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**A generalization of the Nielsen-Hölder theorem about \(\Gamma(z)\).**
*(English)*
Zbl 1054.32003

Introduction: A classical theorem of N. Nielsen [ Die Gammafunktion. Band I. Handbuch der Theorie der Gammafunktion. Band II, Chelsea, New York (1965); Leipzig: Teubner (1906; JFM 37.0450.01)], that generalized a weaker result due to Hölder, says that the gamma function, \(\Gamma(z)\), is not a solution of any nontrivial algebraic differential equation. Nielsen’s theorem is equivalent to saying that the transcendence degree (over \(\mathbb C\)) of the field \(\mathbb C\) with \(z\) and all of the derivatives of \(\Gamma(z)\) adjoined is infinite. We further generalize Hölder’s result to Theorem I. Let \(n\) denote a natural number. There does not exist a nonidentically zero entire function of \(n+2\) complex variables \(F(w_1,w_2,\cdots,w_{n+1},z)\), which is polynomial in \(z\) that vanishes identically when each \(w_j\) is replaced by \((\Gamma(z))^{(j-1)}\), for \(j=1,2,\cdots,n\). Since \(\Gamma(z)\) has no zeros, Theorem I is really just an application of a more general result below, which represents a generalization of the theorem in [F. Gross and C. F. Osgood, Complex Variables Theory Appl. 29, No. 3, 261–263 (1996; Zbl 0856.30022)].

Consider a differential equation of the form (1) \(F(z,y,y^{(1)},\cdots,y^{(n)})=0\), or \(F=0\), where \(F(z,z_0,z_1,\cdots,z_n)\) is a nonzero entire function of \(n+2\) complex variables that is polynomial in \(z\). Suppose that \(f\) is a nonconstant meromorphic function that is a solution of \(F=0\). Suppose further that for some \(\epsilon>0, f\) has a defect \(\delta>\epsilon\) at a complex number \(a\).

Theorem II. Under the above conditions, there exists a nonzero polynomial \(P(z_0,z_1,\cdots,z_n)\) in the \(n+1\) complex variables \(z_0,z_1,\cdots,z_i,\cdots,z_n\) such that \(P(y,y^{(1)},\cdots,y^{(n)})=0\) has \(f\) as a solution. Further, the coefficients of \(P\) can be chosen to not depend explicitly upon ‘\(a\)’ but only upon the degree of \(F\) in \(z\), upon the total degree of the first nonzero term of the expansion of \(F\) at \((a,0,\cdots,0)\), and upon \(\epsilon\).

Consider a differential equation of the form (1) \(F(z,y,y^{(1)},\cdots,y^{(n)})=0\), or \(F=0\), where \(F(z,z_0,z_1,\cdots,z_n)\) is a nonzero entire function of \(n+2\) complex variables that is polynomial in \(z\). Suppose that \(f\) is a nonconstant meromorphic function that is a solution of \(F=0\). Suppose further that for some \(\epsilon>0, f\) has a defect \(\delta>\epsilon\) at a complex number \(a\).

Theorem II. Under the above conditions, there exists a nonzero polynomial \(P(z_0,z_1,\cdots,z_n)\) in the \(n+1\) complex variables \(z_0,z_1,\cdots,z_i,\cdots,z_n\) such that \(P(y,y^{(1)},\cdots,y^{(n)})=0\) has \(f\) as a solution. Further, the coefficients of \(P\) can be chosen to not depend explicitly upon ‘\(a\)’ but only upon the degree of \(F\) in \(z\), upon the total degree of the first nonzero term of the expansion of \(F\) at \((a,0,\cdots,0)\), and upon \(\epsilon\).