# Questions tagged [special-functions]

This tag is for questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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### On Logarithmic Derivative transformation

I am curious about a certain transformation called the logarithmic derivative that seems to appear a lot in different cool ideas, for example: The use in generating functions for recursions of the ...
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### Equations via Lambert's W function

I'm studying Lambert's W function and I came across the equation $2^x = 2x$. Upon inspection it is easy to see that $x = 1$ and $x = 2$ are the real solutions to the equation. Solving for Lambert's W ...
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### A generalized Euler transformation

The Gaussian hyper-geometric function obeys a number of functional identities which are useful in order to speed up the convergence of the series. A glimpse of the Wikipedia page https://en.wikipedia....
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### problem about the $\mathscr{E}_1(z)$

we know that: $$\mathscr{E}_1(z)=\int_z^{+\infty}\frac{e^{-t}}{t}\mathrm{d}t,\quad |\arg(z)|<\pi$$ my question is that : why the $|\mathrm{arg}(z)|<\pi$
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### $\sum_{n=0}^\infty\frac{H_n(x)H_n(y)t^n}{2^nn!}$=$\frac{\exp\left[\frac{2xyt-(x^2+y^2)t^2}{1-t^2}\right]}{\sqrt{1-t^2}}$

I am told to prove that : $$\sum_{n=0}^\infty\frac{H_n(x)H_n(y)t^n}{2^nn!} = \frac{\exp\left[\frac{2xyt-(x^2+y^2)t^2}{1-t^2}\right]}{\sqrt{1-t^2}}$$ where $H_n(x)$ is Hermite polynomial.I am ...
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### Confluent hypergeometric function for integer values in a closed form

I have an issue about finding expression involving finite summation for $_1F_1(mn,n,\delta x)$ where $m$ and $n$ are integer and $m,n>1$ for my case. I found an expression in [T.o.I. 3.383.$1^{11}$]...
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### Express in terms of Euler integrals [closed]

Express in terms of Euler integrals: $$\int_{0}^{+\infty}\frac{x^{m-1}}{(1+x)^n} dx$$
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### Show that $P_n(x) ={}_2F_1\left(-n,n+1;1;\frac{1-x}{2}\right)$.

I am told that $$P_n(x) ={}_2F_1\left(-n,n+1;1;\tfrac{1-x}{2}\right),$$ where $P_n(x)$ is Legendre polynomial and ${}_2F_1\left(a,b;c;z\right)$ is hypergeometric function. I am just wondering how to ...
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### Inverse of $f(x)=x^n(1-x)^k$

I am trying to find an inverse of a function \begin{align} f(x)=x^n(1-x)^k, x \in (0,1) \end{align} where $n$ and $k$ are some positive integers. I know that his function doesn't have a 'pure' ...