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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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About Turan`s problem(inequality) in multivariable
I am sorry. Ottem is right. $P_n$ is a polynomial of degree $n$. In particular, you may assume that $P_n$ is a symmetric function. Let's focus that $P_n(x_1,\, \dots ,x_k) = x_1^n + \cdots + x_k^n$, f …