IJPAM: Volume 77, No. 4 (2012)

A GENERALIZATION OF Y. NOZAKI AND
M. RIESZ'S ULTRAHYPERBOLIC KERNEL

Manuel A. Aguirre
Núcleo Consolidado Matemática Pura y Aplicada
Facultad de Ciencias Exactas
Universidad Nacional del Centro
Tandil, Provincia de Buenos Aires, ARGENTINA


Abstract. In this article, we introduce the distributional family $B_{\alpha +mn-n}(V)$ defined by ([*]). $B_{\alpha mn-n}(V)$ is a generalization of distributional family $R_{\alpha }(u)$ defined by ([*]), due to Y. Nozaki and this is called in [#!T!#] the Marcel Riesz Ultrahyperbolic Kernel.

We obtain the formula $L_{a}^{k}\{B_{\alpha +mn-n}(V)\}=C_{\alpha ,m,n}B_{\alpha +mn-n-2km}(V)$ and evaluate $B_{\alpha +mn-n}(V)$ at $\alpha =-2m(\frac{n}{2}+k-\frac{n}{2m})$ for the cases: $\mu =2ms+m$, $s=1,2,...$, $v=2mt$, $t=0,1,2,...$; $\mu =2ms$, $s=0,1,2,...$, $v=2mt+m$, $t=1,2,...$, $\mu =2ms$, $s=0,1,2,...$, $v=2mt$, $t=0,1,2,..$ and $\mu =2ms+m$ and $v=2mt+m$, $s,t=0,1,2,...$, where $L_{a}$ is the operator defined by ([*]) and the constant $C_{\alpha ,m,n}$ is defined by ([*]).

All the results are generalizations of Marcel Riesz ultrahyperbolic kernel and appear in [#!T!#] and [#!R!#].

Received: December 27, 2011

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 77
Issue: 4