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There is known link text addition theorem for the Legendre functions of the first kind $P_{\nu}^m(x)$, which establishes the connection between $P_{\nu}^m(x)P_{\nu}^{m}(y)$ and $P_{\nu}(F(x,y))$ for some concrete function $F(x,y)$ (which itself deals with the spherical distance, but let's not think about it for a moment). But what about addition theorem, in which $\mu$ does vary, while $x$ is fixed? I mean, say for $m=0$, is there any expression of $P_{\nu}(x)P_{\mu}(x)$ via $P_{f(\nu,\mu)}(g(x))$ for some functions $f$, $g$?

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Dear Fedor Petrov, I have added the classical analysis tag, thinking it could be useful. – Giuseppe Jun 28 '11 at 21:25
Fedor, the Legendre functions are special instances of the $_2F_1$ hypergeometric functions, so there products would normally satisfy the differential equations of order 3 (at least!), hence could hardly be given as another LF. See also my answer… . I'd like to point out that the link in your Q is dead, and I am interested to see what kind of addition formula is meant there. – Wadim Zudilin Jul 3 '11 at 13:42
ops, sorry, hope that this one works: – Fedor Petrov Jul 3 '11 at 17:53

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