scholarly journals Truncated-Exponential-Based Appell-Type Changhee Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1588
Author(s):  
Tabinda Nahid ◽  
Parvez Alam ◽  
Junesang Choi
Keyword(s):  
Recurrence Relations ◽  
Explicit Representation ◽  
Integral Inequalities ◽  
Exponential Polynomials ◽  
Integral Formulas ◽  
Open Questions ◽  
Changhee Polynomials ◽  
Natural Way ◽  
Addition Formulas

The truncated exponential polynomials em(x) (1), their extensions, and certain newly-introduced polynomials which combine the truncated exponential polynomials with other known polynomials have been investigated and applied in various ways. In this paper, by incorporating the Appell-type Changhee polynomials Chn*(x) (10) and the truncated exponential polynomials in a natural way, we aim to introduce so-called truncated-exponential-based Appell-type Changhee polynomials eCn*(x) in Definition 1. Then, we investigate certain properties and identities for these new polynomials such as explicit representation, addition formulas, recurrence relations, differential and integral formulas, and some related inequalities. We also present some integral inequalities involving these polynomials eCn*(x). Further we discuss zero distributions of these polynomials by observing their graphs drawn by Mathematica. Lastly some open questions are suggested.

scholarly journals Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 9 ◽  
Author(s):  
Daeyeoul Kim ◽  
Yilmaz Simsek ◽  
Ji Suk So
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Integral Formulas ◽  
Euler Numbers And Polynomials ◽  
Negative Order ◽  
Changhee Polynomials ◽  
Changhee Numbers

The purpose of this paper is to construct generating functions for negative order Changhee numbers and polynomials. Using these generating functions with their functional equation, we prove computation formulas for combinatorial numbers and polynomials. These formulas include Euler numbers and polynomials of higher order, Stirling numbers, and negative order Changhee numbers and polynomials. We also give some properties of these numbers and polynomials with their generating functions. Moreover, we give relations among Changhee numbers and polynomials of negative order, combinatorial numbers and polynomials and Bernoulli numbers of the second kind. Finally, we give a partial derivative of an equation for generating functions for Changhee numbers and polynomials of negative order. Using these differential equations, we derive recurrence relations, differential and integral formulas for these numbers and polynomials. We also give p-adic integrals representations for negative order Changhee polynomials including Changhee numbers and Deahee numbers.

scholarly journals New Tribonacci recurrence relations and addition formulas

2020 ◽  
Vol 26 (4) ◽  
pp. 164-172
Author(s):  
Kunle Adegoke ◽  
◽  
Adenike Olatinwo ◽  
Winning Oyekanmi ◽  
◽  
...  
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Simple Relation ◽  
Addition Formula ◽  
Lucas Numbers ◽  
Tribonacci Numbers ◽  
Three Term Recurrence Relation ◽  
Special Case ◽  
Addition Formulas

Only one three-term recurrence relation, namely, W_{r}=2W_{r-1}-W_{r-4}, is known for the generalized Tribonacci numbers, W_r, r\in Z, defined by W_{r}=W_{r-1}+W_{r-2}+W_{r-3} and W_{-r}=W_{-r+3}-W_{-r+2}-W_{-r+1}, where W_0, W_1 and W_2 are given, arbitrary integers, not all zero. Also, only one four-term addition formula is known for these numbers, which is W_{r + s} = T_{s - 1} W_{r - 1} + (T_{s - 1} + T_{s-2} )W_r + T_s W_{r + 1}, where ({T_r})_{r\in Z} is the Tribonacci sequence, a special case of the generalized Tribonacci sequence, with W_0 = T_0 = 0 and W_1 = W_2 = T_1 = T_2 = 1. In this paper we discover three new three-term recurrence relations and two identities from which a plethora of new addition formulas for the generalized Tribonacci numbers may be discovered. We obtain a simple relation connecting the Tribonacci numbers and the Tribonacci–Lucas numbers. Finally, we derive quadratic and cubic recurrence relations for the generalized Tribonacci numbers.

scholarly journals On some results concerning the polygonal polynomials

2019 ◽  
Vol 35 (1) ◽  
pp. 01-12
Author(s):  
DORIN ANDRICA ◽  
◽  
OVIDIU BAGDASAR ◽  
Keyword(s):  
Recurrence Relations ◽  
Integer Sequences ◽  
Open Questions ◽  
New Meanings ◽  
Online Encyclopedia ◽  
Integral Formulae

In this paper we define the nth polygonal polynomial and we investigate recurrence relations and exact integral formulae for the coefficients of Pn and for those of the Mahonianpolynomials. We also explore numerical properties of these coefficients, unraveling new meanings for old sequences and generating novel entries to the Online Encyclopedia of Integer Sequences (OEIS). Some open questions are also formulated.

On applications for Mahler expansion associated with p-adic q-integrals

2019 ◽  
Vol 15 (01) ◽  
pp. 67-84 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Explicit Formula ◽  
Gamma Function ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Symmetric Relation ◽  
Euler Constant ◽  
Basic Properties ◽  
Changhee Polynomials

In this paper, we primarily consider a generalization of the fermionic [Formula: see text]-adic [Formula: see text]-integral on [Formula: see text] including the parameters [Formula: see text] and [Formula: see text] and investigate its some basic properties. By means of the foregoing integral, we introduce two generalizations of [Formula: see text]-Changhee polynomials and numbers as [Formula: see text]-Changhee polynomials and numbers with weight [Formula: see text] and [Formula: see text]-Changhee polynomials and numbers of second kind with weight [Formula: see text]. For the mentioned polynomials, we obtain new and interesting relationships and identities including symmetric relation, recurrence relations and correlations associated with the weighted [Formula: see text]-Euler polynomials, [Formula: see text]-Stirling numbers of the second kind and Stirling numbers of first and second kinds. Then, we discover multifarious relationships among the two types of weighted [Formula: see text]-Changhee polynomials and [Formula: see text]-adic gamma function. Also, we compute the weighted fermionic [Formula: see text]-adic [Formula: see text]-integral of the derivative of [Formula: see text]-adic gamma function. Moreover, we give a novel representation for the [Formula: see text]-adic Euler constant by means of the weighted [Formula: see text]-Changhee polynomials and numbers. We finally provide a quirky explicit formula for [Formula: see text]-adic Euler constant.

scholarly journals New classes of recurrence relations involving hyperbolic functions, special numbers and polynomials

2021 ◽  
pp. 15-15
Author(s):  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Umbral Calculus ◽  
Discrete Mathematics ◽  
New Class ◽  
Integral Formulas ◽  
Euler Numbers And Polynomials ◽  
Derivative Formulas ◽  
Special Numbers And Polynomials

By using the calculus of finite differences methods and the umbral calculus, we construct recurrence relations for a new class of special numbers. Using this recurrence relation, we define generating functions for this class of special numbers and also new classes of special polynomials. We investigate some properties of these generating functions. By using these generating functions with their functional equations, we obtain many new and interesting identities and relations related to these classes of special numbers and polynomials, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers. Finally, some derivative formulas and integral formulas for these classes of special numbers and polynomials are given. In general, this article includes results that have the potential to be used in areas such as discrete mathematics, combinatorics analysis and their applications.

scholarly journals EXPLICIT REPRESENTATIONS OF THE INTEGRAL CONTAINING THE ERROR TERM IN THE DIVISOR PROBLEM II

2011 ◽  
Vol 54 (1) ◽  
pp. 133-147
Author(s):  
JUN FURUYA ◽  
YOSHIO TANIGAWA
Keyword(s):  
Error Term ◽  
Divisor Problem ◽  
Explicit Representation ◽  
Integral Sign ◽  
Integral Formulas ◽  
Alternative Approach ◽  
Bernoulli Functions ◽  
Dirichlet Divisor Problem ◽  
Explicit Representations ◽  
Riemann Zeta

AbstractIn our previous paper [2], we derived an explicit representation of the integral ∫1∞t−θΔ(t)logjtdt by differentiation under the integral sign. Here, j is a fixed natural number, θ is a complex number with 1 < θ ≤ 5/4 and Δ(x) denotes the error term in the Dirichlet divisor problem. In this paper, we shall reconsider the same formula by an alternative approach, which appeals to only the elementary integral formulas concerning the Riemann zeta- and periodic Bernoulli functions. We also study the corresponding formula in the case of the circle problem of Gauss.

scholarly journals New families of special numbers for computing negative order Euler numbers and related numbers and polynomials

2018 ◽  
Vol 12 (1) ◽  
pp. 1-35 ◽  
Author(s):  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Bernstein Polynomials ◽  
Fibonacci Numbers ◽  
Stirling Numbers ◽  
Euler Numbers ◽  
Open Questions ◽  
Negative Order ◽  
Central Factorial Numbers ◽  
Probability And Statistics

The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to many well-known numbers, which are Bernoulli numbers, Fibonacci numbers, Lucas numbers, Stirling numbers of the second kind and central factorial numbers. Our other inspiration of this paper is related to the Golombek's problem [15] "Aufgabe 1088. El. Math., 49 (1994), 126-127". Our first numbers are not only related to the Golombek's problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by tables. We give some applications in probability and statistics. That is, special values of mathematical expectation of the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, we derive recurrence relations and some formulas of our numbers. Moreover, we come up with a conjecture with two open questions related to our new numbers. We give two algorithms for computation of our numbers. We also give some combinatorial applications, further remarks on our new numbers and their generating functions.

scholarly journals On generalized Laguerre matrix polynomials

2018 ◽  
Vol 10 (1) ◽  
pp. 32-45
Author(s):  
Raed S. Batahan ◽  
A. A. Bathanya
Keyword(s):  
Differential Equation ◽  
Spectral Property ◽  
Recurrence Relation ◽  
Matrix Function ◽  
Recurrence Relations ◽  
Explicit Representation ◽  
Integral Expression ◽  
Matrix Polynomials

Abstract The main object of the present paper is to introduce and study the generalized Laguerre matrix polynomials for a matrix that satisfies an appropriate spectral property. We prove that these matrix polynomials are characterized by the generalized hypergeometric matrix function. An explicit representation, integral expression and some recurrence relations in particular the three terms recurrence relation are obtained here. Moreover, these matrix polynomials appear as solution of a differential equation.

scholarly journals On Lommel Matrix Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2335
Author(s):  
Ayman Shehata
Keyword(s):  
Entire Function ◽  
Recurrence Relations ◽  
Complex Analysis ◽  
Order Type ◽  
Explicit Representation ◽  
Integral Representations ◽  
Matrix Polynomials ◽  
New Class ◽  
Special Cases ◽  
Matrix Recurrence Relations

The main aim of this paper is to introduce a new class of Lommel matrix polynomials with the help of hypergeometric matrix function within complex analysis. We derive several properties such as an entire function, order, type, matrix recurrence relations, differential equation and integral representations for Lommel matrix polynomials and discuss its various special cases. Finally, we establish an entire function, order, type, explicit representation and several properties of modified Lommel matrix polynomials. There are also several unique examples of our comprehensive results constructed.

scholarly journals The Number of Ways to Assemble a Graph

10.37236/2644 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Andrew Vince ◽  
Miklós Bóna
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Explicit Formulas ◽  
Future Directions ◽  
Macromolecular Assembly ◽  
Open Questions ◽  
Asymptotic Formulae ◽  
Special Cases ◽  
Assembly Problem ◽  
Precise Asymptotic

Motivated by the question of how macromolecules assemble,the notion of an assembly tree of a graph is introduced. Given a graph $G$, the paper is concerned with enumerating the number of assembly trees of $G$, a problem that applies to the macromolecular assembly problem. Explicit formulas or generating functions are provided for the number of assembly trees of several families of graphs, in particular for what we call $(H,\phi)$-graphs.  In some natural special cases, we use a powerful recent result of Zeilberger and Apagodu to provide recurrence relations for the diagonal of the relevant multivariate generating functions, and we use a result of Wimp and Zeilberger to find very precise asymptotic formulae for the coefficients of these diagonals.  Future directions for reseach, as well as open questions, are suggested.

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