APPLICATION OF MULTIHARMONIC NUMBERS FOR THE SYNTHESIS OF CLOSED FORMS OF PARAMETRICALLY MODIFIED FACTORIAL GENERATING SEQUENCES

A toolkit and a method for reducing sequences of integers belonging to the class of factorial-generating recursions to a closed form are presented. The signs and properties of the modified factorial-generating recursion of one and two variables are determined. The best-known factorial-generating recursion of two variables is the sequence of Stirling numbers of the first kind. Modified hyperharmonic numbers are used to synthesize an analytical recursion model. The advantages of these numbers for constructing closed forms of factorial-generating recursions are revealed. An incomplete closed form of the sequence of Stirling numbers of the first kind is synthesized.