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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.

MSC:
12H05 Differential algebra
12H10 Difference algebra
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References:
[1] Cohn, R. M.,Difference algebra. Interscience Tracts in Pure and Applied Math. 17, Wiley, New York, 1965.
[2] Hölder, O.,Über die Eigenschaft der Gammafunction keiner algebraischen Differentialgleichung zu genügen. Math. Ann.28 (1887), 1–13. · JFM 18.0440.02
[3] Kolchin, E. R.,Differential algebra and algebraic groups. Pure and Applied Math.: A Series of Monographs and Textbooks, 54, Academic Press, New York, 1973.
[4] Loxton, J. H. andvan der Poorten, A. J.,A class of hypertranscendental functions. Aequationes Math.16 (1977), 93–106. · Zbl 0384.10014
[5] Moore, E. H.,Concerning transcendentally transcendental functions. Math. Ann.48 (1897), 49–74. · JFM 27.0307.01
[6] Stridsberg, E.,Contributions à l’étude des fonctions algébrico-transcendentes qui satisfont à certaines équations fonctionnelles algébriques. Ark Mat, Astronom Fys.6 (1910), no. 15 and no. 18. · JFM 41.0374.01
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