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The zero-count refutation

Abstract

A two-component Gamma quotient violates the proposed comparison of interior zero counts.

Theorem 1.1 (The counterexample family).

Lean statement: D5/S3/Analytic/MeijerZeroCount/Result.family

Proof. Machine-checked in Lean as D5/S3/Analytic/MeijerZeroCount/Result.family (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. D. B. Karp and E. G. Prilepkina (2016). Completely monotonic gamma ratio and infinitely divisible H-function of Fox. DOI: 10.1007/s40315-015-0128-9. URL: https://arxiv.org/abs/1501.05388.

Commentary.

For every natural n and d at least two and every positive real c, set a=(n+c,n+c) and b=(c,2n+d+c). In the unit interval with endpoints excluded, Meijer’s G has exactly n distinct zeros and the control function has exactly one zero. Mellin inversion identifies G with the Rodrigues kernel, whose zero count follows from Rolle’s theorem and the polynomial degree. The control zero count follows from strict monotonicity of its normalized factor.

Theorem 1.2 (Failure of the comparison).

Lean statement: D5/S3/Analytic/MeijerZeroCount/Result.result

Proof. Machine-checked in Lean as D5/S3/Analytic/MeijerZeroCount/Result.result (✓ std3). ∎

Resolves. Problems/karp-prilepkina-2015-meijer-zero-count (refuted) by D5/S3/Analytic/MeijerZeroCount/Result.result.

Source. Repository-derived.

Acknowledgement. D. B. Karp and E. G. Prilepkina (2016). Completely monotonic gamma ratio and infinitely divisible H-function of Fox. DOI: 10.1007/s40315-015-0128-9. URL: https://arxiv.org/abs/1501.05388.

Commentary.

The proposition claim is false. Taking n=d=2 and c=1 gives nonnegative a=(3,3), b=(1,7), and sum of b(i)-a(i) equal to two, which is greater than one. The interior zero counts are two for G and one for the control function, contradicting the proposed inequality. The family theorem also exhibits arbitrarily large discrepancies while keeping two components and d=2.

References