By Hardy G.H., Wright E.M.

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Any /^-finite function / is a finite sum f = J2ft, where / is of some type m t . Downloaded from University Publishing Online. 250 on Tue Jan 24 03:47:54 GMT 2012. 13 SL2(M), differential operators, and convolution 21 The function / i s infinite or C-finite if the C " / (n e N, C the Casimir operator) are contained in a finite dimensional vector space of functions. This is equivalent to either of the following conditions: (i) there exists a nonconstant polynomial Pin one variable such that P(C)f = 0; (ii) there exists an ideal / of finite codimension of C[C] such that Df = 0 for any Del.

15(b), we may take for Vt — {«;} the image of a Siegel set 6-. Then the 6- are disjoint and have disjoint images in F\X. 14), the complement of {J( V* is relatively compact and is contained in F \X. 15(a). By construction, r ( 6 [ ) fl T(6j) = 0 for i ^ j . 16. Recall that our Siegel sets contain geodesic arcs passing through i. Thus, this fundamental set is a Poincare one if / is not elliptic for F, and is used as the origin in Poincare's construction. Remark. 17 is such that Q fl 3X is a set of representatives for the F-equivalence classes of F-cuspidal points.

2 The group T2 has index 3 in Fi, has as a fundamental domain a geodesic triangle with the three vertices at infinity given by |z| ^ 1 and \lZz\ ^ 1, and is the free product of Z/2Z and Z. The groups T\ and F 2 belong to an interesting 1-parameter family of subgroups F\ (A, € [1, 00)) of PG considered by E. Hecke. Ifk = 2 cos n/m (m ^ 3) then FA is discrete, isomorphic to Z/2Z*Z/mZ, and hence a quotient of F2. For other values of X e (1,2), F^ is isomorphic to F 2 but not discrete; for k > 2 it is discrete, again isomorphic to F 2 , but not of finite covolume.