A unified approach to hypergeometric class functions
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866916828243034112 |
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| author | Dereziński, Jan |
| author_facet | Dereziński, Jan |
| contents | Hypergeometric class equations are given by second
order differential operators in one variable whose coefficient at the second derivative
is a polynomial of degree $\leq2$, at the first derivative of degree
$\leq1$ and the free term is a number. Their solutions, called hypergeometric class functions, include the Gauss hypergeometric function and its various limiting cases. The paper presents a unified approach to these functions. The main structure behind this approach is a family of complex 4-dimensional Lie algebras, originally due to Willard Miller. Hypergeometric class functions can be interpreted as eigenfunctions of the quadratic Casimir operator in a representation of Miller's Lie algebra given by differential operators in three complex variables. One obtains a unified treatment of various properties of hypergeometric class functions such as recurrence relations, discrete symmetries, power series expansions, integral representations, generating functions and orthogonality of polynomial solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00166 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A unified approach to hypergeometric class functions Dereziński, Jan Classical Analysis and ODEs Mathematical Physics Complex Variables Hypergeometric class equations are given by second order differential operators in one variable whose coefficient at the second derivative is a polynomial of degree $\leq2$, at the first derivative of degree $\leq1$ and the free term is a number. Their solutions, called hypergeometric class functions, include the Gauss hypergeometric function and its various limiting cases. The paper presents a unified approach to these functions. The main structure behind this approach is a family of complex 4-dimensional Lie algebras, originally due to Willard Miller. Hypergeometric class functions can be interpreted as eigenfunctions of the quadratic Casimir operator in a representation of Miller's Lie algebra given by differential operators in three complex variables. One obtains a unified treatment of various properties of hypergeometric class functions such as recurrence relations, discrete symmetries, power series expansions, integral representations, generating functions and orthogonality of polynomial solutions. |
| title | A unified approach to hypergeometric class functions |
| topic | Classical Analysis and ODEs Mathematical Physics Complex Variables |
| url | https://arxiv.org/abs/2502.00166 |