A Sufficient and Necessary Condition for the Power-Exponential Function <inline-formula><math display="inline"><semantics><msup><mfenced separators="" open="(" close=")"><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mfenced><mrow><mi>α</mi><mi>x</mi></mrow></msup></semantics></math></inline-formula> to Be a Bernstein Function and Related <i>n</i>th Derivatives

Abstract

In the paper, the authors find a sufficient and necessary condition for the power-exponential function 1+1xαx to be a Bernstein function, derive closed-form formulas for the nth derivatives of the power-exponential functions 1+1xαx and (1+x)α/x, and present a closed-form formula of the partial Bell polynomials Bn,k(H0(x),H1(x),⋯,Hn−k(x)) for n≥k≥0, where Hk(x)=∫0∞eu−1−ueuuk−1e−xudu for k≥0 are completely monotonic on (0,∞).

Keywords

• Bernstein function
• sufficient and necessary condition
• power-exponential function
• completely monotonic function
• partial Bell polynomial
• derivative