The aim of this contribution is twofold. First, by performing the quadratic decomposition of q-Dunkl-Appell sequences, a new lowering q-differential operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {K}_{q^2 ; \theta ;\kappa }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">K</mi> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>;</mo> <mi>θ</mi> <mo>;</mo> <mi>κ</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> (with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\kappa =\pm \frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mo>=</mo> <mo>±</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>) naturally emerges, as the two polynomial sequences lying in the principal diagonal are <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {K}_{q^2 ; \theta ;\kappa }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">K</mi> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>;</mo> <mi>θ</mi> <mo>;</mo> <mi>κ</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>-Appell. Second, triggered by this result, after developing the concept of the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {K}_{q ; \theta ;\kappa }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">K</mi> <mrow> <mi>q</mi> <mo>;</mo> <mi>θ</mi> <mo>;</mo> <mi>κ</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>-Appell sequences, all the orthogonal <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {K}_{q ; \theta ;\kappa }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">K</mi> <mrow> <mi>q</mi> <mo>;</mo> <mi>θ</mi> <mo>;</mo> <mi>κ</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>-Appell sequences are sought, which outcome was the Wall q-polynomials with parameter <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$b=\frac{q^{1-\kappa }}{1-(q^{1/2}-1)\theta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>=</mo> <mfrac> <msup> <mi>q</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>κ</mi> </mrow> </msup> <mrow> <mn>1</mn> <mo>-</mo> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> (resp. the Little q-Laguerre polynomials with parameter <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha =bq^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mi>b</mi> <msup> <mi>q</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>) – they are indeed the unique ones fulfilling both properties, up to a linear transformation. This leads to a new characterization of these polynomial sequences.

On the quadratic decomposition of q-Dunkl-Appell polynomial sequences

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2026年6月27日

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Mohamed Khalfallah

摘要

The aim of this contribution is twofold. First, by performing the quadratic decomposition of q-Dunkl-Appell sequences, a new lowering q-differential operator $\mathcal {K}_{q^2 ; \theta ;\kappa }$ $K_{q^{2}; θ; κ}$ (with $\kappa =\pm \frac{1}{2}$ $κ = \pm \frac{1}{2}$ ) naturally emerges, as the two polynomial sequences lying in the principal diagonal are $\mathcal {K}_{q^2 ; \theta ;\kappa }$ $K_{q^{2}; θ; κ}$ -Appell. Second, triggered by this result, after developing the concept of the $\mathcal {K}_{q ; \theta ;\kappa }$ $K_{q; θ; κ}$ -Appell sequences, all the orthogonal $\mathcal {K}_{q ; \theta ;\kappa }$ $K_{q; θ; κ}$ -Appell sequences are sought, which outcome was the Wall q-polynomials with parameter $b=\frac{q^{1-\kappa }}{1-(q^{1/2}-1)\theta }$ $b = \frac{q^{1 - κ}}{1 - (q^{1 / 2} - 1) θ}$ (resp. the Little q-Laguerre polynomials with parameter $\alpha =bq^{-1}$ $α = b q^{- 1}$ ) – they are indeed the unique ones fulfilling both properties, up to a linear transformation. This leads to a new characterization of these polynomial sequences.