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On a kind of sequence of polynomials

  • Zhang Xiangde
Session 7A: Combinatorics
Part of the Lecture Notes in Computer Science book series (LNCS, volume 959)

Abstract

In this paper, motivated by a conjecture concerning the q-derangement polynomials Dn(q) which are defined by Brenti, we study a kind of sequence of polynomials {Sn(q)}. The Sn(q) is a polynomial of degree n with nonnegative real coefficients and satisfy some initial conditions and the following recurrence relation:
$$S_n (q) = a_n qS_{n - 1} (q) + b_n q(1 + c_n q)S'_{n - 1} (q) + d_n qS_{n - 2} (q),$$
where an>0,bn>0,dn>0 and n>0. We show that Sn(q) has n distinct real roots(≤ 0), separated by the roots of Sn−1(q). As a consequence, the conjecture is proved.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Zhang Xiangde
    • 1
  1. 1.Northeastern UniversityShenyangChina

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