Difference Equations, Binomial Coefficients and Exponential Functions

2000 ◽  
Vol 84 (501) ◽  
pp. 468 ◽  
Author(s):  
Geza Makay
Keyword(s):  
Difference Equations ◽  
Binomial Coefficients ◽  
Exponential Functions

scholarly journals On Bleimann-Butzer-Hahn operators for exponential functions

2007 ◽  
Vol 75 (3) ◽  
pp. 409-415 ◽  
Author(s):  
Ulrich Abel ◽  
Mircea Ivan
Keyword(s):  
Exponential Type ◽  
Binomial Coefficients ◽  
Monotone Functions ◽  
Exponential Functions ◽  
Approximation Process

Some inequalities involving the binomial coefficients are obtained. They are used to determine the domain of convergence of the Bleimann, Butzer and Hahn approximation process for exponential type functions. An answer to Hermann's conjecture related to the Bleimann, Butzer and Hahn operators for monotone functions is given.

scholarly journals Certain Hadamard Proportional Fractional Integral Inequalities

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 504 ◽  
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad
Keyword(s):  
Integral Operator ◽  
Difference Equations ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Exponential Functions ◽  
Non Local ◽  
Positive Functions

In this present paper we study the non-local Hadmard proportional integrals recently proposed by Rahman et al. (Advances in Difference Equations, (2019) 2019:454) which containing exponential functions in their kernels. Then we establish certain new weighted fractional integral inequalities involving a family of n ( n ∈ N ) positive functions by utilizing Hadamard proportional fractional integral operator. The inequalities presented in this paper are more general than the inequalities existing in the literature.

scholarly journals On Binomial Coefficient Residues

1957 ◽  
Vol 9 ◽  
pp. 363-370 ◽  
Author(s):  
J . B. Roberts
Keyword(s):  
Prime Number ◽  
Difference Equations ◽  
Binomial Coefficient ◽  
Binomial Coefficients ◽  
Linear Difference Equations ◽  
Constant Coefficients

The number of binomial coefficients , which are congruent to j , 0 ≤ j ≤ p − 1, modulo the prime number p is denoted by θj(n). In this paper we give systems of simultaneous linear difference equations with constant coefficients whose solutions would yield the quantities θj(n) explicitly.

scholarly journals The asymptotic behaviour of q-difference equations with multiple delays

2009 ◽  
Vol 43 (1) ◽  
pp. 41-50
Author(s):  
Jan Čermák
Keyword(s):  
Asymptotic Behaviour ◽  
Difference Equations ◽  
Asymptotic Properties ◽  
Continuous Case ◽  
Multiple Delays ◽  
Exponential Functions

Abstract We investigate asymptotic properties of solutions of a class of linear q-difference equations. We relate their behaviour to the asymptotics of some q-exponential functions and mention possible connections with the corresponding continuous case.

scholarly journals INTEGRAL REPRESENTATIONS AND INEQUALITIES OF EXTENDED CENTRAL BINOMIAL COEFFICIENTS

2021 ◽  
Author(s):  
Chunfu Wei
Keyword(s):  
Binomial Coefficient ◽  
Integral Representations ◽  
Binomial Coefficients ◽  
Exponential Functions ◽  
Central Binomial Coefficient

In the paper, the author presents three integral representations of extended central binomial coefficient, proves decreasing and increasing properties of two power-exponential functions involving extended (central) binomial coefficients, derives several double inequalities for bounding extended (central) binomial coefficient, and compares with known results.

scholarly journals A Study on Some New Results Arising from (p,q)-Calculus

2018 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Exponential Function ◽  
Chain Rule ◽  
Binomial Coefficients ◽  
Final Part ◽  
Trigonometric Functions ◽  
Elementary Functions ◽  
Exponential Functions ◽  
Hyperbolic Functions ◽  
Quantum Calculus ◽  
A Chain

This paper includes some new investigations and results for post quantum calculus, denoted by (p,q)-calculus. A chain rule for (p,q)-derivative is developed. Also, a new (p,q)-analogue of the exponential function is introduced and some its properties including the addition property for (p,q)-exponential functions are investigated. Several useful results involving (p,q)-binomial coefficients and (p,q)-antiderivative are discovered. At the final part of this paper, (p,q)-analogue of some elementary functions including trigonometric functions and hyperbolic functions are considered and some properties and relations among them are analyzed extensively.

scholarly journals Generalized quantum exponential function and its applications

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4907-4922
Author(s):  
Burcu Silindir ◽  
Ahmet Yantir
Keyword(s):  
Exponential Function ◽  
Difference Equations ◽  
Taylor Series ◽  
Infinite Product ◽  
Existence And Uniqueness Theorem ◽  
Exponential Functions ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
First Order ◽  
Dynamic Wave

This article aims to present (q; h)-analogue of exponential function which unifies, extends hand q-exponential functions in a convenient and efficient form. For this purpose, we introduce generalized quantum binomial which serves as an analogue of an ordinary polynomial. We state (q,h)-analogue of Taylor series and introduce generalized quantum exponential function which is determined by Taylor series in generalized quantum binomial. Furthermore, we prove existence and uniqueness theorem for a first order, linear, homogeneous IVP whose solution produces an infinite product form for generalized quantum exponential function. We conclude that both representations of generalized quantum exponential function are equivalent. We illustrate our results by ordinary and partial difference equations. Finally, we present a generic dynamic wave equation which admits generalized trigonometric, hyperbolic type of solutions and produces various kinds of partial differential/difference equations.

scholarly journals Exponential functions of discrete fractional calculus

2013 ◽  
Vol 7 (2) ◽  
pp. 343-353 ◽  
Author(s):  
Nihan Acar ◽  
Ferhan Atici
Keyword(s):  
Fractional Calculus ◽  
Difference Equations ◽  
Linear Independence ◽  
Discrete Fractional Calculus ◽  
Linear Difference Equations ◽  
Exponential Functions ◽  
Real Roots ◽  
Nabla Operator ◽  
Constant Coefficients ◽  
Discrete Functions

In this paper, exponential functions of discrete fractional calculus with the nabla operator are studied. We begin with proving some properties of exponential functions along with some relations to the discrete Mittag-Leffler functions. We then study sequential linear difference equations of fractional order with constant coefficients. A corresponding characteristic equation is defined and considered in two cases where characteristic real roots are same or distinct. We define a generalized Casoratian for a set of discrete functions. As a consequence, for solutions of sequential linear difference equations, their nonzero Casoratian ensures their linear independence.

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