In this paper, we deal with a lowering operator, denoted by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Lambda _{k, l}^{\varvec{\theta }, \varvec{q}, \varvec{a}, \varvec{b}},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="normal">Λ</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> </mrow> <mrow> <mrow> <mi mathvariant="bold-italic">θ</mi> </mrow> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">q</mi> </mrow> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">a</mi> </mrow> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">b</mi> </mrow> </mrow> </msubsup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> defined as a finite product of lowering operators in terms of the q-Dunkl operator. Our first goal is to characterize polynomial sequences that satisfy an Appell relation with respect to such an operator. Second, taking into account a cubic decomposition of a q-Dunkl–Appell sequence, we prove that sequences of their polynomial components exhibit <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Lambda _{k, l}^{\varvec{\theta }, \varvec{q}, \varvec{a}, \varvec{b}}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Λ</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> </mrow> <mrow> <mrow> <mi mathvariant="bold-italic">θ</mi> </mrow> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">q</mi> </mrow> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">a</mi> </mrow> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">b</mi> </mrow> </mrow> </msubsup> </math></EquationSource> </InlineEquation>-Appell properties, but expressed through a product of three operators related to the q-Dunkl operator. Generating functions of the principal components in the cubic decomposition of a q-Dunkl–Appell monic polynomial sequence are deduced.

Lowering operators associated with the cubic decomposition of q-Dunkl–Appell polynomial sequences

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发表日期：

2026年4月21日

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Mohamed Khalfallah,
Francisco Marcellán

摘要

In this paper, we deal with a lowering operator, denoted by $\Lambda _{k, l}^{\varvec{\theta }, \varvec{q}, \varvec{a}, \varvec{b}},$ $Λ_{k, l}^{θ, q, a, b},$ defined as a finite product of lowering operators in terms of the q-Dunkl operator. Our first goal is to characterize polynomial sequences that satisfy an Appell relation with respect to such an operator. Second, taking into account a cubic decomposition of a q-Dunkl–Appell sequence, we prove that sequences of their polynomial components exhibit $\Lambda _{k, l}^{\varvec{\theta }, \varvec{q}, \varvec{a}, \varvec{b}}$ $Λ_{k, l}^{θ, q, a, b}$ -Appell properties, but expressed through a product of three operators related to the q-Dunkl operator. Generating functions of the principal components in the cubic decomposition of a q-Dunkl–Appell monic polynomial sequence are deduced.