The Euler-poisson-darboux Equation in the Exceptional Cases1

where in the EPD equation (1.1) it is understood that Aw = 2Zf=i «*,•*< and k is a parameter, — °° < k < °°. We designate a solution of Li[w]=0 by um(x, t), writing x in place of *i, ■ • • , xm. While particular results for this famous equation go back to Poisson and even to Euler, a systematic investigation of the Cauchy problem covering all values of k and m is of recent date. The results may be summarized as follows. (i) If k = m — 1, the solution, as given explicitly by Asgeirsson [l], is