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Some properties of classes of real self-reciprocal polynomials.
- Source :
-
Journal of Mathematical Analysis & Applications . Jan2016, Vol. 433 Issue 2, p1290-1304. 15p. - Publication Year :
- 2016
-
Abstract
- The purpose of this paper is twofold. Firstly we investigate the distribution, simplicity and monotonicity of the zeros around the unit circle and real line of the real self-reciprocal polynomials R n ( λ ) ( z ) = 1 + λ ( z + z 2 + ⋯ + z n − 1 ) + z n , n ≥ 2 and λ ∈ R . Secondly, as an application of the first results we give necessary and sufficient conditions to guarantee that all zeros of the self-reciprocal polynomials S n ( λ ) ( z ) = ∑ k = 0 n s n , k ( λ ) z k , n ≥ 2 , with s n , 0 ( λ ) = s n , n ( λ ) = 1 , s n , n − k ( λ ) = s n , k ( λ ) = 1 + k λ , k = 1 , 2 , … , ⌊ n / 2 ⌋ when n is odd, and s n , n − k ( λ ) = s n , k ( λ ) = 1 + k λ , k = 1 , 2 , … , n / 2 − 1 , s n , n / 2 ( λ ) = ( n / 2 ) λ when n is even, lie on the unit circle, solving then an open problem given by Kim and Park in 2008. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 0022247X
- Volume :
- 433
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Journal of Mathematical Analysis & Applications
- Publication Type :
- Academic Journal
- Accession number :
- 109553435
- Full Text :
- https://doi.org/10.1016/j.jmaa.2015.08.038