Zeros of a table of polynomials satisfying a four-term contiguous relation

  • Jack Luong

    University of California, Los Angeles, USA
  • Khang Tran

    California State University, Fresno, USA
Zeros of a table of polynomials satisfying a four-term contiguous relation cover
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Abstract

For any A(z),B(z),C(z)C[z]A(z),B(z),C(z)\in\mathbb{C}[z], we study the zero distribution of a table of polynomials {Pm,n(z)}m,nN0\{P_{m,n}(z)\}_{m,n\in\mathbb{N}_0} satisfying the recurrence relation

Pm,n(z)=A(z)Pm1,n(z)+B(z)Pm,n1(z)+C(z)Pm1,n1(z)P_{m,n}(z)=A(z)P_{m-1,n}(z)+B(z)P_{m,n-1}(z)+C(z)P_{m-1,n-1}(z)

with the initial conditions P0,0(z)=1P_{0,0}(z)=1 and Pm,n(z)=0P_{-m,-n}(z)=0 for all m,nNm,n\in\mathbb{N}. We show that the zeros of Pm,n(z)P_{m,n}(z) lie on a curve whose equation is given explicitly in terms of A(z)A(z), B(z)B(z), and C(z)C(z). We also study the zero distribution of a case with a general initial condition.

Cite this article

Jack Luong, Khang Tran, Zeros of a table of polynomials satisfying a four-term contiguous relation. Z. Anal. Anwend. 41 (2022), no. 1/2, pp. 81–91

DOI 10.4171/ZAA/1698