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        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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        47 views

        GUE, tau-function of Painlevé II, and an article of Forrester-Witte

        Let $ \mu $ be the Gaussian measure $ d\mu(x) = e^{-x^2/2} \frac{dx}{\sqrt{2\pi} } $. I am interested in the following random matrix integral defined for all $ s \in \mathbb{R} $, $ N \geq 1 $ and $ a ...
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        112 views

        A complex integration formula

        I'm trying to integrate a formula but I can’t figure it out. Its calculation involves an error function. Here's the formula: $f(a, b, c)=\int_{0}^{+\pi} d \theta \exp (a \cos \theta) \operatorname{...
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        2answers
        343 views

        Сlosed formula for $(g\partial)^n$

        The objective is to obtain a closed formula for: $$ \boxed{A(n)=\big(g(z)\,\partial_z\big)^n,\qquad n=1,2,\dots} $$ where $g(z)$ is smooth in $z$ and $\partial_z$ is a derivative with respect to $z$. ...
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        140 views

        Calculating $\int_1^{\infty}\frac{\operatorname{ali}(x)}{x^3}dx$, where $\operatorname{ali}(x)$ is the inverse function of the logarithmic integral

        It is well-known that we can compute the closed-form of the integrals $$\int_1^{\infty}\frac{\log x}{x^2}dx$$ and $$\int_1^{\infty}\frac{\operatorname{li} (x)}{x^3}dx,$$ where $\operatorname{li} (x)$ ...
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        1answer
        71 views

        Asymptotic behaviour of function using Fox $H$-function representation

        In equation (9) of this paper, it is claimed that the limiting behaviour $$ \int_0^\infty \frac{1-\cos(kx_0)}{s+Dk^\alpha}dk \sim \frac{\Gamma(2-\alpha)\sin(\pi(2-\alpha)/2)x_0^{\alpha-1}}{(\alpha-1)D}...
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        3answers
        293 views

        Lambert W function multiplied by a constant

        Consider the equation $$y=x e^x.$$ Its real solution is given by $$x=W(y),$$ where $W$ is the Lambert W function (or product log). Can the function $$f(x) = W(-\frac{1}{r}xe^x)$$ be written in ...
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        59 views

        Recurrence involving families of orthogonal polynomials

        Let $ \forall n \in N, n\geq 1$ $$ R_n(x)=(-1)^n n! \displaystyle \frac{(x-1)...(x-n)}{(x(x+1)..(x+n))^2}$$ thus by decomposition in simple element it's easy to see that $$ (1): \quad R_n(x)= \...
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        1answer
        36 views

        Joint density of a quadratic function of entries of orthogonal matrix

        $U=(U_{ij})_{1\leq i,j\leq m},V=(V_{ij})_{1\leq i,j\leq m}$ are independently and uniformly distributed on the orthogonal group $O(m)$. For any positive integer $k,n$ such that $1\leq k\leq n\leq m$, ...
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        1answer
        161 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}\...
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        35 views

        Specify modified error function in form of error functions

        How can we express $\mathrm{erf}(\frac{t-a}{m})$ as a sum of functions of the form $\mathrm{erf}(t)$? I am developing a fitting routine and I encounter this integral: $$\int_{0}^{x}\mathrm{erf}\left(\...
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        55 views

        Differential equation with Fresnel integral

        We have $\frac{y'(x)}{\cos(x)}=C(x)$ and need to find y(x). Generally we should express $y(x)$ through $C(x)$ and elementary functions. I can only do it through $C(x)$ and $S(x)$, or through $\Phi(x)$....
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        0answers
        51 views

        Positivity and zeros of Heun's function

        I am interested in understanding where in the complex plane a Heun function might vanish, or where it (or its real part) is positive. Consider the case where the Frobenius indices at $0$ are $(0,1- \...
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        2answers
        92 views

        Non-asymptotic upper bound of right tail of Gamma function

        I'm wondering if there is any non-asymptotic upper bound for the following Gamma function: $$f_a(x)=\int_{x}^{\infty}t^a\exp(-t)dt$$ for $x>0,a>0$? Something like $x^a\exp(-x)$?
        2
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        1answer
        91 views

        Ratio of hypergeometric function

        Given $a>b>0$, is there any upper bound of the following ratio of hypergeometric function? $$\frac{_2F_1(a,1-b;a+1;x)}{_2F_1(a,1-b;a+1;y)}$$ for $1>x>y>0$ ideally in the form like some ...
        3
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        1answer
        168 views

        Ratio of Selberg integral

        I'm considering a ratio of incomplete Selberg integral: $$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{...

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