IJPAM: Volume 58, No. 4 (2010)

RELATIONS BETWEEN THE ULTRAHYPERBOLIC
OPERATOR AND -TH DERIVATIVE OF
DIRAC'S DELTA IN AND $P(x)-c^{2}$

Manuel A. Aguirre
Núcleo Consolidado Matemática Pura y Aplicada (NuCOMPA)
Facultad de Ciencias Exactas
Universidad Nacional del Centro
Pinto 399, Tandil, Provincia de Buenos Aires, 7000, ARGENTINA
e-mail: maguirre@exa.unicen.edu.ar

Abstract.In this article we obtain formulae between the operator ultrahyperbolic iterated times defined by () and the -th derivative of Dirac's delta defined in $P(x)=x_{1}^{2}+\dots x_{\mu }^{2}-x_{\mu +1}^{2}\dots -x_{\mu +\nu }^{2}$ and $P(x)-c^{2}.$ In particular we obtain that $\delta ^{(k-1)}(P_{+})$ is homogeneous solution of the ultrahyperbolic operator iterated times if $\frac{n}{2}-l\leq k<\frac{n}{2}$ and $E_{n,r,\mu ,\nu}$ defined in (23) is elementary solution of ultrahyperbolic operator iterated times. By putting in (18) we have that $\delta ^{(s)}(P_{+})$ is homogeneous solution of the ultrahyperbolic operator if $s=% \frac{n-4}{2}.$