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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Official URL: http://sankhya.isical.ac.in/search/30a2/30a2019.ht...

## 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 a_{1}, ...,a_{r} are column vectors of A and β_{1}, ..., β_{r} are those of B then the new product AΟB is the partitioned matrix (a_{1}⊗β _{1}a_{2}⊗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'_{u}t)+^{r}∑f=1 bf Φ(a'_{i}t)=gf (constant), j=1, ...,q
involving a vector variable t where e_{u} are unit vectors of an identity matrix of order p, a_{1} 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 L_{1},...,L_{q} and M_{1},...M_{p} of independent random variables random variables X_{1},...,X_{n} with the condition
for i= 1,...,q. when L_{f} and M_{j} are linear, the X_{1} 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.

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ID Code: | 96501 |

Deposited On: | 10 Jan 2013 10:55 |

Last Modified: | 10 Jan 2013 10:55 |

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