In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. We investigate their some properties and derive several relations among q-Bernoulli numbers and polynomials, and newly de…ned (q; r; w)Stirling numbers of the second kind. We also obtain q-Bernstein polynomials as a linear combination of (q...

Peter Kovesi Eun-Jung Holden Jason C. Wong Centre for Exploration Targeting Centre for Exploration Targeting Centre for Exploration Targeting The University of Western Australia The University of Western Australia The University of Western Australia 35 Stirling Highway. Crawley WA. 6009 35 Stirling Highway. Crawley WA. 6009 35 Stirling Highway. Crawley WA. 6009 [email protected] eun-jung....

We employ the generalized factorials to define a Stirling-type pair {s n, k;α,β, r , S n, k;α,β, r } which unifies various Stirling-type numbers investigated by previous authors. We make use of the Newton interpolation and divided differences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function. The g...

The Jacobi-Stirling numbers of the first and second kinds were introduced in 2006 in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial...

The function y = Φα(x), the solution of y e = x for x and y large enough, has a series expansion in terms of lnx and ln lnx, with coefficients given in terms of Stirling cycle numbers. It is shown that this expansion converges for x > (αe) for α ≥ 1. It is also shown that new expansions can be obtained for Φα in terms of associated Stirling numbers. The new expansions converge more rapidly and ...

Cauchy coeecient integrals and Hankel contours provide a natural generalization of Stirling numbers for unrestricted complex values of their arguments. Many classical identities survive such an extension. Sur les nombres de Stirling d'indice complexe et les contours de Hankel R esum e : Les int egrales de Cauchy et les contours de Hankel fournissent une g en eralisation naturelle des nombres de...

Keywords: Stirling numbers q-Stirling numbers Multiparameter non-central Stirling numbers Comtet numbers q-Analogue Harmonic numbers Generalized q-harmonic numbers a b s t r a c t In this paper we derive q-analogues of the multiparameter non-central Stirling numbers of the first and second kind, introduced by El-Desouky. Moreover, recurrence relations, explicit formulas and a connection between...

are called a weighted Stirling pair if f(g(x)) = g(f(x)) = x, i.e.,/and g are reciprocal. When W(x) = l, B^in.k) and B2(n,k) reduce to a Stirling type pair whose properties are exhibited in [7]. In this paper, we shall present a weighted Stirling pair that includes some previous generalizations of Stirling numbers as particular cases. Some related combinatorial and arithmetic properties are als...

In this paper, we introduce Stirling permutations of the second kind. In particular, we count Stirling permutations of the second kind by their cycle ascent plateaus, fixed points and cycles. Moreover, we get an expansion of the ordinary derangement polynomials in terms of the Stirling derangement polynomials. Finally, we present constructive proofs of a kind of combinatorial expansions of the ...