ABSTRACT
U(a, c, z ) and V(a, c, z) are related in the following way. By a transformation of variables, which is suggested by comparing the right-hand sides of (10.01) and (10.02), we find that ezU(c-a, c, - z ) satisfies equation (9.02). This solution is recessive as z -+ - m, hence it is a constant multiple of V(a, c, z). Inspection of leading terms shows that this multiple is unity; thus
V(a, c, z) = ezU(c - a, c, - z). (10.03)
An integral representation for U(a, c,z) analogous to (9.07) is given by
[ta-'(1 +t)"-'-'e-"dt (Iphzl c +a, Rea > 0). U(a, c, z) = - I-(a)
This can be verified by showing that the integral satisfies the confluent hypergeometric equation, and then comparing the asymptotic form yielded by Watson's lemma with (10.01). 10.2 The transformation leading to (10.03) also shows that eZM(c-a, c, -z) satisfies (9.02). When Re c > 1 or c = 1, this solution is recessive at z = 0, and since it assumes the value I/T(c) at this point, we deduce that
Analytic continuation removes all restrictions on the parameters in this result, which is known as Kummer's transformation. The transformation can also be established by multiplying the series (9.04) by the power series for e-' and using Vandermonde's theorem.
