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Every shift strictly between zero and one

Abstract

Negative reciprocal Gamma Turanian coefficients at boundary shifts.

Theorem 1.1 (The second coefficient is strictly negative).

Lean statement: D5/S3/Analytic/ReciprocalGammaTuran/Boundary.d_two_neg

Proof. Machine-checked in Lean as D5/S3/Analytic/ReciprocalGammaTuran/Boundary.d_two_neg (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every real t strictly between zero and one, the constant unit sequence at mu=-t and alpha=beta=t has a strictly negative second coefficient. Gamma recurrence evaluates this coefficient as 1-(1+t^2)/(t(1-t^2)Gamma(t)Gamma(1-t)). Euler reflection reduces its sign to a strict lower bound for sin(pi t).

The finite Euler product is bounded below by (1-t^2)(1-t^2/4)(1+t^2/(n+2))/(1+t^2/2). The induction uses the exact difference u(1-u)/(a(a+1)^2) between (1+u/a)(1-u/(a+1)^2) and 1+u/(a+1), with u=t^2. Passing to the sine-product limit gives sin(pi t) at least pi t times (1-t^2)(1-t^2/4)/(1+t^2/2), which is strictly greater than pi t(1-t^2)/(1+t^2). This statement concerns degree two and does not assert a formula for every degree.

References