Some recursion relations for solutions of a class of equations
This work presents some recursion relations for solutions of a class of singular partial differential equations. The equations consist of iterated differential operators each of which involves real parameters.
On a singular differential operator
SummaryStudy of the properties of a singular differential operator in one independent variable. Applications to ordinary and partial differential equations.
The Euler-Poisson-Darboux equation in the exceptional cases
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