Generalized higher order Bernoulli number pairs and generalized Stirling number pairs

Abstract

From a delta series f ( t ) and its compositional inverse g ( t ) , Hsu defined the generalized Stirling number pair ( S ˆ ( n , k ) , s ˆ ( n , k ) ) . In this paper, we further define from f ( t ) and g ( t ) the generalized higher order Bernoulli number pair ( B ˆ n ( z ) , b ˆ n ( z ) ) . Making use of the Bell polynomials, the potential polynomials as well as the Lagrange inversion formula, we give some explicit expressions and recurrences of the generalized higher order Bernoulli numbers, present the relations between the generalized higher order Bernoulli numbers of both kinds and the corresponding generalized Stirling numbers of both kinds, and study the relations between any two generalized higher order Bernoulli numbers. Moreover, we apply the general results to some special number pairs and obtain series of combinatorial identities. It can be found that the introduction of generalized Bernoulli number pair and generalized Stirling number pair provides a unified approach to lots of sequences in mathematics, and as a consequence, many known results are special cases of ours.