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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