459 research outputs found
Extended Bernoulli and Stirling matrices and related combinatorial identities
In this paper we establish plenty of number theoretic and combinatoric
identities involving generalized Bernoulli and Stirling numbers of both kinds.
These formulas are deduced from Pascal type matrix representations of Bernoulli
and Stirling numbers. For this we define and factorize a modified Pascal matrix
corresponding to Bernoulli and Stirling cases.Comment: Accepted for publication in Linear Algebra and its Application
A note on degenerate Bell numbers and polynomials
Recently, several authors have studied the degenerate Bernoulli and Euler
polynomials and given some intersting identities of those polynomials. In this
paper, we consider the degenerate Bell numbers and polynomials and derive some
new identities of those numbers and polynomials associated with special numbers
and polynomials. In addition, we investigate some properties of the degenerate
Bell polynomials which are derived by using the notion of composita. From our
investigation, we give some new relations between the degenerate Bell
polynomials and the special polynomials.Comment: 12 page
Generalized Stirling Numbers and Generalized Stirling Functions
Here presented is a unified approach to Stirling numbers and their
generalizations as well as generalized Stirling functions by using generalized
factorial functions, -Gamma functions, and generalized divided difference.
Previous well-known extensions of Stirling numbers due to Riordan, Carlitz,
Howard, Charalambides-Koutras, Gould-Hopper, Hsu-Shiue, Tsylova Todorov,
Ahuja-Enneking, and Stirling functions introduced by Butzer and Hauss, Butzer,
Kilbas, and Trujilloet and others are included as particular cases of our
generalization. Some basic properties related to our general pattern such as
their recursive relations and generating functions are discussed. Three
algorithms for calculating the Stirling numbers based on our generalization are
also given, which include a comprehensive algorithm using the characterization
of Riordan arrays.Comment: 26 pages, Presented on May 30, 2011, at City University of Hong Kong,
International Conference on Asymptotics and Special Functions, 30 May - 03
June, 201
On umbral extensions of Stirling numbers and Dobinski-like formulas
Umbral extensions of the stirling numbers of the second kind are considered
and the resulting dobinski-like various formulas including new ones are
presented. These extensions naturally encompass the two well known
q-extensions. The further consecutive umbral extensions q-stirling numbers are
therefore realized here in a two-fold way. The fact that the umbral q-extended
dobinski formula may also be interpreted as the average of powers of random
variable with the q-poisson distribution singles out the q-extensions which
appear to be a kind of singular point in the domain of umbral extensions as
expressed by corresponding two observations. Other relevant possibilities are
tackled with the paper`s closing down questions and suggestions with respect to
other already existing extensions while a brief limited survey of these other
type extensions is being delivered. There the newton interpolation formula and
divided differences appear helpful and inevitable along with umbra symbolic
language in describing properties of general exponential polynomials of
touchard and their possible generalizations. Exponential structures or
algebraically equivalent prefabs with their exponential formula appear to be
also naturally relevant.Comment: 40 page
Identities And Relations On The Hermite-based Tangent Polynomials
In this note, we introduce and investigate the Hermite-based Tangent numbers
and polynomials, Hermite-based modifieed degenerate- Tangent polynomials,
poly-Tangent polynomials. We give some identities and relations for these
polynomials.Comment: 10 page
A note on type 2 Changhee and Daehee polynomials
In recent years, many authors have studied Changhee and Dae- hee polynomials
in connection with many special numbers and polynomials. In this paper, we
investigate type 2 Changhee and Daehee numbers and polynomials and give some
identities for these numbers and polynomials in relation to type 2 Euler and
Bernoulli numbers and polynomials. In addition, we express the central
factorial numbers of the second kind in terms of type 2 Bernoulli, type 2
Changhee and type 2 Daehee numbers of negative integral orders.Comment: 10 page
Some identities of degeenrate ordered Bell polynomials and numbers arising from umbral calculus
In this paper, we study degenerate ordered Bell polynomials with the
viewpoint of Carlitz's degenerate Bernoulli and Euler polynomials and derive by
using umbral calculus some properties and new identities for the degenerate
ordered Bell polynomials associated with special polynomials.Comment: 19 page
On degenerate q-Euler polynomials
In this paper, we consider degenerate Carlitz's type q-Euler polynmials and
numbers and we investigate some identities arising from the fermionic p-adic
integral equations and the generating function of thoe polynomials.Comment: 7 page
Some relations of two type 2 polymomilas and discrete harmonic numbers and polynomials
The aim of this paper is twofold. The first one is to find several relations
between the type 2 higher-order degenerate Euler polynomials and the type 2
higher-order Changhee polynomials in connection with the degenerate stirling
numbers of both kinds and Jindalrae-stirling numbers of both kinds. The second
one is to introduce the discrete harmonic numbers and some related polynomials
and numbers, and to derive their explicit expressions and an identity.Comment: 12 page
The -adic Analysis of Stirling Numbers via Higher Order Bernoulli Numbers
In this paper, we use our previous study of the higher order Bernoulli
numbers to investigate the -adic properties of the Stirling
numbers of the second kind . For example, we give a new, greatly
simplified proof of the formula if ,
and generalize this result to arbitrary primes . We also consider the
Stirling numbers of the first kind , with new results analogous to
those for the Stirling numbers of the second kind. New mod congruences for
Stirling numbers of both kinds are also given.Comment: 22 pages, under consideration by the International Journal of Number
Theory. A new unified method of studying Stirling numbers of both kinds is
developed. and classical theorems for the prime have new, simpler proofs.
The results are generalized to arbitrary primes, with essentially the same
proof
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