Hausdorff moment sequences and hypergeometric functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909379406594048 |
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| author | Sugawa, Toshiyuki Wang, Li-Mei |
| author_facet | Sugawa, Toshiyuki Wang, Li-Mei |
| contents | Pólya in 1926 showed that the hypergeometric function $F(z)=\null_2F_1(a,b;c;z)$ has a totally monotone sequence as its coefficients; that is, $F$ is the generating function of a Hausdorff moment sequence, when $0\le a\le 1$ and $0\le b\le c.$ In this paper, we give a complete characterization of such hypergeometric functions $F$ in terms of complex parameters $a,b,c.$ To this end, we study the class of general properties of generating functions of Hausdorff moment sequences and, in particular, we provide a sufficient condition for the class by making use of a Phragmèn-Lindelöf type theorem. As an application, we give also a necessary and sufficient condition for a shifted hypergeometric function to be universally starlike. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_04345 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hausdorff moment sequences and hypergeometric functions Sugawa, Toshiyuki Wang, Li-Mei Complex Variables Primary 33C05, Secondary 30E05 Pólya in 1926 showed that the hypergeometric function $F(z)=\null_2F_1(a,b;c;z)$ has a totally monotone sequence as its coefficients; that is, $F$ is the generating function of a Hausdorff moment sequence, when $0\le a\le 1$ and $0\le b\le c.$ In this paper, we give a complete characterization of such hypergeometric functions $F$ in terms of complex parameters $a,b,c.$ To this end, we study the class of general properties of generating functions of Hausdorff moment sequences and, in particular, we provide a sufficient condition for the class by making use of a Phragmèn-Lindelöf type theorem. As an application, we give also a necessary and sufficient condition for a shifted hypergeometric function to be universally starlike. |
| title | Hausdorff moment sequences and hypergeometric functions |
| topic | Complex Variables Primary 33C05, Secondary 30E05 |
| url | https://arxiv.org/abs/2411.04345 |