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.
914 questions
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Are shifted Dirichlet L-functions $\sum_{n=1}^\infty \chi(n)(n+a)^{-s}$ studied in the literature?
I am studying generalizations of Dirichlet L-functions through the framework of LC-functions (a generalization of the Hurwitz zeta function). In this context, I have encountered naturally the ...
- 481
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A system of T-functions on $(0,\infty)$
Let $r>1$ be real and
\begin{align*}
f_1(x) &= 1,\\
f_2(x) &= (x+4)^r,\\
f_3(x) &=(x+4)^r(x+3)^r,\\
f_4(x) &= (x+4)^r(x+3)^r(x+2)^r,\\
f_5(x) &=(x+4)^r(x+3)^r(x+2)^r(x+1)^r.
\...
- 258
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37
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Characterizing exponential families with elementary normalizing transformations
Let $f_\theta(x) = \exp(\theta T(x) - K(\theta))$ be a one-parameter exponential family of probability density functions with respect to the Lebesgue measure on $\mathbb{R}$, for $\theta$ in an open ...
- 763
2
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Show the monotonicity of a function involving the difference of the digamma function
Denoting the digamma function by $\psi$, define
\begin{equation*}
g(n,s) = \psi\left(\frac{n+2}{s}\right)+\psi\left(\frac{n}{s}\right)-2\psi\left(\frac{n+1}{s}\right)+\frac{s}{n(n+1)}
\end{...
- 123
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682
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The Bring quintic and the Baby Monster?
I. Quintic
The general quintic was reduced to the Bring form $x^5+ax+b=0$ in the 1790s, while the Baby Monster was found in the 1970s.
We combine the two together using the McKay-Thompson series (...
- 13.1k
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116
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Asymptotic double integral Airy functions
I am back with some tough asymptotic expansion that I would like to share with experts.
I suspect the following identity is true (at least is some sense, maybe as a distribution):
\begin{equation}
...
- 61
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509
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Given the Ramanujan $G_n$ function, why is the quintic $x^5+5x^4+40x^3 = 4^3\left(\frac{4}{G_n^{16}}-G_n^{8}\right)^3$ solvable in radicals?
I. Definitions. Given the nome $q = e^{\pi i\tau}\,$ and $\tau=\sqrt{-n}\,$ for positive integer $n$, then the Ramanujan G and g functions are,
$$\begin{align}2^{1/4}G_n &= q^{-\frac{1}{24}}\prod_{...
- 13.1k
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Why does the general quintic factor over the Rogers-Ramanujan continued fraction $R(q)$?
I. Let $q = e^{2\pi i\tau}$ and $r=R(q)$ be the Rogers-Ramanujan continued fraction. Then the j-function $j(\tau)$ has the formula using polynomial invariants of the icosahedron,
$$j(\tau) = -\frac{(r^...
- 13.1k
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699
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With what probability does an inscribed/circumscribed triangle contain a point?
Three points uniformly selected on the unit circle form a triangle containing a point $R$ at distance $r \in [0;1]$ from its center with probability $P(r) = \frac{1}{4} - \frac{3}{2 \pi^2}\textrm{Li}...
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3
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Are there known explicit closed-form expressions for the Taylor polynomials of $1 / (1-q)^n$?
Let
$$
P_{n,d}(q) := \sum_{k=0}^d \binom{n+k-1}{k} q^k
$$
denote the Taylor polynomials (of degree $d$) of $\frac{1}{(1-q)^n}$ (truncated binomial series, the coefficients are the multiset ...
- 7,933
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The inequality for the Bessel functions $J_\nu(x)^2 \leq J_{\nu-1/2}(x)^2 + J_{\nu+1/2}(x)^2$
(Cross posted from MSE https://math.stackexchange.com/questions/5075724/)
Let $J_\nu$ be the Bessel function of the first kind of order $\nu$.
Does the inequality
\begin{equation} \label{eq:1} \tag{1}
...
- 329
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249
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Can we compute the Gaussian tail quickly with decent precision? (Former "Is this logistic approximation to the Gaussian integral valid?")
The original text is below. I suggest the following edit and reopening:
In some practical applications there is a need to evaluate the antiderivative for the Gaussian function on the line quickly ...
- 41
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Upper bounds for Gegenbauer functions of non-integer degree
(Cross posted from MSE https://math.stackexchange.com/questions/5073844/)
Let $C_{\nu}^{\lambda} \colon (-1, 1] \to \mathbb{R}$ be the (normalized) Gegenbauer function of order $\lambda \geq 0$ and ...
- 329
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336
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Does the hypergeometric function ${}_1F_2(n;1+\frac{n}{2},\frac{3}{2}+\frac{n}{2};-\frac{x^2}{4})$ have an elementary form, or other simplified form?
I am interested in an elementary or simplified form of the hypergeometric function $f(n,x)={}_1F_2(n;1+\frac{n}{2},\frac{3}{2}+\frac{n}{2};-\frac{x^2}{4})$ for integer $n\geq1$. I would be satisfied ...
- 63
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$L^p$-norms of Hermite polynomials with variance different from 1
Let's consider the Hermite polynomials $H_n$ orthogonal with respect to $d\gamma(x) = (2\pi)^{-1/2} e^{-x^2/2} dx$. It is well-known that $\mathbb{E}[H_n(X)^2] = n!$ for a standard normal r.v. $X$.
I ...
- 63
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3
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851
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Proving identity involving dilogarithms and $\pi/9$
I saw this identity in a textbook "The dilogarithm in algebraic fields" and on Mathworld, but no proof:
$$\operatorname{Li}_2(-a) + \operatorname{Li}_2(a^2) - \frac{1}{3} \operatorname{Li}_2(...
- 587
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Closed-form Expression for Spherical Integral $I(n, m)$
I am investigating the following integral over the unit sphere $\mathbb{S}^{N-1}$:
$
I(n, m) = \int_{\mathbb{S}^{N-1}} (1 + \langle u, x \rangle)^n (1 + \langle v, x \rangle)^m \, d\omega(x),
$
where:
...
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What is this modified arithmetico-geometric mean function?
I acquired a vintage programmable calculator and thought I'd give it a spin by computing some interesting transcendental function. I wanted to compute the arithmetico-geometric mean but the calculator ...
- 8,733
4
votes
1
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263
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Infinite sum over pairs of Legendre polynomials
I am trying to obtain a closed-form expression for the following sum over pairs of Legendre polynomials:
$$F(t,x,y)=\sum_{n=1}^\infty\frac{\Gamma(n)^2}{\Gamma(2n)}[P_{n-1}(y)-(n+1-ny)P_n(y)]P_n(x)t^n$$...
- 1,125
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votes
2
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365
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Bounding the digamma function's asymptotic remainder in a vertical strip
Let $\psi(z) = \Gamma'(z)/\Gamma(z)$ be the digamma function and $\log z$ the principal branch of the logarithm. Define the remainder term
$$R(z) = \psi(z) - \log z + \frac{1}{2z}$$
I am looking for a ...
- 703
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254
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An integral of Bessel function
The question is related to Improper integrals of Bessel function
I wonder if there is a closed form for the integral:
$$\int_0^\infty e^{-cx^2} I_\nu(ax)I_\nu(bx) dx,$$
where $I_\nu$ is the modified ...
- 181
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Positivity of integral
We consider the function $F:(0,\infty) \to \mathbb R$. I am trying to show that for all $\alpha>0$ the integral
$$F(r):=\int_0^{\infty} \frac{1-e^{- q}}{1+\alpha q} J_0(rq) \ dq$$
is positive for ...
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Examples of functions that vanish on a closed convex region and are positive outside
Question:
given a convex region $\mathcal{D}\subset\mathbb{R}^n$ i.e. a region for which $x, y\in\mathcal{D}$, implies $\alpha x+(1-\alpha)y\in\mathcal{D}$ for all $\alpha\in[0,1]$, what are examples ...
- 14k
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847
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Missing factor of 10 in derivation for integral form of ζ(3)
While playing around with the divergence theorem to find a new form for $ζ(3)$, I stumbled across this form:
$$\iint_{[0,1] \times [0,1]} \frac{\ln(1-xy)+\ln(1-\sqrt{xy})}{xy} \ dx \ dy$$
The main ...
- 149
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Can the diagram of relations between special functions be expanded more? [closed]
The problem is to expand a diagram of relations between special functions, similar to the one found in John D. Cook's blog, to include more functions and highlight relationships through Meijer G-...
user555829
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Improper integrals of Bessel function
Let $I_\nu$ be the modified Bessel function of first kind. I wonder if there is some formulas for the integrals:
$\int_0^\infty e^{-cx} I_\nu(a \sqrt{x})I_\nu(b \sqrt{x})dx$,
$\int_0^\infty x e^{-cx}...
- 181
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A constant appearing in Dwork's work on the Bessel equation
In Dwork's paper "Bessel functions as $p$-adic functions of the argument", a certain constant $\gamma$ arises as a matrix entry in his calculations of a Frobenius structure that he ...
- 4,046
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Do these series expressions for $\zeta(s)$ and $\eta(s)$ also exist for the Dirichlet $\beta(s)$ function?
The following two series expressions are both entire functions (note that $\xi(s)$ is taken without the factor $\frac12$):
\begin{align}
\xi(s) =s\,(s-1)\,\pi^{-s/2}\,\Gamma\left(\frac{s}{2}\right)\,\...
- 169
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votes
2
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385
views
Partial differential equations for the Weierstrass sigma function
The Wolfram Functions Site
gives for the Weierstrass sigma function two partial differential equation w.r.t. $g_2,g_3$. The first one is
$$ z\frac{\partial \sigma(z; g_2, g_3)}{\partial z}
- 4g_2\...
- 13.5k
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Hopf-algebraic approach to special functions
In my edition of Andrews', Roy's and Askey's "Special functions", they mention a certain new approach to hypergeometric series which they couldn't include into the book, and which deals with ...
- 635
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365
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Nonlinear ODE $y''= k \tan(y).$
The nonlinear second-order differential equation
\begin{equation}
y'' = k \tan(y)
\end{equation}
bears some resemblance to the "real pendulum equation" $ y'' = k \sin(y) $, whose solutions ...
- 79
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votes
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Bessel function $J_\nu(x)$ asymptotics for $\nu\approx x$
If one considers the Bessel function $J_\nu(x)$ with a fixed $x$ and changing $\nu$, one gets a graph which is oscillating rapidly until $\nu\approx x$ and then a quick exponential decay. However, I ...
- 3,808
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Seeking expressions for some integrals in terms of special functions
Consider the following three functions:
$$F_1(z)=\int_{-\pi/2}^{\pi/2}e^{\lambda \phi}\sin \phi\, e^{z e^{\lambda \phi}(A\sin \phi+B\cos \phi)}d\phi,$$
$$F_2(z)=\int_{-\pi/2}^{\pi/2}e^{\lambda \phi}\...
- 11
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Origins of the Bessel function (particularly of the 1st kind)
Crossposted on History of Science and Mathematics SE
I'm working on an exploratory essay that uses the Skellam Distribution and I found that it involves the Bessel function (specifically of the first ...
- 113
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0
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Closed form of a sum involving special functions
I am new to heat kernels for elliptic operators. When using the eigenfunction expansion method, I came across several expressions involving the Hermite polynomials $H_n$. After some routine ...
- 119
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The Hurwitz zeta function on the line $\mathrm{Re}(s) = 1$
For $\mathrm{Re}(s) > 1$ and $0 < a \leqslant 1$, let $\zeta(s,a) = \sum_{n \geqslant 0} 1/(n+a)^s$ denote the Hurwitz zeta function. As a function of $s$, $\zeta(s,a)$ has a meromorphic ...
-1
votes
1
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Derivation involving Hermite polynomials and an exponential function
In the paper https://www.ams.org/journals/mcom/2011-80-274/S0025-5718-2010-02426-3/S0025-5718-2010-02426-3.pdf , it is stated on page 10003 that the Hermite polynomials $H_k(x)$ can be defined via the ...
- 540
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Equivalence of an iterated integral and a multiple zeta value
As I understand, the multiple zeta function is related to Chen's iterated integral in the following way:
$$
\zeta(s_1, s_2, \dots, s_k) = \int_{0}^1 \frac{1}{t_1^{s_1 - 1}} \int_{0}^{t_1} \frac{1}{1 - ...
5
votes
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Algebraic independence relating to Airy functions
Let $u(z)$ be a nontrivial solution of the Airy differential equation $u''(z) = z u(z)$. By using a classical result of Siegel, it is known (see, e.g., S.B. Bank, Mh. Math. 94, 179-200, 1982, Theorem ...
- 1,032
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How to find Closed Form for an Expression Involving Lerch transcendent and Polylogarithms?
this question has been asked on MSE
I tried to prove that
$$ \Omega=\Im \Phi\left(\frac{i}{2},2,\frac{3}{2} \right)-\Re \Phi\left(\frac{i}{2},2,\frac{3}{2} \right)+8\Im\text{Li}_2\left(\frac{-1+i}{2}\...
- 711
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0
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104
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Analytic extension of $\sin(\pi x)\int_{0}^{1}\frac{\text{Li}_0^{(1,0)}(-t)}{t^{x+1}}\mathrm{d}t$
I'm adding a new different question because the approach is different (and much quicker) than my first question (I leave the first question open just for completeness)
\begin{align*}x!_{(\infty)}=&...
- 119
0
votes
2
answers
275
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Asymptotic behavior of an integral
I came up with the following integrations
$$\int_0^\infty \frac{1}{\sqrt{1+a^2\cosh ^2x}}dx,\quad a>0$$
I want to know the precise asymptotic behavior as $a\to 0^+$. I think it is related to the ...
- 805
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0
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109
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Clarification on Calculations for $v_{1,z} $ in "A Method for Verifying the Generalized Riemann Hypothesis"
I am reading the paper "A Method for Verifying the Generalized Riemann Hypothesis" by Ghaith A. Hiary, Summer Ireland, and Megan Kyi. On page 18, $ v_{1,z} $ is calculated using Corollary 9 ...
- 540
1
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1
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226
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Evaluating $\int_0^{2\pi} \sqrt{(a - b \sin t)^2 + 1} \, dt $; connections to elliptic integrals?
I am investigating the integral
$$\int_0^{2\pi} \sqrt{(a - b \sin t)^2 + 1} \, dt$$
where $a > 0$ and $b>0$. This integral arises in a context involving periodic functions, and I suspect it has ...
1
vote
0
answers
210
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Analytic continuation of $\sin(\pi x)\cdot\text{PV}\int_0^1\frac{s^{x-1}}{s+t}\mathrm{d}s$, for $t\in(0,1]$
Introduction
I want to write the function
$$x!_{(\infty)}=\prod_{k=1}^{\infty}k^{\text{sinc}(x-k)}\qquad\text{where}\;\;\text{sinc}(x):=\begin{cases}
\frac{\sin(\pi x)}{\pi x}&x\neq 0\\
1&x=0
\...
- 119
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1
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q->0 limits of q-special functions and q-identities - interesting examples and general principles?
There are many beautiful facts in "q-analogs world": q-special functions (e.g. basic hypergeometric series , q-identities (e.g. quantum dilogarithm related). Typically q->1 we return to ...
- 26.4k
3
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2
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351
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How to calculate the double sine function via Sage or Pari/GP to high precision?
From a paper of Ruijsenaars, we can extract the following integral representation for the logarithm of the double sine function:
$$\log(S_2(w,z)) = \int_0^\infty \left( \frac{e^{-y(1+w-z)}-e^{-yz}}{(1-...
- 2,115
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vote
0
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166
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Algebraic relations for $\Gamma$ function
Let $N$ and $n$ be positive integers with $\mathrm{GCD}(n,N)\ne1$. I want to prove the following claim:
$\Gamma\left(\frac nN\right)$, $\pi$ and the $\Gamma\left(\frac uN\right)$ ($u\in[1,N-1]$, $\...
- 4,306
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0
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200
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A closed formula for a sum involving hypergeometric functions
Can we find a closed formula for this sum:
$$\sum_{p,q\geq 0} (p+q+1)r^{p+q} \frac{{}_1F_1(1+p;2+p+q;r^2)}{{}_1F_1(1+p;2+p+q;1)}$$
where
$$_1F_1(a;c;z) = \sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n n!} z^...
- 571
1
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1
answer
105
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Need bound for absolute value of complex-valued special functions (Taylor coefficients of Faddeeva's w(z))
To guarantee accuracy for code [1] that computes Faddeeva's w(z) [2] using Taylor expansions around different centers, I would need upper bounds for the absolute values $|w_n(z)|$ of the coefficients
$...
- 111