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.

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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 ...
minhtoan's user avatar
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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 ...
Max Muller's user avatar
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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-...
FractalScout's user avatar
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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, $$...
Tito Piezas III's user avatar
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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 ...
Tito Piezas III's user avatar
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1 answer
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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 $x$ ($\nabla u$ ...
Matchmaticians's user avatar
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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 ...
Anixx's user avatar
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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 ...
Luis Mendo's user avatar
6 votes
4 answers
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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 ...
Cosmonut's user avatar
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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}...
Zhi-Wei Sun's user avatar
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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 $\psi_k(.)$ ...
Z. Alfata's user avatar
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2 answers
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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 ...
Jean-Philippe Burelle's user avatar
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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 ...
Ramanasa's user avatar
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2 answers
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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 ...
Michael Hardy's user avatar
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)\...
Julian's user avatar
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1 answer
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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 ...
Joe's user avatar
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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-...
Iosif Pinelis's user avatar
6 votes
3 answers
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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 ...
sigma_z_1980's user avatar
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')\...
Math_Y's user avatar
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1 answer
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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 ...
vito-ვიტო's user avatar
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 ...
djbinder's user avatar
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1 answer
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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-...
Twi's user avatar
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6 votes
1 answer
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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}}\...
Claude Leibovici's user avatar
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 \...
MarkCurrant's user avatar
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. ...
forwhole's user avatar
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}\...
Tragen's user avatar
  • 65
6 votes
1 answer
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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 ...
Ismail Bello's user avatar
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}{...
J. M. isn't a mathematician's user avatar
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 ...
maliesen's user avatar
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2 answers
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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(...
Zurab Silagadze's user avatar
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 ...
Bullmoose's user avatar
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6 votes
1 answer
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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 ...
Aleksey Pichugin's user avatar
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\: \...
Remy's user avatar
  • 447
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 ...
Abhishek Halder's user avatar
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[...
Zurab Silagadze's user avatar
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) \...
genideal's user avatar
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-...
Tito Piezas III's user avatar
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 (...
Zhi-Wei Sun's user avatar
  • 14.4k
6 votes
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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 ...
Matteo Beccaria's user avatar
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 ...
Matematleta's user avatar
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 ...
Timothy J. Doyle's user avatar
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 ...
Bjørn Kjos-Hanssen's user avatar
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(...
Alexey Ustinov's user avatar
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 (...
Alexander Moll's user avatar
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) =...
Campello's user avatar
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 ...
Ofer's user avatar
  • 151
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 ...
Moritz Firsching's user avatar
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....
jtkw's user avatar
  • 63
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 ...
Stanley Yao Xiao's user avatar

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