Solutions to some functional equations and the applications

Khatri, C. G. ; Rao, C. R. (1968) Solutions to some functional equations and the applications Sankhya, 30 (2). pp. 167-180. ISSN 0972-7671

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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, ...,ar are column vectors of A and β1, ..., βr are those of B then the new product AΟB is the partitioned matrix (a1⊗β 1a2⊗β:...:ar⊗βr) 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 pu=1 cfu ψu(e'ut)+r∑f=1 bf Φ(a'it)=gf (constant), j=1, ...,q involving a vector variable t where eu 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,...,Lq and M1,...Mp of independent random variables random variables X1,...,Xn with the condition for i= 1,...,q. when Lf and Mj are linear, the X1 have normal distributions. When Lf are linear in the reciprocals of the variables and Mj are linear in the variables, the Xi have gamma or conjugate gamma distributions. When the Xf variables, are non-negative, Lf are linear in the variables and Mj are linear in the logarithms of the variables, the Xf 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 Xf are identically distributed.

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