1998 Fiscal Year Final Research Report Summary
THEORY OF MAHLER FUNCTIONS AND APPLICATIONS TO REGULAR SEQUENCES
| Project/Area Number |
09640060
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | KEIO UNIVERSITY |
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Principal Investigator |
NISHIOKA Kumiko KEIO UNIV., ECONOMICS,PROF., 経済学部, 教授 (80144632)
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| Co-Investigator(Kenkyū-buntansha) |
NISHIOKA Keiji KEIO UNIV., ENV.INFO., PROF., 環境情報学部, 教授 (10228158)
KOMIYA Hidetoshi KEIO UNIV., COMMERCE,PROF., 商学部, 教授 (90153676)
WATABE Mutsuo KEIO UNIV., COMMERCE,PROF., 商学部, 教授 (30080493)
HIKARI Michitaka KEIO UNIV., ECONOMICS,PROF., 経済学部, 教授 (30056296)
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| Project Period (FY) |
1997 – 1998
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| Keywords | transcendental number / algebraic independence / Mahler function / Fibonacci number / Theta series / finite automaton |
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| Research Abstract |
If a function f(z) satisfies a functional equation under the trasformation z <does not result in>z^d, f(z) is called a Mahler function. The transcendence and the algebraic independence of the values of Mauler functions have been studied for a long time. Until recently we have satisfactory results only in the case Mahler functions satisfy functional equations under a common trasformation z <does not result in>z^d. If we simultaneously treate Mahler functions under various transformations, a lot of difficult problems occur. Our research provides a solution to these problems. First we have obtained a widely applicable theorem on algebraic independence. Second we applied it to algebraic independence of reciprocal sums of Fibonacci numbers. This result gives an almost complete solution to a problem proposed by Erdos and Graham. We also proved that 2-adic expansions of real algebraic numbers can not be represented by finte automata, which means that 2-adic expansions of real algebraic numbers are complex in a certain sense.
On the other hand, applying Nesterenko's theorem on the algebraic independence of the values of modular functions, we obtained a transcendence result of the reciprocal sums of Fibonacci numbers which can not be proved by using Mahler functions. We also obtained the transcendence of Theta series and Rogers-Ramanujan continued fraction.
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