Generalizing the Index of the $u$-Deformed Homogeneous Polynomials and Generating Functions for $\mathrm{R}_{-n}(x,y;q|u)$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908812219252736 |
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| author | López, Ronald Orozco |
| author_facet | López, Ronald Orozco |
| contents | This paper introduces the $u$-deformed homogeneous functions $\mathrm{R}_α(x,y;u|q)$, for all $α\in\mathbb{C}$. Basic properties of the functions $\mathrm{R}_α(x,y;u|q)$ are given, along with recurrence relations, their $q$-difference equation, and representations. Generating functions for the functions $\mathrm{R}_{-n}(x,y;u|q)$ via the $u$-deformed $q$-exponential operator E$(qD_{q}|u)$, when $u=q^b,q^{b+1/2}$, $b\geq1$, are obtained. This allows us to obtain some $q^b$-series and $q^{b+1/2}$-series. Additionally, transformation formulas for basic hypergeometric series $_{1}ϕ_{1}$ and $_{1}ϕ_{2}$ are derived. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08943 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalizing the Index of the $u$-Deformed Homogeneous Polynomials and Generating Functions for $\mathrm{R}_{-n}(x,y;q|u)$ López, Ronald Orozco Combinatorics Primary 05A30. Secondary 11B39, 33D15, 33D45 This paper introduces the $u$-deformed homogeneous functions $\mathrm{R}_α(x,y;u|q)$, for all $α\in\mathbb{C}$. Basic properties of the functions $\mathrm{R}_α(x,y;u|q)$ are given, along with recurrence relations, their $q$-difference equation, and representations. Generating functions for the functions $\mathrm{R}_{-n}(x,y;u|q)$ via the $u$-deformed $q$-exponential operator E$(qD_{q}|u)$, when $u=q^b,q^{b+1/2}$, $b\geq1$, are obtained. This allows us to obtain some $q^b$-series and $q^{b+1/2}$-series. Additionally, transformation formulas for basic hypergeometric series $_{1}ϕ_{1}$ and $_{1}ϕ_{2}$ are derived. |
| title | Generalizing the Index of the $u$-Deformed Homogeneous Polynomials and Generating Functions for $\mathrm{R}_{-n}(x,y;q|u)$ |
| topic | Combinatorics Primary 05A30. Secondary 11B39, 33D15, 33D45 |
| url | https://arxiv.org/abs/2408.08943 |