Some transcendental functions over function fields with positive characteristic. (English. Abridged French version) Zbl 1074.11038
Summary: We define two families of functions over function fields with positive characteristic and show that such a function is transcendental if and only if its generating sequence is not ultimately zero. As a result, the Carlitz exponential and the Carlitz logarithm are transcendental functions. Our proof is elementary in the sense that we only use a theorem due to H. Sharif and C. Woodcock [J. Lond. Math. Soc. (2) 37, No. 3, 395–403 (1988; Zbl 0612.12018)], and to T. Harase [Isr. J. Math. 63, No. 3, 281–288 (1988; Zbl 0675.13015)] which generalizes the theorem of Christol about automatic sequences [G. Christol, Theor. Comput. Sci. 9, No. 1, 141–145 (1979; Zbl 0402.68044); G. Christol et al., Bull. Soc. Math. Fr. 108, No. 4, 401–419 (1980; Zbl 0472.10035)].
MSC:
| 11J81 | Transcendence (general theory) |
| 11B85 | Automata sequences |
| 11J91 | Transcendence theory of other special functions |
References:
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