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International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 2, Pages 267-276

A generalization of Lucas' theorem to vector spaces

Mathematics Department, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received 2 June 1988; Revised 15 August 1991

Copyright © 1993 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The classical Lucas' theorem on critical points of complex-valued polynomials has been generalized (cf. [1]) to vector-valued polynomials defined on K-inner product spaces. In the present paper, we obtain a generalization of Lucas' theorem to vector-valued abstract polynomials defined on vector spaces, in general, which includes the above result of the author [1] in K-inner product spaces. Our main theorem also deduces a well-known result due to Marden on linear combinations of polynomial and its derivative. At the end, we discuss some examples in support of certain claims.