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 $$\epsilon$$.

MSC:

 32A60 Zero sets of holomorphic functions of several complex variables 32A15 Entire functions of several complex variables

Citations:

JFM 37.0450.01; Zbl 0856.30022
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