A Sufficient and Necessary Condition for the Power-Exponential Function 1+1xαx to Be a Bernstein Function and Related Nth Derivatives
Fractal and fractional(2023)
摘要
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,∞).
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关键词
Bernstein function,sufficient and necessary condition,power-exponential function,completely monotonic function,partial Bell polynomial,derivative,closed-form formula,Descartes ' rule of signs,zero of polynomial
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