Polynomials with r-Lah coefficient and hyperharmonic numbers

In this paper, we take advantage of the Mellin type derivative to produce some new families of polynomials whose coefficients involve r-Lah numbers. One of these polynomials leads to rediscover many of the identities of r-Lah numbers. We show that some of these polynomials and hyperharmonic numbers are closely related. Taking into account of these connections, we reach several identities for harmonic and hyperharmonic numbers

Triangular Recurrences, Generalized Eulerian Numbers, and Related Number Triangles

Many combinatorial and other number triangles are solutions of recurrences of the Graham-Knuth-Patashnik (GKP) type. Such triangles and their defining recurrences are investigated analytically. They are acted on by a transformation group generated by two involutions: a left-right reflection and an upper binomial transformation, acting row-wise. The group also acts on the bivariate exponential generating function (EGF) of the triangle. By the method of characteristics, the EGF of any GKP triangle has an implicit representation in terms of the Gauss hypergeometric function. There are several parametric cases when this EGF can be obtained in closed form. One is when the triangle elements are the generalized Stirling numbers of Hsu and Shiue. Another is when they are generalized Eulerian numbers of a newly defined kind. These numbers are related to the Hsu-Shiue ones by an upper binomial transformation, and can be viewed as coefficients of connection between polynomial bases, in a manner that generalizes the classical Worpitzky identity. Many identities involving these generalized Eulerian numbers and related generalized Narayana numbers are derived, including closed-form evaluations in combinatorially significant cases.Comment: 62 pages, final version, accepted by Advances in Applied Mathematic

Some formulas for the restricted r-Lah numbers

The r-Lah numbers, which we denote here by `(r)(n, k), enumerate partitions of an (n+r)-element set into k+r contents-ordered blocks in which the smallest r elements belong to distinct blocks. In this paper, we consider a restricted version `(r) m (n, k) of the r-Lah numbers in which block cardinalities are at most m. We establish several combinatorial identities for `(r) m (n, k) and obtain as limiting cases for large m analogous formulas for `(r)(n, k). Some of these formulas correspond to previously established results for `(r)(n, k), while others are apparently new also in the r-Lah case. Some generating function formulas are derived as a consequence and we conclude by considering a polynomial generalization of `(r) m (n, k) which arises as a joint distribution for two statistics defined on restricted r-Lah distributions. Keywords: restricted Lah numbers, polynomial generalization, r-Lah numbers, combinatorial identities MSC: 11B73, 05A19, 05A1