2017 Fiscal Year Final Research Report
Study on irreducibility of algebraic difference equations and differential transcendence of solutions
| Project/Area Number |
26800049
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Basic analysis
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| Research Institution | Yamagata University |
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Principal Investigator |
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| Project Period (FY) |
2014-04-01 – 2018-03-31
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| Keywords | 差分方程式 / 差分代数 / 既約性 / 超超越性 / 差分リッカチ方程式 |
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| Outline of Final Research Achievements |
We say that a function is differentially transcendental when it does not satisfy any algebraic differential equation. There are various studies on differential transcendence of functions satisfying difference equations. O. Hoelder's study on Gamma function is typical. In 1905, H. Tietze studied differential transcendence of solutions of difference Riccati equations, and obtained a sufficient condition. In this study, H. Tietze's result is made purely algebraic, and applicable to difference Riccati equations with other transforming operators such as one of q-difference, Mahler type, etc. As an application, it is seen that a solution of q-Airy equation is differentially transcendental when q is not a root of unity. The theory and the method for proving irreducibility of q-Painleve euqations are found to be applicable to a d-Painleve equation.
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| Free Research Field |
差分代数
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