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 ...
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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.
\...
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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
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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{...
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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 (...
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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}
...
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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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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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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 ...
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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}
...
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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 ...
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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 ...
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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 ...
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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 ...
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