Solutions to some functional equations and the applications

Abstract

Three sets of results are contained in this paper. The first is on a new matrix product. If A and B are two matrices of orders p×r and q×r respectively, and if a₁, ...,a_r are column vectors of A and β₁, ..., β_r are those of B then the new product AΟB is the partitioned matrix (a₁⊗β ₁a₂⊗β:...:a_r⊗β_r) where ⊗ denotes the Kronecker product. Propositions involving the new product of matrices are stated. The second is on the solution of functional equations of two types. One is of the form ^p∑_u=1 cfu ψu(e'_ut)+^r∑f=1 bf Φ(a'_it)=gf (constant), j=1, ...,q involving a vector variable t where e_u are unit vectors of an identity matrix of order p, a₁ are given column vectors and ψu,Φt are unknown continuous functions. Another is of the form ⁿ∑f=1 dfΦ(bjt)=gt, i=I,...,q involving an unknown function Φ of a single variable t. Conditions under which the unknown functions in these two types of equations are polynomials of an assigned degree are given. The third, on the characterization of normal and gamma distributions, extends the earlier work of the authors (Rao, 1967 and Khatri and Rao, 1968*). We consider two sets of functions L₁,...,L_q and M₁,...M_p of independent random variables random variables X₁,...,X_n with the condition for i= 1,...,q. when L_f and M_j are linear, the X₁ have normal distributions. When L_f are linear in the reciprocals of the variables and M_j are linear in the variables, the X_i have gamma or conjugate gamma distributions. When the X_f variables, are non-negative, L_f are linear in the variables and M_j are linear in the logarithms of the variables, the X_f have gamma distributions;. These results are proved under some conditions on the compounding coefficients for p>1, and in the case of p=1 with the further condition that the X_f are identically distributed.