The distributional products on spheres and Pizetti’s formula

The distribution δ ( k ) ( r − a ) concentrated on the sphere O a with r − a = 0 is defined as ( δ ( k ) ( r − a ) , ϕ ) = ( − 1 ) k a n − 1 ∫ O a ∂ k ∂ r k ( ϕ r n − 1 ) d σ . Taking the Fourier transform of the distribution and the integral representation of the Bessel function, we obtain asymptotic expansions of δ ( k ) ( r − a ) for k = 0 , 1 , 2 , … in terms of △ j δ ( x 1 , … , x n ) , in order to show the well-known Pizetti formula by a new method. Furthermore, we derive an asymptotic product of ϕ ( x 1 , … , x n ) δ ( k ) ( r − a ) , where ϕ is an infinitely differentiable function, based on the formula of △ m ( ϕ ψ ) , and hence we are able to characterize the distributions focused on spheres, which can be written as the sums of multiplet layers in the Gel’fand sense.