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| Format: | Preprint |
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2019
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| Online Access: | https://arxiv.org/abs/1910.06817 |
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| _version_ | 1866912474703331328 |
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| author | Fischler, S. Rivoal, T. |
| author_facet | Fischler, S. Rivoal, T. |
| contents | Siegel defined in 1929 two classes of power series, the E-functions and G-functions, which generalize the Diophantine properties of the exponential and logarithmic functions respectively. In 1949, he asked whether any E-function can be represented as a polynomial with algebraic coefficients in a finite number of hypergeometric E-functions with rational parameters. The case of E-functions of differential order less than 2 was settled in the affirmative by Gorelov in 2004, but Siegel's question is open for higher order. We prove here that if Siegel's question has a positive answer, then the ring G of values taken by analytic continuations of G-functions at algebraic points must be a subring of the relatively "small" ring H generated by algebraic numbers, $1/π$ and the values of the derivatives of the Gamma function at rational points. Because that inclusion seems unlikely (and contradicts standard conjectures), this points towards a negative answer to Siegel's question in general. As intermediate steps, we first prove that any element of G is a coefficient of the asymptotic expansion of a suitable E-function, which completes previous results of ours. We then prove (in two steps) that the coefficients of the asymptotic expansion of an hypergeometric E-function with rational parameters are in H. Finally, we prove a similar result for G-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_06817 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On Siegel's problem for E-functions Fischler, S. Rivoal, T. Number Theory Siegel defined in 1929 two classes of power series, the E-functions and G-functions, which generalize the Diophantine properties of the exponential and logarithmic functions respectively. In 1949, he asked whether any E-function can be represented as a polynomial with algebraic coefficients in a finite number of hypergeometric E-functions with rational parameters. The case of E-functions of differential order less than 2 was settled in the affirmative by Gorelov in 2004, but Siegel's question is open for higher order. We prove here that if Siegel's question has a positive answer, then the ring G of values taken by analytic continuations of G-functions at algebraic points must be a subring of the relatively "small" ring H generated by algebraic numbers, $1/π$ and the values of the derivatives of the Gamma function at rational points. Because that inclusion seems unlikely (and contradicts standard conjectures), this points towards a negative answer to Siegel's question in general. As intermediate steps, we first prove that any element of G is a coefficient of the asymptotic expansion of a suitable E-function, which completes previous results of ours. We then prove (in two steps) that the coefficients of the asymptotic expansion of an hypergeometric E-function with rational parameters are in H. Finally, we prove a similar result for G-functions. |
| title | On Siegel's problem for E-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/1910.06817 |