamma(b+s)}{\Gamma(3+a-s)}\cos(-1+a+s)\,
{}_2F_1\Big(-1-a+s,-1+a+s;\frac{1}{2};z\Big) y^s\: \...
6
votes
1
answer
419
views
Definite integral involving inverse regularized incomplete beta functions
In my research I encountered the following integral
$$J = \int_0^1 I_t^{-1}(a_1,b_1) \: I_t^{-1}(a_2,b_2) \: \mathrm{d}t$$
which I would like to evaluate as a closed form expression, that is, as a ...
6
votes
1
answer
697
views
Аrе thеsе integrals known?
While studying some dark matter related stuff, I came across to the following interesting identities:
$$\int\limits_0^\infty\sqrt{\frac{y}{xp}}\,e^{-y}\left(K(p)-E(p)\right)dy=
\frac{\pi x}{4} \left[...
6
votes
1
answer
239
views
An identity of complicated combinations of gamma functions (related to hypergeometric functions)
Can somebody help me in proving the following equation?
\begin{align*}&\textstyle \sum _{d=0} ^{n} \frac{1}{d!(n-d)!} \frac{\Gamma (b+d) \Gamma (b+n-d) \Gamma (c-n+d) \Gamma (c-b+1-n + 2d) \...
6
votes
1
answer
250
views
On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence
In a previous MO post, H. Cohen suggested Gorodetsky's 2021 paper which discussed $6+6+3=15$ "sporadic sequences". The first 6 are Zagier's sporadic sequences, the second 6 are by Almkvist-...
6
votes
0
answers
266
views
On the determinant $\det[\sec2\pi\frac{jk}p]_{0\le j,k\le(p-1)/2}$
On the basis of my computation, I have the following conjecture involving the secant function.
Conjecture. Let $p$ be an odd prime and define
$$S_p:=\det\left[\sec2\pi\frac{jk}p\right]_{0\le j,k\le (...
6
votes
0
answers
183
views
Sum of squared hypergeometric polynomials
$\sum_{m=1}^\infty \frac{1}{m} \bigg[{}_2F_1(-m,m,2,u)\bigg]^2 = \frac 1 4 -\frac 1 2 \log u$
I have very strong numerical support that this is true when
$0<u\le1$.
Can anyone help proving or ...
6
votes
0
answers
389
views
Recurrence Formula for Zernike polynomials
I'm not sure if this is research level, so if this result is known, please excuse the intrusion. I am trying to find a relation between solutions of the Laplacian equation in $4$ dimensions and those ...
6
votes
0
answers
458
views
Conway's box function iterated to produce a hierarchy of nested sets of real numbers
Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). When the ...
6
votes
0
answers
199
views
Elementary function relative to erf
The modified Bessel function of the 1st kind $I_0$ is defined by
$$
I_0(z)=\frac1\pi\int_0^{2\pi}e^{z\cos\theta}\,d\theta
$$
and arises, among other places, in the probability density function of a ...
6
votes
0
answers
288
views
Legendre polynomials and formal groups
Let $P_n(x)$ be Legendre polynomials:
$$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$
Usual arguments from the theory of formal groups allow to
prove that for any $n$
$$P_n(x)=Q_n(...
6
votes
0
answers
224
views
Evaluating an infinite product of q-exponentials
For the $q$-exponential $$e_q(u) = \sum_{n=0}^{\infty} \frac{u^n}{[n]_q!}$$ with $[k]_q=\frac{1-q^k}{1-q}$ and $[n]_q! = [n]_q [n-1]_q \cdots [1]_q$, we don't have the property $e_q(u) e_q(v) = e_q (...
6
votes
0
answers
947
views
An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function
I think, here, I found
$$
P_x(s)=\sum_{p < x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n}
\sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}}
\left[
{\rm li}(t^{\frac zn-s})
\right]^{x}_2
\tag{7}
$...
6
votes
0
answers
3k
views
hypergeometric function $_2F_1(-n;-r;1;2)$
The hypergeometric function $_2F_1(-n;-r;1;2)$ appears in many different situations. For instance, it counts the number of integer points within a sphere in the $l_1$ norm, i.e.,
$$_2F_1(-n;-r;1;2) =...
5
votes
3
answers
5k
views
Spherical Harmonics - a bunch of questions about them
Hi there,
Please tell me if I should divide these into individual questions next time.
Short intro:
Spherical Harmonics are a nice collection of functions. They are orthogonal and allow you to take ...
5
votes
2
answers
3k
views
Nicer expression for 2.1369288...?
In Drift Analysis and Evolutionary Algorithms Revisited by Johannes Lengler and Angelika Steger in Theorem 10, there is mention of a constant "$2.2$", and in the proof it becomes apparent ...
5
votes
1
answer
739
views
Any name for this special function?
We know
$$
\sum_{m=0}^\infty \frac{x^m}{(a-m)!m!} = \frac{1}{a!}(1+x)^m
$$
where we understand the factorial as Gamma function $\Gamma(x)$ such that it is divergent if the argument is negative integer....
5
votes
2
answers
1k
views
Evaluating elliptic integrals
I am interested in evaluating some elliptic integrals, and I have not been able to secure a reference to do exactly what I need. Most of the references I've found seem to focus on reducing more ...