An Integral Representation Formula for Multi Meta-φ-Monogenic Functions of Second Class

Consider the following operator D-phi(j),D-lambda(j) = phi((j))(0)partial derivative((j))(chi 0) + Sigma(mj)(i=1)phi((j))(i) (chi)e(i+aj) partial derivative(i+aj) - lambda((j)) where phi((j)) is a Clifford-valued function and lambda((j)) is a Clifford-constant defined by phi((j)) = phi((j))(0) + Sigma(mj)(i=1) phi((j))(i)e(i+aj), lambda((j)) = lambda((j))(0) + Sigma(mj)(i=1) lambda((j))(i)e(i+aj) with m = m(1) + center dot center dot center dot + m(n), a(1) = m(0) = 0 and a(j) = m(1) + center dot center dot center dot + m(j-1) for j = 2,...,n; and phi((j))(i) can be real-valued functions defined in Rm1+1 x Rm2+1 x center dot center dot center dot x Rmn+1 lambda((j))(i) i are real numbers for i = 0, 1,...,m(j) and j = 1,..., n. A function u is multi meta-phi-monogenic of second class, in several variables x((j)), for j = 1,...,n, if D-phi(j),D-lambda(j) u=0. In this paper we give a Cauchy-type integral formula for multi meta-phi-monogenic of second class operator in one way by iteration and in the second way by the use of the construction of the Levi function. Also, in this work, we define a multi meta-phi-monogenic function of first class with the help of the Clifford type algebras depending on parameters.