Preprint Article Version 1 This version not peer reviewed

On Multi-Order Logarithmic Polynomials and their Explicit Formulas, Recurrence Relations, and Inequalities

Version 1 : Received: 9 September 2017 / Approved: 11 September 2017 / Online: 11 September 2017 (04:22:18 CEST)

How to cite: Qi, F. On Multi-Order Logarithmic Polynomials and their Explicit Formulas, Recurrence Relations, and Inequalities. Preprints 2017, 2017090034 (doi: 10.20944/preprints201709.0034.v1). Qi, F. On Multi-Order Logarithmic Polynomials and their Explicit Formulas, Recurrence Relations, and Inequalities. Preprints 2017, 2017090034 (doi: 10.20944/preprints201709.0034.v1).

Abstract

In the paper, the author introduces the notions "multi-order logarithmic numbers" and "multi-order logarithmic polynomials", establishes an explicit formula, an identity, and two recurrence relations by virtue of the Faa di Bruno formula and two identities of the Bell polynomials of the second kind in terms of the Stirling numbers of the fi rst and second kinds, and constructs some determinantal inequalities, product inequalities, logarithmic convexity for multi-order logarithmic numbers and polynomials by virtue of some properties of completely monotonic functions.

Subject Areas

multi-order logarithmic number; multi-order logarithmic polynomial; explicit formula; identity; recurrence relation; inversion theorem; Bell polynomial of the second kind; Stirling number; determinantal inequality; product inequality; completely monotonic function; logarithmic convexity; Faa di Bruno formula