Twin MRM-triples in multiplicative renormalization method

Abstract

MRM-applicable measures for MRM-factors 0 F 0 (−; −; x) = e x , 1 F 0 (κ; −; x) = (1 − x) −κ and 1 F 1 (1; 2; x) = 1 x (e x − 1) have already been completely determined. Conversely, MRM-factors for which those measures are MRM-applicable have also been completely determined. Some measures have couples of MRM-triples h(x), ρ(t), B(t) and h 0 (x), ρ 0 (t), B 0 (t) such that h 0 (x) = d dx xh(x) , B(t) = 1 t ρ(t), ρ 0 (t) = ρ(t), B 0 (t) = ρ (t) = d dt tB(t). Such triples are called twin MRM-triples. We determine in this paper all twin MRM-triples for symmetric measures and for measures with the Jacobi-Szegö parameters αn = a 2 n 2 + a 1 n + a 0 .