On Polar Jacobi Polynomials
Abstract
:1. Introduction
2. Algebraic Properties of the Polar Jacobi Polynomials
- 1.
- Three-term recurrence relation.
- 2.
- First structure relation.
- 3.
- Second structure relation.
- 4.
- Squared Norm. For every ,
- 5.
- Second-order difference equation. For every ,
- 6.
- Forward shift operator.
- 7.
- Asymptotic formula. Let . Put where the branch of the square root is chosen so that for . Then,
3. Zero Location
- 1.
- If is a zero of , then is a zero of .
- 2.
- If is a zero of , then ζ is a zero of .
- 3.
- The zeros of have multiplicity of at most 2 and their multiple zeros are located on .
- 4.
- All the zeros of are located on the curve
- Observe that the zeros of do not have to be simple. Let or ; then, the polar polynomial of degree two , or .
- When the parameters are not standard, i.e., or then, by Corollary 2, statement 3 of Theorem 6 is no longer true. For example, if , , and , then .
- 1.
- All zeros of the polar Jacobi polynomials with pole ξ are contained in .
- 2.
- If , the zeros of the polar Jacobi polynomials with pole ξ are simple and contained in the exterior of the ellipse , where .
Funding
Data Availability Statement
Conflicts of Interest
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Costas-Santos, R.S. On Polar Jacobi Polynomials. Mathematics 2024, 12, 2767. https://doi.org/10.3390/math12172767
Costas-Santos RS. On Polar Jacobi Polynomials. Mathematics. 2024; 12(17):2767. https://doi.org/10.3390/math12172767
Chicago/Turabian StyleCostas-Santos, Roberto S. 2024. "On Polar Jacobi Polynomials" Mathematics 12, no. 17: 2767. https://doi.org/10.3390/math12172767
APA StyleCostas-Santos, R. S. (2024). On Polar Jacobi Polynomials. Mathematics, 12(17), 2767. https://doi.org/10.3390/math12172767