On an extremal problem of Selberg

We consider the minimum value m_p of the product under the conditions that and . This problem stems from estimation of the Selberg integral appeared in the study of integral-valued entire functions [A. Selberg, Über einen Satz von A. Gelfond, Arch. Math. Naturvid. 44 (1941) 159–170]. We give upper and lower estimates of m_p : . This, in particular, slightly improves the earlier lower estimate given by Selberg: logm_p>p·0.31654924….     

On an extremal problem of Selberg II

Let m_p be the minimum of the product under the conditions that and . In our previous paper [J. Kaneko, On an extremal problem of Selberg, J. Approx. Theory 142 (2006) 129–137], we showed that the following estimates hold. provided p255. In this note, we prove that the limit of as p→∞ exists and is expressed by the (unique) solution of some simultaneous transcendental equations. By using this expression we obtain numerically.     

Pre-evaluation of metal ions as a catalyst on chemiluminometric sequential injection analysis with luminol-H2O2 system

Catalytic effect of metal ions on luminol chemiluminescence (CL) was investigated by sequential injection analysis (SIA). The SIA system was set up with two solenoid micropumps, an eight-port selection valve, and a photosensor module with a fountain-type chemiluminescence cell. The SIA system was controlled and the CL signals were collected by a LabVIEW program. Aqueous solutions of luminol, H₂O₂, and a sample solution containing metal ion were sequentially aspirated to the holding coil, and the zones were immediately propelled to the detection cell. After optimizing the parameters using 1 × 10⁻⁵ M Fe³⁺ solution, catalytic effect of some metal species was compared. Among 16 metal species examined, relatively strong CL responses were obtained with Fe³⁺, Fe²⁺, VO²⁺, VO₃⁻, MnO₄⁻, Co²⁺, and Cu²⁺. The limits of detection by the present SIA system were comparable to FIA systems. Permanganate ion showed the highest CL sensitivity among the metal species examined; the calibration graph for MnO₄⁻ was linear at the concentration level of 10⁻⁸ M and the limit of detection for MnO₄⁻ was 4.0 × 10⁻¹⁰ M (S/N = 3).     

A 1Ψ1 Summation Theorem for Macdonald Polynomials

In this note we extend the Ramanujan's 11 summation formula to the case of a Laurent series extension of multiple q-hypergeometric series of Macdonald polynomial argument [7]. The proof relies on the elegant argument of Ismail [5] and the q-binomial theorem for Macdonald polinomials. This result implies a q-integration formula of Selberg type [3, Conjecture 3] which was proved by Aomoto [2], see also [7, Appendix 2] for another proof. We also obtain, as a limiting case, the triple product identity for Macdonald polynomials [8].