6,123 research outputs found
On a Generalization of Bernoulli and Euler Polynomials
Universal Bernoulli polynomials are introduced and their number theoretical
properties discussed. Generalized Euler polynomials are also proposed.Comment: 28 pages, no figure
On -Appell Polynomials
We introduce polynomial sets of -Appell type and give some of their
characterizations. The algebraic properties of the set of all polynomial
sequences of -Appell type are studied. Next, we give a recurrence
relation and a -difference equation for those polynomials. Finally, some
examples of polynomial sequences of -Appell type are given,
particularly, a set of -Hermite polynomials is given and their
three-term recurrence relation and a second order homogeneous
-difference equation are provided.Comment: 12 page
A -Umbral Approach to -Appell Polynomials
In this paper we aim to specify some characteristics of the so called family
of -Appell Polynomials by using -Umbral calculus. Next in our study, we
focus on -Genocchi numbers and polynomials as a famous member of this
family. To do this, firstly we show that any arbitrary polynomial can be
written based on a linear combination of -Genocchi polynomials. Finally, we
approach to the point that similar properties can be found for the other
members of the class of -Appell polynomials
Computation Methods for combinatorial sums and Euler type numbers related to new families of numbers
The aim of this article is to define some new families of the special
numbers. These numbers provide some further motivation for computation of
combinatorial sums involving binomial coefficients and the Euler kind numbers
of negative order. We can show that these numbers are related to the well-known
numbers and polynomials such as the Stirling numbers of the second kind and the
central factorial numbers, the array polynomials, the rook numbers and
polynomials, the Bernstein basis functions and others. In order to derive our
new identities and relations for these numbers, we use a technique including
the generating functions and functional equations. Finally, we not only give a
computational algorithm for these numbers, but also some numerical values of
these numbers and the Euler numbers of negative order with tables. We also give
some combinatorial interpretations of our new numbers.Comment: arXiv admin note: text overlap with arXiv:1604.0560
Duals of Bernoulli Numbers and Polynomials and Euler Number and Polynomials
A sequence inverse relationship can be defined by a pair of infinite inverse
matrices. If the pair of matrices are the same, they define a dual
relationship. Here presented is a unified approach to construct dual
relationships via pseudo-involution of Riordan arrays. Then we give four dual
relationships for Bernoulli numbers and Euler numbers, from which the
corresponding dual sequences of Bernoulli polynomials and Euler polynomials are
constructed. Some applications in the construction of identities of Bernoulli
numbers and polynomials and Euler numbers and polynomials are discussed based
on the dual relationships
On a new -analogue of Appell polynomials
A new -analogue of Appell polynomial sequences and their generalizations
are introduced and their main characterizations are proved. As consequences new
-analogue of Bernoulli and Euler polynomials and numbers is introduced,
their main representations are given.Comment: arXiv admin note: text overlap with arXiv:1712.0132
Derivative Polynomials for tanh, tan, sech and sec in Explicit Form
The derivative polynomials for the hyperbolic and trigonometric tangent,
cotangent and secant are found in explicit form, where the coefficients are
given in terms of Stirling numbers of the second kind. As application, some
integrals are evaluated and the reflection formula for the polygamma function
is written in explicit form.Comment: A similar version in The Fibonacci Quarterly, 200
Polynomial-Value Sieving and Recursively-Factorable Polynomials
We identify a recursive structure among factorizations of polynomial values
into two integer factors. Polynomials for which this recursive structure
characterizes all non-trivial representations of integer factorizations of the
polynomial values into two parts are here called recursively-factorable
polynomials. In particular, we prove that and the prime-producing
polynomials and are recursively-factorable.
For quadratics, the we prove that this recursive structure is equivalent to a
Diophantine identity involving the product of two binary quadratic forms. We
show that this identity may be transformed into geometric terms, relating each
integer factorization to a lattice point of the conic section
, and vice versa
Enumeration of snakes and cycle-alternating permutations
Springer numbers are an analog of Euler numbers for the group of signed
permutations. Arnol'd showed that they count some objects called snakes, that
generalize alternating permutations. Hoffman established a link between
Springer numbers, snakes, and some polynomials related with the successive
derivatives of trigonometric functions.
The goal of this article is to give further combinatorial properties of
derivative polynomials, in terms of snakes and other objects: cycle-alternating
permutations, weighted Dyck or Motzkin paths, increasing trees and forests. We
obtain the generating functions, in terms of trigonometric functions for
exponential ones and in terms of J-fractions for ordinary ones. We also define
natural q-analogs, make a link with normal ordering problems and combinatorial
theory of differential equations.Comment: 22 page
On the modified q-Genocchi numbers and polynomials and their applications
The main objective of this paper is to introduce the modified q-Genocchi
polynomials and to define their generating function. In the paper, we show new
relations, which are explicit formula, derivative formula, multiplication
formula, and some others, for mentioned q-Genocchi polynomials. By applying
Mellin transformation to the generating function of the modified q-Genocchi
polynomials, we define q-Genocchi zeta-type functions which are interpolated by
the modified q-Genocchi polynomials at negative integers.Comment: 10 pages; submitte
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