On hyperbolic polynomials with four-term recurrence and linear coefficients
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On hyperbolic polynomials with four‑term recurrence and linear coefficients Richard Adams1 Received: 30 April 2019 / Revised: 20 May 2020 / Accepted: 23 July 2020 / Published online: 4 August 2020 © Istituto di Informatica e Telematica (IIT) 2020
Abstract For any real numbers a, b , and c, we form the sequence of polynomials {Pn (z)}∞ n=0 satisfying the four-term recurrence
Pn (z) + azPn−1 (z) + bPn−2 (z) + czPn−3 (z) = 0, n ∈ ℕ, with the initial conditions P0 (z) = 1 and P−n (z) = 0 . We find necessary and sufficient conditions on a, b , and c under which⋃the zeros of Pn (z) are real for all n, and pro∞ vide an explicit real interval on which n=0 Z(Pn ) is dense, where Z(Pn ) is the set of zeros of Pn (z). Keywords Zeros of polynomials · Generating function · Sequences of polynomials Mathematics Subject Classification 26C10 · 12D10 · 30C15
1 Introduction Recursively-defined sequences of polynomials have been studied extensively since the 18th century. Of particular interest are orthogonal polynomials, such as Hermite polynomials, Laguerre polynomials, and Jacobi polynomials, which have numerous applications in differential equations, mathematical and numerical analysis, and approximation theory (see[1, 9]). Common orthogonal polynomials such as the Chebychev polynomials can be defined by a three-term recurrence relation, which in modern times has been generalized significantly. For recent work on three-term recurrences, see[2, 5–7, 12, 13]. Not much is known, however, about four-term recurrences. Consider the sequence of polynomials {Pn (z)}∞ satisfying the fourn=0 term recurrence relation
* Richard Adams [email protected] 1
California State University, Fresno, Fresno, USA
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Pn (z) + A(z)Pn−1 (z) + B(z)Pn−2 (z) + C(z)Pn−3 (z) = 0, where P0 (z) = 1 , P−n (z) = 0 , and A(z), B(z) , and C(z) are some linear complexvalued polynomials with real coefficients. With certain initial conditions, one may wish to find where the zeros of Pn (z) lie on the complex plane for each n ∈ ℕ . Let Z(Pn ) = {z ∈ ℂ|Pn (z) = 0} . We say that a polynomial is hyperbolic if all of its zeros are real, and we are interested in finding necessary and sufficient conditions under which Pn (z) is hyperbolic for all n ∈ ℕ . In[14] the authors characterized the case where A(z) = a , B(z) = b , and C(z) = z . In this paper, we characterize the case where A(z) = az , B(z) = b , and C(z) = cz for some nonzero real numbers a, b , and c. Next we present the main result of this paper. Theorem 1 The zeros of the sequence {Pn (z)}∞ satisfying the recurrence relation n=0
Pn (z) + azPn−1 (z) + bPn−2 (z) + czPn−3 (z) = 0, a, b, c ∈ ℝ⧵{0}, where P0 (z) = 1 and P−n (z) = 0 , are real √ if and only if b > 0 and which case they lie on the interval I ∶= ab ⋅ (−𝜆, 𝜆) , where
4
𝜆 ∶= � Furthermore,
c ab
⋃∞
n=0
√ � 32 � 3𝛼+1+ 9𝛼 2 −10𝛼+1 √ −5𝛼 −5𝛼+1+ 9𝛼 2 −10𝛼+1
∶= 𝛼 ≤ 91 , in .
� √ + 1 + 9𝛼 2 − 10𝛼 + 1
Z(Pn ) is dense on I, where Z(Pn ) is the set of zeros of Pn (z).
This paper is orga
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