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A diagonal recurrence relation for the Stirling numbers of the first kind

Qi Feng (Henan Polytechnic University, Institute of Mathematics, Jiaozuo City, Henan Province, China + Inner Mongolia University for Nationalities, College of Mathematics, Tongliao City, Inner, Mongolia Autonomous Region China + Tianjin Polytechnic University, Col)
Guo Bai-Ni (Henan Polytechnic University, School of Mathematics and Informatics, Jiaozuo City, Henan Province, China)

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

Keywords: diagonal recurrence relation, inhomogeneous linear ordinary differential equation, Stirling numbers of the first kind, Lerch transcendent, complete monotonicity