# 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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### First-order non-linear differential equation and transcendental equation

I'm trying to solve this differential equation : $$\frac{dy}{dx}= \frac{-2 y^3}{(y+1)^2(y+2)^2}$$ with the boundary condition $y(x_0)=x_0$, $x>0$, and $y(x)$ being a positive function. The ...
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### On an inequality involving the Lambert $W$ function and the sum of divisors function

Let $W(n)$ be the principal/main branch of the Lambert $W$ function (this is the Wikipedia related to this special function). I was inspired in Robin equivalence to the Riemann hypothesis (see ) ...
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### Reciprocal expansion of modified Bessel function

I am reading Sherstyukov and Sumin - Reciprocal expansion of modified Bessel function in simple fractions and obtaining general summation relationships containing its zeros. The authors say they are ...
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If $x^3+y^3-3\alpha xy=1$, is there an expression for the integral $$\int_0^z \frac{\mathrm dx}{y^2-\alpha x}$$ in terms of more familiar functions? A.C. Dixon introduced the elliptic functions $\... 1answer 199 views ### Gegenbauer's addition theorem for Jacobi polynomials I have the following identity, $$\int_{-1}^{1} \! dz \, j_0\bigl(\sqrt{x^2 + y^2 - 2xy z}\bigr) \, P_n(z) = 2 \, j_n(x) \, j_n(y) \;,$$ where$x, y > 0$,$P_n$is a Legendre polynomial, and$...
I would like to simplify the following Meijer G-function: $G_{1,2}^{1,1} \left(z\mid \binom{0}{0,a}\right)$ into a new Meijer G-function of lower order. In other words, I would like to "simplify" ...