The Euler-Poisson-Darboux equation in the exceptional cases

Abstract

where in the EPD equation (1.1) it is understood that Au= EJ=1 u_ixi and k is a parameter, -oo < k < oo. We designate a solution of Lk[u]=O by U(k)(X, t), writing x in place of xi, * , x.. 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 [1], is