Article SOME PROPERTIES OF A SOLUTION TO A FAMILY OF INHOMOGENEOUS LINEAR ORDINARY DIFFERENTIAL EQUATIONS

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. 1. Main results In [7, Section 3, Theorem 1], by inductive argument, it was proved that the family of inhomogeneous linear ordinary differential equations (√ 1− 4t − 1 ) F (t) + n ∑ i=1 an,i (1− 4t)n−i+1/2 F (i−1)(t) = an,0 (1− 4t)n (1.1) for n ∈ N have a solution F (t) =  1 2 ln(1− 4t) √ 1− 4t − 1 , 0 6= t < 14 ; 1, t = 0, (1.2) E-mail addresses: [email protected], [email protected], [email protected], [email protected]. 2010 Mathematics Subject Classification. 11B73, 11M35, 26A48, 26D07, 34A05, 44A10.