Some Properties of the Fuss–Catalan Numbers

Feng Qi,Pietro Cerone

doi:10.3390/math6120277

Abstract

In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the Fuss–Catalan numbers.

Highlights

Introduction and Main ResultsThe Catalan numbers Cn for n ≥ 0 constitute a sequence that is one of the most fascinating sequences in combinatorial number theory with over fifty significant combinatorial interpretations.For details, please refer to monographs [1,2] and closely related references therein.The Catalan numbers Cn have a generating function √ ∞ 1 − 1 − 4x == ∑ Cn x n = 1 + x + 2x2 + 5x3 + 14x4 + 42x5 + 132x6 + · · · . 1 + 1 − 4x n =0Two explicit formulas for Cn with n ≥ 0 read that Cn =
C ( a, b; x ), we find several analytic properties, including monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan sequence { An ( p, r )}n≥0 and related ones
By Lemma 2, the identity (7), and A0 ( p, r ) = 1 for all p > 1 and r > 0, we conclude that the sequence of the Fuss–Catalan numbers { An ( p, r )}n≥0 is increasing and logarithmically convex

Summary

Introduction and Main Results

The Catalan numbers Cn for n ≥ 0 constitute a sequence that is one of the most fascinating sequences in combinatorial number theory with over fifty significant combinatorial interpretations. In the papers [11,12,13,14,15,16,17,18,19,20,21,22,23], the authors discovered many analytic properties, including the monotonicity, a general expression of the asymptotic expansion (2), Schur-convexity, a generalization of the expansion (2), minimality, (logarithmically) complete monotonicity, product inequalities, a generating function, logarithmic convexity, exponential representations, determinantal inequalities, series identities, integral representations, and connections with the Bessel polynomials and the Bell polynomials of the second kind, of the Catalan numbers and function Cn and Cx and the Catalan–Qi function C ( a, b; x ).

Lemmas

Proofs of Theorems 1–3

Remarks

Conclusions