Monotonicity Properties of Determinants of Special Functions

Abstract

We prove the absolute monotonicity or complete monotonicity of some determinant functions whose entries involve $$\psi^{(m)}(x)=({d^m}/{dx^m}) [\Gamma'(x)/\Gamma(x)],$$ modified Bessel functions Iν, Kν, the confluent hypergeometric function Φ, and the Tricomi function Ψ. Our results recover and generalize some known determinantal inequalities. We also show that a certain determinant formed by the Fibonacci numbers are nonnegative while determinants involving Hermite polynomials of imaginary arguments are shown to be completely monotonic functions.