Conjugate $(1/q, q)$-harmonic Polynomials in $q$-Clifford Analysis
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arXiv
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| Format: | Preprint |
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2025
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| author | Bernstein, Swanhild Altavilla, Amedeo Zimmermann, Martha Lina |
| author_facet | Bernstein, Swanhild Altavilla, Amedeo Zimmermann, Martha Lina |
| contents | We consider the problem of constructing a conjugate $(1/q, q)$-harmonic homogeneous polynomial $V_k$ of degree $k$ to a given $(1/q, q)$-harmonic homogeneous polynomial $U_k$ of degree $k.$ The conjugated harmonic polynomials $V_k$ and $U_k$ are associated to the $(1/q, q)$-mono\-genic polynomial $F = U_k + \overline{e}_0V. $ We investigate conjugate $(1/q, q)$-harmonic homogeneous polynomials in the setting of $q$-Clifford analysis. Starting from a given $(1/q, q)$-harmonic polynomial $U_k$ of degree $k$, we construct its conjugate counterpart $V_k$, such that the Clifford-valued polynomial $F = U_k + e_0 V_k$ is $(1/q, q)$-monogenic, i.e., a null solution of a generalized $q$-Dirac operator. Our construction relies on a combination of Jackson-type integration, Fischer decomposition, and the resolution of a $q$-Poisson equation. We further establish existence and uniqueness results, and provide explicit representations for conjugate pairs, particularly when $U_k$ is real-valued. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_09585 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Conjugate $(1/q, q)$-harmonic Polynomials in $q$-Clifford Analysis Bernstein, Swanhild Altavilla, Amedeo Zimmermann, Martha Lina Complex Variables 30G35, 39A70, 81A99 We consider the problem of constructing a conjugate $(1/q, q)$-harmonic homogeneous polynomial $V_k$ of degree $k$ to a given $(1/q, q)$-harmonic homogeneous polynomial $U_k$ of degree $k.$ The conjugated harmonic polynomials $V_k$ and $U_k$ are associated to the $(1/q, q)$-mono\-genic polynomial $F = U_k + \overline{e}_0V. $ We investigate conjugate $(1/q, q)$-harmonic homogeneous polynomials in the setting of $q$-Clifford analysis. Starting from a given $(1/q, q)$-harmonic polynomial $U_k$ of degree $k$, we construct its conjugate counterpart $V_k$, such that the Clifford-valued polynomial $F = U_k + e_0 V_k$ is $(1/q, q)$-monogenic, i.e., a null solution of a generalized $q$-Dirac operator. Our construction relies on a combination of Jackson-type integration, Fischer decomposition, and the resolution of a $q$-Poisson equation. We further establish existence and uniqueness results, and provide explicit representations for conjugate pairs, particularly when $U_k$ is real-valued. |
| title | Conjugate $(1/q, q)$-harmonic Polynomials in $q$-Clifford Analysis |
| topic | Complex Variables 30G35, 39A70, 81A99 |
| url | https://arxiv.org/abs/2504.09585 |