Absolute convergence of the fractional integral series for Re(β) > 0

Prove that for all complex parameters α and β with Re(α) > 0 and Re(β) > 0, the integral ∫_0^1 ∑_{n=0}^∞ (-1)^n binom(α−1, n) t^{n+β−1} log Γ(t) dt is absolutely convergent. Establishing this would extend Lemma BetaAbsConvLemma, which currently proves absolute convergence only for Re(β) ≥ 1, and would justify interchanging the order of summation and integration in the derivations of Theorem BetaLeftRaabe and Theorem BetaRightRaabe for the wider parameter range Re(β) > 0.

Background

The paper establishes Raabe-type formulas for the gamma function via Riemann-Liouville fractional integrals. A key technical step is to justify exchanging the order of summation and integration by verifying absolute convergence.

Lemma BetaAbsConvLemma proves that, for Re(α) > 0 and Re(β) ≥ 1, the integral ∫0^1 ∑{n=0}^∞ (−1)^n binom(α−1, n) t^{n+β−1} log Γ(t) dt is absolutely convergent, enabling the use of Fubini–Tonelli to swap sum and integral. The authors remark that numerical evidence suggests the same absolute convergence should hold for the broader range Re(β) > 0, which would in turn expand the applicability of Theorem BetaLeftRaabe and Theorem BetaRightRaabe, but they do not have a proof.