Note on r-central Lah numbers and r-central Lah-Bell numbers

The r-Lah numbers generalize the Lah numbers to the r-Stirling numbers in the same sense. The Stirling numbers and the central factorial numbers are one of the important tools in enumerative combinatorics. The r-Lah number counts the number of partitions of a set with n+r elements into k+r ordered blocks such that r distinguished elements have to be in distinct ordered blocks. In this paper, the r-central Lah numbers and the r-central Lah-Bell numbers (r is an element of N) are introduced parallel to the r-extended central factorial numbers of the second kind and r-extended central Bell polynomials. In addition, some identities related to these numbers including the generating functions, explicit formulas, binomial convolutions are derived. Moreover, the r-central Lah numbers and the r-central Lah-Bell numbers are shown to be represented by Riemann integral, respectively.