Complete Monotonicity of Special Functions

Abstract

In this work we prove that if an entire function $f(z)$ is of order strictly less than one and it has only negative zeros, then for each nonnegative integer $k,m$ the real function $\left(-\frac{1}{x}\right)^{m}\frac{d^{k}}{dx^{k}}\left(x^{k+m}\frac{d^{m}}{dx^{m}}\left(\frac{f'(x)}{f(x)}\right)\right)$ is completely monotonic on $(0,\infty)$. Applications to Askey-Wilson polynomials, Bessel functions, Ramanujan's entire function, Riemann-xi function and character Riemann-xi functions are also provided.