Associated Lah numbers and r-Stirling numbers

We introduce the associated Lah numbers. Some recurrence relations and convolution identities are established. An extension of the associated Stirling and Lah numbers to the r-Stirling and r-Lah numbers are also given. For all these sequences we give combinatorial interpretation, generating functions, recurrence relations, convolution identities. In the sequel, we develop a section on nested sums related to binomial coefficient.