Rational approximation of Euler's constant using multiple orthogonal polynomials
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908380801531904 |
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| author | Wolfs, Thomas Van Assche, Walter |
| author_facet | Wolfs, Thomas Van Assche, Walter |
| contents | We construct new rational approximants of Euler's constant that improve those of Aptekarev et al. (2007) and Rivoal (2009). The approximants are given in terms of certain (mixed type) multiple orthogonal polynomials associated with the exponential integral. The dual family of multiple orthogonal polynomials leads to new rational approximants of the Gompertz constant that improve those of Aptekarev et al. (2007). Our approach is motivated by the fact that we can reformulate Rivoal's construction in terms of type I multiple Laguerre polynomials of the first kind by making use of the underlying Riemann-Hilbert problem. As a consequence, we can drastically simplify Rivoal's approach, which allows us to study the Diophantine and asymptotic properties of the approximants more easily. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_09799 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rational approximation of Euler's constant using multiple orthogonal polynomials Wolfs, Thomas Van Assche, Walter Number Theory Classical Analysis and ODEs 11J13, 11J72, 33C45, 42C05 We construct new rational approximants of Euler's constant that improve those of Aptekarev et al. (2007) and Rivoal (2009). The approximants are given in terms of certain (mixed type) multiple orthogonal polynomials associated with the exponential integral. The dual family of multiple orthogonal polynomials leads to new rational approximants of the Gompertz constant that improve those of Aptekarev et al. (2007). Our approach is motivated by the fact that we can reformulate Rivoal's construction in terms of type I multiple Laguerre polynomials of the first kind by making use of the underlying Riemann-Hilbert problem. As a consequence, we can drastically simplify Rivoal's approach, which allows us to study the Diophantine and asymptotic properties of the approximants more easily. |
| title | Rational approximation of Euler's constant using multiple orthogonal polynomials |
| topic | Number Theory Classical Analysis and ODEs 11J13, 11J72, 33C45, 42C05 |
| url | https://arxiv.org/abs/2404.09799 |