<p>In this paper we consider interlacing of the zeros of polynomials from different sequences <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{p_n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{g_n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>g</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. In our main result we consider a mixed recurrence equation necessary for existence of a linear term <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(x-A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> so that the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(n+1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> zeros of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(x-A)g_n(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>g</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> interlace with the <i>n</i> zeros of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. We apply our result to Meixner-Pollaczek, Pseudo-Jacobi and Continuous Hahn polynomials to obtain new interlacing results for the zeros of polynomials of the same degree from different polynomial sequences.</p>

Interlacing of zeros from different sequences of Meixner-Pollaczek, Pseudo-Jacobi and Continuous Hahn polynomials

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2025年5月14日

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A. S. Jooste,
K. Jordaan

摘要

In this paper we consider interlacing of the zeros of polynomials from different sequences $\{p_n\}$ ${p_{n}}$ and $\{g_n\}$ ${g_{n}}$ . In our main result we consider a mixed recurrence equation necessary for existence of a linear term $(x-A)$ $(x - A)$ so that the $(n+1)$ $(n + 1)$ zeros of $(x-A)g_n(x)$ $(x - A) g_{n} (x)$ interlace with the n zeros of $p_n$ $p_{n}$ . We apply our result to Meixner-Pollaczek, Pseudo-Jacobi and Continuous Hahn polynomials to obtain new interlacing results for the zeros of polynomials of the same degree from different polynomial sequences.