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A note on differentially algebraic solutions of first order linear difference equations. (English) Zbl 0542.12012
In 1887, O. Hölder [Math. Ann. 28, 1–13 (1886; JFM 18.0440.02] showed that the \(\Gamma\)-function satisfies no algebraic differential equation over \(\mathbb C(x)\), that is, it satisfies no equation of the form \(P(x,y,y',\ldots,y^{(n)})=0\) where \(P\) is a polynomial with complex coefficients. He did this by showing that the difference equation \(f(x+1)=f(x)+1/x\) (satisfied by \(f=\Gamma '/\Gamma)\) has no such solution. Using similar methods, E. H. Moore showed, in [Math. Ann. 48, 49–74 (1897; JFM 27.0307.01)], that \(f(nx)=f(x)-e^ x\) has no solution that also satisfies an algebraic differential equation over \(\mathbb C(x,e^ x).\)
In this paper the author puts this in difference and differential algebraic terms and gives necessary and sufficient conditions for a first order linear difference equation over a general differential-difference field to have a solution differentially algebraic over that field. He then shows how Hölder’s and Moore’s results can be derived from this theorem.

12H05 Differential algebra
12H10 Difference algebra
Full Text: DOI EuDML
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