In this paper, we propose the another yet generalization of Stirling numbers of the rst kind for non-integer values of their arguments. We discuss the analytic representations of Stirling numbers through harmonic numbers, the generalized hypergeometric function and the logarithmic beta integral. We present then in nite series involving Stirling numbers and demonstrate how they are related to Eu...

Say an integer n is exceptional if the maximum Stirling number of the second kind S(n, k) occurs for two (of necessity consecutive) values of k. We prove that the number of exceptional integers less than or equal to x is O(x), for any ! > 0. We derive a similar result for partitions of n into exactly k integers.

The dish-Stirling technology for power generation, using a Stirling engine fed with a renewable energy such as solar, means a promising development regarding electricity generation. The dish-Stirling technology presents the highest efficiency, around 30%, in relation to other solar energy generation systems. However the technology status has not achieved enough advancement to be competitive wit...

Purely combinatorial methods are used to show that Stirling numbers, defined combinatorially, satisfy orthogonality relations. The proofs are extended to several generalizations of Stirling numbers.

In this paper we prove that some results concerned the generalized Stirling numbers are the consequence of the results of Toscano and Chak. The new explicit expressions for generalized Stirling numbers are also given.

We show how the generating function for signed Stirling numbers of the first kind can be proved using the involution principle and a natural combinatorial interpretation based on cycle-colored permuations. We seek an involution-based proof of the generating function for signed Stirling numbers of the first kind, written here as ∑ k (−1)c(n, k)x = (−1)(x)(x− 1) · · · (x− n+ 1) where c(n, k) is t...

Say that an integer n is exceptional if the maximum Stirling number of the second kind S(n, k) occurs for two (of necessity consecutive) values of k. We prove that the number of exceptional integers less than or equal to x is O(x), for any ǫ > 0.

The efficiency of the ideal stirling cycle assumes, that the cylinder volumes are isothermal and that the temperatures in the cylinder volumes are identical to the temperatures in the heat sink and the heat source. The efficiency is then identical to the efficiency of the Carnot cycle. However, in a real stirling machine the cylinder volumes are nearly adiabatic instead of isothermal, and the h...

Recent developments of Baxter algebras have lead to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first kind and the second kind, partitions and multinomial coefficients. This allows us to apply congruences from number theory to obtain congruences in Baxter algebras.