Difference Independence of the Euler Gamma Function

Abstract

In this paper, the authors established a sharp version of the difference analogue of the celebrated Hölder’s theorem concerning the differential independence of the Euler gamma function Γ. More precisely, if P is a polynomial of n + 1 variables in ℂ[X, Y ₀, ⋯, Y _n−1] such that

$P(s,\Gamma (s + {a_0}), \cdots ,\Gamma (s + {a_{n - 1}})) \equiv 0$

for some (a ₀, ⋯, a _n−1) ∈ ℂ ⁿ and a _i − a _j ∉ ℤ for any 0 ≤ i ≤ j ≤ n − 1, then they have

$P \equiv 0.$.

Their result complements a classical result of algebraic differential independence of the Euler gamma function proved by Hölder in 1886, and also a result of algebraic difference independence of the Riemann zeta function proved by Chiang and Feng in 2006.

Keywords

Algebraic difference independence / Euler gamma function / Algebraic difference equations

References

[1] Adamczewski B, Bell J P, Delaygue E. Algebraic independence of G-functions and congruences à la Lucas. Ann. Sci. Éc. Norm. Supér., 2019, 52(4): 515-559

[2] Bank S B, Kaufman R P. A note on Hölder’s theorem concerning the gamma function. Math. Ann., 1978, 232(2): 115-120

[3] Bank S B, Kaufman R P. On differential equations and functional equations. J. Reine Angew. Math., 1979, 311(312): 31-41

[4] Chiang Y M, Feng S J. Difference independence of the Riemann zeta function. Acta Arith., 2006, 125(4): 317-329

[5] Hardouin C. Hypertranscendance des systèmes aux différences diagonaux. Compos. Math., 2008, 144(3): 565-581

[6] Hayman W K. Meromorphic Functions, 1964, Oxford: Clarendon Press

[7] Hilbert D. Mathematical problems. Bull. Amer. Math. Soc., 1902, 8(10): 437-479

[8] Hölder O. Über die eigenschaft der γ-funktion, keiner algebraischen differentialgleichung zu genügen. Math. Ann., 1886, 28(10): 1-13

[9] Li B Q, Ye Z. On differential independence of the Riemann zeta function and the Euler gamma function. Acta Arith., 2008, 135(4): 333-337

[10] Li B Q, Ye Z. Algebraic differential equations concerning the Riemann zeta function and the Euler gamma function. Indiana Univ. Math. J., 2010, 59(4): 1405-1415

[11] Li B Q, Ye Z. Algebraic differential equations with functional coefficients concerning ζ and Γ. J. Differential Equations, 2016, 260(2): 1456-1464

[12] Liao L W, Yang C C. On some new properties of the gamma function and the Riemann zeta function. Math. Nachr., 2003, 257: 59-66

[13] Markus L. Differential independence of Γ and ζ. J. Dynam. Differential Equations, 2007, 19(1): 133-154

[14] Ostrowski A. Über Dirichletsche Reihen und algebraische Differentialgleichungen. Math. Z., 1920, 8(3–4): 241-298

[15] Steuding J. Value-distribution of L-function, 2007, Berlin: Springer-Verlag

[16] Voronin S M. The distribution of the nonzero values of the Riemann ζ-function. Trudy Mat. Inst. Steklov., 1972, 128: 131-150

[17] Voronin S M. Theorem on the “universality” of the Riemann zeta-function. Izv. Akad. Nauk SSSR Ser. Mat., 1975, 39(3): 475-486