Self-reciprocal polynomials with related maximal zeros
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For each real number n > 6, we prove that there is a sequence {pk(n, z)}∞ k=1 of fourth degree self-reciprocal polynomials such that the zeros of pk(n, z) are all simple and real, and every pk+1(n, z) has the largest (in modulus) zero αβ where α and β are the first and the second largest (in modulus) zeros of pk(n, z), respectively. One such sequence is given by pk(n, z) so that [수식] where q0(n) = 1 and other qk(n)’s are polynomials in n defined by the severely nonlinear recurrence [수식] for m ≥ 1, with the usual empty product conventions, [수식]
Abstract
For each real number n > 6, we prove that there is a sequence {pk(n, z)}∞ k=1 of fourth degree self-reciprocal polynomials such that the zeros of pk(n, z) are all simple and real, and every pk+1(n, z) has the largest (in modulus) zero αβ where α and β are the first and the second largest (in modulus) zeros of pk(n, z), respectively. One such sequence is given by pk(n, z) so that [수식] where q0(n) = 1 and other qk(n)’s are polynomials in n defined by the severely nonlinear recurrence [수식] for m ≥ 1, with the usual empty product conventions, [수식]
참고문헌 (1)
- T. Sheil-Small, "Complex Polynomials, In Cambridge Studies in Advaced Mathematics 73" (2002), Cambridge University Press