Application of a new integral expansion for solving a class of functional equations

In a recent paper [M. Masjed-Jamei, H.M. Srivastava, An integral expansion for analytic functions based upon the remainder values of the Taylor series expansions, Appl. Math. Lett. 22 (2009) 406–411], a new type of integral expansions for analytic functions was introduced and investigated. In this sequel to our earlier paper, we make use of the aforementioned expansion in order to explicitly obtain the general solutions of the following functional equation: f ( k + 1 ) ( a ) + b k f ( k ) ( a ) = c k ( k ∈ N 0 ≔ N ∪ { 0 } ; N ≔ { 1 , 2 , 3 , ⋯ } ) for various kinds of real sequences { b k } and { c k } , where (as usual) f ( k ) ( a ) is the k th derivative of the unknown function f ( x ) at x = a . We also present some illustrative examples in this sense.