For all integers n and nonnegative integers t, the harmonic logarithms λ_n^(t)(x) of order t and degree n are defined as the unique functions satisfying 1.λ_0^(t)(x) = (ln x)^t, 2.λ_n^(t)(x) has no constant term except λ_0^(0)(x) = 1, 3.d/(d x) λ_n^(t)(x) = ⌊n⌉ λ_(n - 1)^(t)(x), where the "Roman symbol" ⌊n⌉ is defined by ⌊n⌉ congruent {n | for n!=0 1 | for n = 0 auto right match (Roman 1992).

logarithm | Roman factorial