The dual Hahn polynomials <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{u_i(x)\}_{i=0}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>u</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>d</mi> </msubsup> </math></EquationSource> </InlineEquation> are a family of discrete orthogonal polynomials involving two real parameters r and s. Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L,L^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>,</mo> <msup> <mi>L</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> denote the corresponding Leonard pair. Assume that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$r\not =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$r+s=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>+</mo> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We show that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L,(L^+\frac{r-d}{2})^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>,</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mo>∗</mo> </msup> <mo>+</mo> <mfrac> <mrow> <mi>r</mi> <mo>-</mo> <mi>d</mi> </mrow> <mn>2</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> is a Leonard pair. According to the theory of Leonard pairs, the polynomials <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\{u_i(x)\}_{i=0}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>u</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>d</mi> </msubsup> </math></EquationSource> </InlineEquation> are simultaneously dual Hahn polynomials and Racah polynomials with respect to the same inner product.

A way to treat dual Hahn polynomials as Racah polynomials via the theory of Leonard pairs

期刊论文
发表日期：

2026年6月16日

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Hau-Wen Huang

摘要

The dual Hahn polynomials $\{u_i(x)\}_{i=0}^d$ ${u_{i} (x)}_{i = 0}^{d}$ are a family of discrete orthogonal polynomials involving two real parameters r and s. Let $L,L^*$ $L, L^{*}$ denote the corresponding Leonard pair. Assume that $r\not =0$ $r \neq 0$ and $r+s=0$ $r + s = 0$ . We show that $L,(L^*+\frac{r-d}{2})^{2}$ $L, {(L^{*} + \frac{r - d}{2})}^{2}$ is a Leonard pair. According to the theory of Leonard pairs, the polynomials $\{u_i(x)\}_{i=0}^d$ ${u_{i} (x)}_{i = 0}^{d}$ are simultaneously dual Hahn polynomials and Racah polynomials with respect to the same inner product.