Generalizing Tran's Conjecture

2020/01/25 by Bögvad, Rikard, Ndikubwayo, Innocent, Shapiro, Boris
#2010 Primary 12D10 #30C15 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Secondary 26C10

paper · doi:10.48550/arxiv.2001.09248

Abstract

A conjecture of Khang Tran [6] claims that for an arbitrary pair of polynomials A(z) and B(z), every zero of every polynomial in the sequence \P_n(z)\_n=1^∞ satisfying the three-term recurrence relation of length k P_n(z)+B(z)P_n-1(z)+A(z)P_n-k(z)=0 with the standard initial conditions P₀(z)=1, P_-1(z)=…=P_-k+1(z)=0 which is not a zero of A(z) lies on the real (semi)-algebraic curve \mathcal C ⊂ \mathbb C given by \Im ((B^k(z))/(A(z)))=0 \rm and 0≤ (-1)^k\Re ((B^k(z))/(A(z)))≤ \frack^k(k-1)^k-1. In this short note, we show that for the recurrence relation (generalizing the latter recurrence of Tran) given by P_n(z)+B(z)P_n-ℓ(z)+A(z)P_n-k(z)=0, with coprime k and ℓ and the same standard initial conditions as above, every root of P_n(z) which is not a zero of A(z)B(z) belongs to the real algebraic curve \mathcal C_ℓ,k given by \Im ((B^k(z))/(A^ℓ(z)))=0.

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