Related papers: Gaussian Mersenne Lucas numbers and polynomials
The primary goal of this paper is to introduce and investigate generalized incomplete exponential functions with matrix parameters. Integral representation, differential formula, addition formula, multiplication formula, and recurrence…
The Stirling numbers of the first kind can be represented in terms of a new class of polynomials that are closely related to the Bernoulli polynomials. Recursion relations for these polynomials are given.
In this paper, we study the Carlitz's degenerate Bernoulli numbers and polynomials and give some formulae and identities related to those numbers and polynomials.
We present here some new identities for generalizations of Fibonacci and Lucas numbers by combinatorially interpreting these numbers in terms of numbers of certain tilings of a $1 \times m$ board. As a consequence, some new interesting…
A full characterization of $(p,q)$-deformed Fibonacci and Lucas polynomials is given. These polynomials obey non-conventional three-term recursion relations. Their generating functions and Fourier integral transforms are explicitly computed…
The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials. We formulate some basic properties of unitary cyclotomic polynomials and study how they are connected with cyclotomic, inclusion-exclusion and…
The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study…
Recently the new q-Euler numbers are defined. In this paper we derive the the Kummer type congruence related to q-Euler numbers and we introduce some interesting formulae related to these q-Euler numbers.
Recently, Komastu introduced the concept of poly-Cauchy numbers and polynomials which generalize Cauchy numbers and polynomials. In this paper, we introduce new generaliza- tion of poly-Cauchy and poly-Bernoulli numbers and polynomials.…
Studying degenerate versions of various special polynomials have become an active area of research and yielded many interesting arithmetic and combinatorial results. Here we introduce a degenerate version of polylogarithm function, called…
Two new expansions for partial sums of Gauss' triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function $\bar{p}(n)$ and for the partition function $p_1(n)$ counting…
In this paper, we first give new generalizations for third-order Jacobsthal $\{J_{n}^{(3)}\}_{n\in \mathbb{N}}$ and third-order Jacobsthal-Lucas $\{j_{n}^{(3)}\}_{n\in \mathbb{N}}$ sequences for Jacobsthal and Jacobsthal-Lucas numbers.…
In this paper one extends the binomial and trinomial coefficients to the concept of 'k-nomial' coefficients, and one obtains some properties of these. As an application one generalizes Pascal's triangle.
We study the interplay between recurrences for zeta related functions at integer values, `Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and…
Polycosecant numbers and polycotangent numbers are introduced as level two analogues of poly-Bernoulli numbers. It is shown that polycosecant numbers and polycotangent numbers satisfy many formulas similar to those of poly-Bernoulli…
The main purpose of this paper is to establish some new properties of Horadam numbers in terms of binomial sums. By that, we can obtain these special numbers in a new and direct way. Moreover, some connections between Horadam and…
Motivated by the work on hypergeometric summation theorems (recorded in the table III of Prudnikov et al. pp. 541-546), we have established some new summation theorems for Clausen's hypergeometric functions with unit argument in terms of…
In this paper we focus on two new families of polynomials which are connected with exponential polynomials and geometric polynomials. We discuss their generalizations and show that these new families of polynomials and their generalizations…
MacMahon's definition of self-inverse composition is extended to $n$-colour self-inverse composition. This introduces four new sequences which satisfy the same recurrence relation with different initial conditions like the famous Fibonacci…
In the recent paper the interesting q-Euler numbers and polynomials introduced in JMAA. The purpose of this paper is to construct the modified q-Euler numbers and polynomiasl. Finally we will give the interesting many identities related to…