Papers by Roberto Corcino
We use rook placements to prove Spivey's Bell number formula and other identities related to it, ... more We use rook placements to prove Spivey's Bell number formula and other identities related to it, in particular, some convolution identities involving Stirling numbers and relations involving Bell numbers. To cover as many special cases as possible, we work on the generalized Stirling numbers that arise from the rook model of Goldman and Haglund. A alternative combinatorial interpretation for the Type II generalized q-Stirling numbers of Remmel and Wachs is also introduced in which the method used to obtain the earlier identities can be adapted easily.
In this paper, we develop the theory of a p, q-analogue of the binomial coefficients. Some proper... more In this paper, we develop the theory of a p, q-analogue of the binomial coefficients. Some properties and identities parallel to those of the usual and q-binomial coefficients will be established including the triangular, vertical, and the horizontal recurrence relations, horizontal generating function, and the orthogonality and inverse relations. The construction and derivation of these results give us an idea of how to handle complex computations involving the parameters p and q. This may be a good start in developing the theory of p, q-analogues of some special numbers in combinatorics. Furthermore, several interesting special cases will be disclosed which are analogous to some established identities of the usual binomial coefficients.
It is shown that the sequence of the generalized Bell polynomials Sn(x) is convex under some re... more It is shown that the sequence of the generalized Bell polynomials Sn(x) is convex under some restrictions of the parameters involved. A kind of recurrence relation for Sn(x) is established, and some numbers related to the generalized Bell numbers and their properties are investigated.
In this paper we investigate special generalized Bernoulli polynomials with a, b, c parameters th... more In this paper we investigate special generalized Bernoulli polynomials with a, b, c parameters that generalize classical Bernoulli numbers and polynomials. The present paper deals
with some recurrence formulae for the generalization of poly-Bernoulli numbers and polynomials with a, b, c parameters. Poly-Bernoulli numbers satisfy certain recurrence relationships
which are used in many computations involving poly-Bernoulli numbers. Obtaining a closed formula for generalization of poly-Bernoulli numbers with a, b, c parameters therefore seems
to be a natural and important problem. By using the generalization of poly-Bernoulli polynomials with a, b, c parameters of negative index we define symmetrized generalization of poly-Bernoulli polynomials with a; b; c parameters of two variables and we prove duality property for
them. Also by Stirling numbers of the second kind we will find a closed formula for them. Furthermore we generalize the Arakawa-Kaneko Zeta functions and by using the Laplace-Mellin
integral, define generalization of Arakawa-Kaneko Zeta functions with a, b, c parameters and obtain an interpolation formula for the generalization of poly- Bernoulli numbers and polynomials with a, b, c parameters. Furthermore we present a link between this type of Zeta functions and Dirichlet series. By our interpolation formula, we will interpolate the generalization of Arakawa-Kaneko Zeta functions with a, b, c parameters.
The r-Whitney numbers of the second kind are a generalization of all the Stirling-type numbers of... more The r-Whitney numbers of the second kind are a generalization of all the Stirling-type numbers of the second kind which are in
line with the unified generalization of Hsu and Shuie. In this paper, asymptotic formulas for r-Whitney numbers of the second kind with integer and real parameters are obtained and the range of validity of each formula is established.
In this paper we introduce the generalization of Multi Poly-Euler polynomials and we investigate ... more In this paper we introduce the generalization of Multi Poly-Euler polynomials and we investigate some relationship involving
Multi Poly-Euler polynomials. Obtaining a closed formula for
generalization of Multi Poly-Euler numbers therefore seems to be a natural and important problem.
We establish two types of convolution identities for the limit of the diff erences of the general... more We establish two types of convolution identities for the limit of the diff erences of the generalized factorial. One is derived using the \three-step method" of R.Corcino and L.C. Hsu and the other one makes use of the explicit formula in symmetric function form. Moreover, using the former, a certain congruence relation is obtained.
Asymptotic formulas for the generalized Stirling numbers of the second kind with integer and real... more Asymptotic formulas for the generalized Stirling numbers of the second kind with integer and real parameters are obtained and
ranges of validity of the formulas are established. The generalizations of Stirling numbers considered here are generalizations along the line of Hsu and Shuie’s unified generalization.
The r-Bell numbers are generalized using the concept of the Hankel contour. Some properties paral... more The r-Bell numbers are generalized using the concept of the Hankel contour. Some properties parallel to those of the ordinary Bell numbers are established. Moreover, an asymptotic approximation for 𝑟-Bell numbers with real arguments is obtained.
In this paper, we defi ne a p,q-difference operator and obtain an explicit formula which is used ... more In this paper, we defi ne a p,q-difference operator and obtain an explicit formula which is used to express the p,q-analogue of the uni ed generalization of Stirling numbers and its exponential generating function in terms of the p,q-difference operator. Explicit formulas for the non-central q-Stirling numbers of the second kind and non-central q-Lah numbers are derived using the new q-analogue of Newton's interpolation formula. Moreover, a p,q-analogue of Newton's interpolation formula is
established.
A q-analogue of Rucinski-Voigt numbers is defined by means of a recurrence relation, and some pro... more A q-analogue of Rucinski-Voigt numbers is defined by means of a recurrence relation, and some properties including the orthogonality and inverse relations with the q-analogue of the limit of the differences of the generalized factorial are obtained.
Ars Combinatoria, 2006
ABSTRACT Stirling numbers (of the seco2nd kind) can be defined as entries in the transition matri... more ABSTRACT Stirling numbers (of the seco2nd kind) can be defined as entries in the transition matrix from the polynomial basis 1,x,x(x-1),x(x-1)(x-2),⋯ to the canonical basis 1,x,x 2 ,x 3 ,⋯. The (r,β)-Stirling numbers correspond similarly to the basis 1,(x-r),(x-r)(x-r-β),(x-r)(x-r-β)(x-r-2β),⋯. The authors establish unimodality, certain asymptotic approximations and asymptotic normality of these numbers.
In this paper, we establish more properties for the q-analogue of the unified generalization of S... more In this paper, we establish more properties for the q-analogue of the unified generalization of Stirling numbers including the vertical and horizontal recurrence relations, and the rational generating function. This generating function plays an important role in deriving one of the explicit formulas in symmetric function form which will be used in giving combinatorial interpretations of the q-analogue in the context of 0-1 tableau. Moreover, using the combinatorics of 0-1 tableaux, we obtain certain generalization of Carlitz identity.
Asymptotic formulas for the generalized Stirling numbers of the second kind with integer and real... more Asymptotic formulas for the generalized Stirling numbers of the second kind with integer and real parameters are obtained and ranges of validity of the formulas are established. The generalizations of Stirling numbers considered here are generalizations along the line of Hsu and Shuie's unified generalization.
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Papers by Roberto Corcino
with some recurrence formulae for the generalization of poly-Bernoulli numbers and polynomials with a, b, c parameters. Poly-Bernoulli numbers satisfy certain recurrence relationships
which are used in many computations involving poly-Bernoulli numbers. Obtaining a closed formula for generalization of poly-Bernoulli numbers with a, b, c parameters therefore seems
to be a natural and important problem. By using the generalization of poly-Bernoulli polynomials with a, b, c parameters of negative index we define symmetrized generalization of poly-Bernoulli polynomials with a; b; c parameters of two variables and we prove duality property for
them. Also by Stirling numbers of the second kind we will find a closed formula for them. Furthermore we generalize the Arakawa-Kaneko Zeta functions and by using the Laplace-Mellin
integral, define generalization of Arakawa-Kaneko Zeta functions with a, b, c parameters and obtain an interpolation formula for the generalization of poly- Bernoulli numbers and polynomials with a, b, c parameters. Furthermore we present a link between this type of Zeta functions and Dirichlet series. By our interpolation formula, we will interpolate the generalization of Arakawa-Kaneko Zeta functions with a, b, c parameters.
line with the unified generalization of Hsu and Shuie. In this paper, asymptotic formulas for r-Whitney numbers of the second kind with integer and real parameters are obtained and the range of validity of each formula is established.
Multi Poly-Euler polynomials. Obtaining a closed formula for
generalization of Multi Poly-Euler numbers therefore seems to be a natural and important problem.
ranges of validity of the formulas are established. The generalizations of Stirling numbers considered here are generalizations along the line of Hsu and Shuie’s unified generalization.
established.
with some recurrence formulae for the generalization of poly-Bernoulli numbers and polynomials with a, b, c parameters. Poly-Bernoulli numbers satisfy certain recurrence relationships
which are used in many computations involving poly-Bernoulli numbers. Obtaining a closed formula for generalization of poly-Bernoulli numbers with a, b, c parameters therefore seems
to be a natural and important problem. By using the generalization of poly-Bernoulli polynomials with a, b, c parameters of negative index we define symmetrized generalization of poly-Bernoulli polynomials with a; b; c parameters of two variables and we prove duality property for
them. Also by Stirling numbers of the second kind we will find a closed formula for them. Furthermore we generalize the Arakawa-Kaneko Zeta functions and by using the Laplace-Mellin
integral, define generalization of Arakawa-Kaneko Zeta functions with a, b, c parameters and obtain an interpolation formula for the generalization of poly- Bernoulli numbers and polynomials with a, b, c parameters. Furthermore we present a link between this type of Zeta functions and Dirichlet series. By our interpolation formula, we will interpolate the generalization of Arakawa-Kaneko Zeta functions with a, b, c parameters.
line with the unified generalization of Hsu and Shuie. In this paper, asymptotic formulas for r-Whitney numbers of the second kind with integer and real parameters are obtained and the range of validity of each formula is established.
Multi Poly-Euler polynomials. Obtaining a closed formula for
generalization of Multi Poly-Euler numbers therefore seems to be a natural and important problem.
ranges of validity of the formulas are established. The generalizations of Stirling numbers considered here are generalizations along the line of Hsu and Shuie’s unified generalization.
established.