Some generalized Durfee square identities

AbstractA generalized q-binomial Vandermonde convolution of Sulanke is proved using a generalization of the Durfee square of a partition

Multiplicate inverse forms of terminating hypergeometric series

The multiplicate form of Gould--Hsu's inverse series relations enables to investigate the dual relations of the Chu-Vandermonde-Gau{\ss}'s, the Pfaff-Saalsch\"utz's summation theorems and the binomial convolution formula due to Hagen and Rothe. Several identitity and reciprocal relations are thus established for terminating hypergeometric series. By virtue of the duplicate inversions, we establish several dual formulae of Chu-Vandermonde-Gau{\ss}'s and Pfaff-Saalsch\"utz's summation theorems in Section (3)\cite{ChuVanGauss} and (4)\cite{PfaffSaalsch}, respectively. Finally, the last section is devoted to deriving several identities and reciprocal relations for terminating balanced hypergeometric series from Hagen-Rothe's convolution identity in accordance with the duplicate, triplicate and multiplicate inversions.Comment: 15 page

Combinatorial identities associated with new families of the numbers and polynomials and their approximation values

Recently, the numbers $Y_{n}(\lambda )$ and the polynomials $Y_{n}(x,\lambda)$ have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating functions. By using these generating functions with their functional equations and derivative equations, we derive various identities and relations including two recurrence relations, Vandermonde type convolution formula, combinatorial sums, the Bernstein basis functions, and also some well known families of special numbers and their interpolation functions such as the Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Stirling numbers of the first kind, and the zeta type function. Finally, by using Stirling's approximation for factorials, we investigate some approximation values of the special case of the numbers $Y_{n}\left( \lambda \right)$.Comment: 17 page

A Continuous Analogue of Lattice Path Enumeration: Part II

Here are exhibited some additional results about the continuous binomial coefficients as introduced by L. Cano and R. Diaz in [1].Comment: second versio