LOWER ESTIMATES OF POLYNOMIALS

OF THE VALUES OF E-FUNCTIONS

V. V. Zudilin (Moscow)

Assume that the system of homogeneous linear differential equations

\frac{d}{dz} y_{l} = ∑_{j=1}^{m} Q_{lj}y_{j}, l = 1, …, m, m ⩾ 2, Q_{lj} = Q_{lj}(z) ∈ ℂ(z),

(1)

has a fundamental matrix of solutions with entries homogeneously algebraically independent over ℂ(z). Let f_{1}(z), …, f_{m}(z) be a collection of E-functions with rational coefficients of Taylor series that satisfies the system (1). Also let α ∈ ℚ \ {0} be an arbitrary non-singular point of this system and let d ∈ ℕ. Then there exist positive constants γ and C dependent on the E-functions in question, α, and d such that

|P (f_{1}(α), …, f_{m}(α))| > C · |h_{1} ⋯ h_{w}|^{−1}H^{1−γ(log log H)^{−1/(m^{2}−m+2)}},

H = max_{1⩽i⩽w}{|h_{i}|},

for each non-trivial homogeneous polynomial P ∈ ℤ[y_{1}, …, y_{m}] of degree d. Here the h_{1}, …, h_{w} are the non-zero coefficients of P (y_{1}, …, y_{m}).

This general result can be applied to the values of generalized hypergeometric functions with rational parameters.

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