Charting the $q$-Askey scheme. III. Verde-Star scheme for $q=1$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915093788229632 |
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| author | Koornwinder, Tom H. |
| author_facet | Koornwinder, Tom H. |
| contents | Following Verde-Star, Linear Algebra Appl. 627 (2021), we label families of orthogonal polynomials in the $q=1$ Askey scheme together with their hypergeometric representations by three sequences $x_k, h_k, g_k$ of polynomials in $k$, two of degree 2 and one of degree 4, satisfying certain constraints. Except for the Hermite polynomials, this gives rise to a precise classification and a very simple uniform parametrization of these families together with their limit transitions. This is displayed in a graphical scheme. We also discuss limits from the $q$-case to the case $q=1$, although this cannot be done in a uniform way. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_06668 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Charting the $q$-Askey scheme. III. Verde-Star scheme for $q=1$ Koornwinder, Tom H. Classical Analysis and ODEs 33C45 Following Verde-Star, Linear Algebra Appl. 627 (2021), we label families of orthogonal polynomials in the $q=1$ Askey scheme together with their hypergeometric representations by three sequences $x_k, h_k, g_k$ of polynomials in $k$, two of degree 2 and one of degree 4, satisfying certain constraints. Except for the Hermite polynomials, this gives rise to a precise classification and a very simple uniform parametrization of these families together with their limit transitions. This is displayed in a graphical scheme. We also discuss limits from the $q$-case to the case $q=1$, although this cannot be done in a uniform way. |
| title | Charting the $q$-Askey scheme. III. Verde-Star scheme for $q=1$ |
| topic | Classical Analysis and ODEs 33C45 |
| url | https://arxiv.org/abs/2307.06668 |