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2 added explanation of matrix coefficient functions.

Expanding on Qiaochu's and David's comments, from the point of view of automorphic forms and automorphic representations, a good reference is "Special Functions and the Theory of Group Representations" by N. Ja. Vilenkin and published by the AMS.

"Special Functions" by Andrews, Askey, and Roy published by CUP is another good reference with an anayltic number theory slant.

EDIT: The point here is that for a finite group $G$ and a representation $\pi$ into a finite dimensional vector space $V$ with orthonormal basis $\{\vec{e_i}\}$, the function $g\to \langle \vec{e_i},\pi(g)\vec{e_j}\rangle$ gives the $i,j$ coefficient of the matrix associated to $\pi(g)$, or 'matrix coefficient function'. For a Lie group $G$ with an infinite dimensional representation $\pi$ in a Hilbert space $H$, there is a natural analog, and the functions which arise this way play an obviously important role in harmonic analysis in $G$. Bessel functions can be interpreted this way.

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Expanding on Qiaochu's and David's comments, from the point of view of automorphic forms and automorphic representations, a good reference is "Special Functions and the Theory of Group Representations" by N. Ja. Vilenkin and published by the AMS.

"Special Functions" by Andrews, Askey, and Roy published by CUP is another good reference with an anayltic number theory slant.