Run-length encodings (Corresp.)

explicitly evaluable functions. For example, the M-ary error probability is expressed as a quadrature in Lindsey's equation (17), PE(M) = 1 [I-2 lrn Qi(h, $;) exp (-g) dz] z/d eeL s m =22/;;moe-(1+d)s~41-'2@3(1, 1 + M, s, sL) d.s, (5) where, following Lindsey, h2/2 has been replaced by L to simplify the notation. From the series form of @3, it is obvious that the integral gives an additional double series numerator parameter: PE(M) = di eeL z ~1 (1 + d)-"-1'2 i (61 A complete set of recursion relations for F1 when one parameter at a time changes has been given by Le Vavasseur [S]. It is a simple matter to derive the necessary change for this two-parameter case but Le Vavasseur has included this as one of several examples, so that we have at once (8) (9)-I, e (a "-' a " '-'_ [eLP,(l)] = (1 + &)Y,(l), which is equivalent to a result of Price [9], who has derived a number of expressions for these and related integrals. Note that the derivation above is, thus far, much simpler and more straightforward than the admirably executed tours de force of previous derivations. However, the last step, viz., recognizing the form of the result, is automatically accomplished in the other derivations, and is much the harder part in the hypergeometric case. To obtain the reduction, we use operational relations [lo] to get The integral with the special parameters of (11) has been previously recognized as a Q function [12]-[14] so that the reduction is essentially complete.