# 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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### What's the convergence condition for the generating function formula of Legendre polynomials?

What is the convergence condition of the next infinite series about the Legendre polynomials $P_n(x)$? $$\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^\infty P_n(x)t^n$$ I know it is convergent at least ...
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### Can a general quintic be solved using Inverse Beta Regularized function?

Tyma Gaidash has recently posted solutions to some quintics in terms of Inverse Beta Regularized function. He also found the closed form for the equation $\cos x=x$ using the same Inverse Beta ...
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### Is it possible to solve sextic equations using the Fox H function?

Although the Kampé de Fériet function can solve the sextic equation, the details about it are shrouded in the fog of more than a century ago. In contrast, we know more about the Fox H function, and we ...
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### Generalization of identity for terminating hypergeometric function

Let ${}_2F_1(a,b;c;z)$ be the ordinary hypergeometric function for $z \in \mathbb{C}$ \begin{equation} {}_2F_1(a,b;c;z) = \sum_{k=0}^{\infty} \frac{z^k}{k!} \frac{(a)_{k} (b)_k}{(c)_k}\,, \end{...
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### What's the fastest way to compute $\log n$ for $n>1$?

As it is well known, if $|x|<1$ then we can compute $\log(1+x)$ by the Taylor series $$\log(1+x)=x-\frac{x^2}2+\frac{x^3}3-\cdots.$$ Thus, to compute $\log n$ with $n>1$, we may employ the ...
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