Orthogonal Polynomials and Special Functions by Richard Askey

By Richard Askey

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1; c > 0 For Re(s) > 1 and any such that E(z,s,m) - an(y,s,m) « as 1 £ i <^ n y. -*• co. e -cy. 2: Let f(z) be a T-automorphic eigenfunction with eigenvalues s . ( l - s . 5) above. Also assume that some f(z) grows at most polynomially as y. -*» ». Then for a, f(z) = a • E(z,s,m) . yn Therefore f(z) - a-E(z,s,m) € L 2 (F) . Moreover, this is an eigenfunction with eigenvalues But the A. f s are positive self-adjoint operators on s . ( l - s . ,n. ,n s. = % + it, t €r ]R, or s. € [-1,1]. ) > 1, we see that f(z) -a*E(z,s,m) E 0 .

Where we choose exactly one for each conjugacy class, and where y 6 T . 5: anc has more then two fixed points, which cannot happen. I—I Consider the set of all pairs representative Y-J " Y O # Yi The map ({T },Y) *-* M is a correspondence. THE SELBERG TRACE FORMULA Proof: The map is clearly well defined and onto. and ({T one, let ({F },Y-i) Tl = {Yi} T 2 T Then there is a l r To prove it is one to },Yo) be in our set of pairs and assume 2 a 6 T = r , = a " F a , so that Y o yxo l y {T Wo}* 35 such that {T } = {T } Y Y l 2 -1 Yo - a~ Yi a « Thus or equivalently } = {T }.

Bu. i an d t. +/5u. ,m, generate the group of units in 0_ ,, of Z JJ,a relative norm 1. ,n so that _ _1 (j) and similarly 1_ 4 3 ^ = 4 s i n 2 ^ ^ ) = 4 - t^j) Also, vol(F ) = d e t ( l o g p ^ } ) . . M y i = det 2 l o g - i which we call the regulator of 0 Letting k k y = y, ... ,m , and denote by R(D,d). _, ,. . /2 J nm—^m -u j=m+l 1) gl n k. logp e J J Observe that the , . . , n k. ef^l 0 fs and x f m n .. r u . /z^*. -e J ) • ' --'-2 m ,. du m+1 n 0_ , is associated. /2 on the particular class we sum over.

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