Lie Theory and Special Functions by Willard Miller Jr.

By Willard Miller Jr.

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Definition Let G be a local Lie transformation group acting on an 0 E U , and let LZ be the set of all complexopen neighborhood U of cm, 1-3. LOCAL TRANSFORMATION GROUPS 17 valued functions on U analytic in a neighborhood of 0. of G on 6Z with multiplier v, consists of a mapping P(g) of Gl onto 6Z defined for g E G, f E Gl by xE [ T w f l ( x ) = 4x9 g)f(xg), where V(X, g) u 9 is a complex-valued function analytic in x and g, such that (1) v(x,e ) = 1, all X E U , (2) v(x,g1g2) = v(x, g,)v(xg, g2), 9 g1 gz g1g2 E G.

RtSUME OF LIE THEORY SL(2)that is purely local; it is defined only in a neighborhood of e, not on all of SL(2). T h e connection between addition theorems for special functions and matrix elements of local multiplier representations presented here has general validity. I n fact, we will show that the basic addition theorems for hypergeometric functions can be obtained from the study of multiplier representations of SL(2),the basic addition theorems for confluent hypergeometric functions follow from the study of multiplier representations of G(0, I), and addition theorems for Bessel functions can be obtained from multiplier representations of T,.

50) x(0) = xo. Since the right-hand side of these equations is independent of w , the action of the 1-parameter group exp art on x takes the form x exp at = (x’ exp at, q(x’, exp at) +w) 16 1. RESUME OF LIE THEORY where q is a complex-valued analytic function of its arguments. T h e action of G on U is completely determined by the group elements exp at, so we have xg = (x’g,dx’,g ) + w), g E G. 52) where v(x’,g) = Moreover, if g = e , then q(x’,e ) = 0, v(x’,e) = 1, for all x E U. , e-”f(x) is independent of w.

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