scholarly journals A determinantal expression and a recursive relation of the Delannoy numbers

2021 ◽  
Vol 13 (2) ◽  
pp. 442-449 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Recursive Relation ◽  
Differentiable Functions ◽  
Delannoy Numbers ◽  
Derivatives Of ◽  
Determinantal Expression

Abstract In the paper, by a general and fundamental, but non-extensively circulated, formula for derivatives of a ratio of two differentiable functions and by a recursive relation of the Hessenberg determinant, the author finds a new determinantal expression and a new recursive relation of the Delannoy numbers. Consequently, the author derives a recursive relation for computing central Delannoy numbers in terms of related Delannoy numbers.

scholarly journals Explicit, Determinantal, and Recurrent Formulas of Generalized Eulerian Polynomials

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 37
Author(s):  
Yan Wang ◽  
Muhammet Cihat Dağli ◽  
Xi-Min Liu ◽  
Feng Qi
Keyword(s):  
General Formula ◽  
Bell Polynomials ◽  
Eulerian Polynomials ◽  
Differentiable Functions ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

In the paper, by virtue of the Faà di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors establish explicit, determinantal, and recurrent formulas for generalized Eulerian polynomials.

scholarly journals The best, by coefficients, weighted quadrature formulas of the form $\sum\limits_{k=1}^n \sum\limits_{i=0}^{r-1} A_{ki} f^{(i)}_{jk}$ for some classes of differentiable functions

1987 ◽  
pp. 37
Author(s):  
Ye.Ye. Dunaichuk
Keyword(s):  
Quadrature Formula ◽  
Sharp Estimate ◽  
Quadrature Formulas ◽  
Differentiable Functions ◽  
Derivatives Of ◽  
Weighted Quadrature

For the quadrature formula (with non-negative, integrable on $[0,1]$ function) that is defined by the values of the function and its derivatives of up to and including $(r-1)$-th order, we find the form of the best coefficients $A^0_{ki}$ ($k = \overline{1, n}$, $i = \overline{0, r-1}$) for fixed nodes $\gamma_k$ ($k = \overline{1, n}$) and we give the sharp estimate of the remainder of this formula on the classes $W^r_p$, $r = 1, 2, \ldots$, $1 \leqslant p \leqslant \infty$.

Lagrangian Formulation of Lorenz and Chen Systems

2021 ◽  
Vol 31 (04) ◽  
pp. 2150055
Author(s):  
Palanisamy Vijayalakshmi ◽  
Zhiheng Jiang ◽  
Xiong Wang
Keyword(s):  
Fractional Derivatives ◽  
Lagrangian Function ◽  
Conserved Quantity ◽  
Lagrangian Formulation ◽  
Lagrangian Functions ◽  
Differentiable Functions ◽  
Derivatives Of

This paper presents the formulation of Lagrangian function for Lorenz, Modified Lorenz and Chen systems using Lagrangian functions depending on fractional derivatives of differentiable functions, and the estimation of the conserved quantity associated with the respective systems.

scholarly journals Fractional Variational Problems Depending on Fractional Derivatives of Differentiable Functions with Application to Nonlinear Chaotic Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Matheus Jatkoske Lazo
Keyword(s):  
Dynamical Systems ◽  
Fractional Derivatives ◽  
Chaotic Systems ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Necessary Condition ◽  
Chaotic Dynamical Systems ◽  
Differentiable Functions ◽  
Fractional Variational Problems ◽  
Derivatives Of

We formulate a necessary condition for functionals with Lagrangians depending on fractional derivatives of differentiable functions to possess an extremum. The Euler-Lagrange equation we obtained generalizes previously known results in the literature and enables us to construct simple Lagrangians for nonlinear systems. As examples of application, we obtain Lagrangians for some chaotic dynamical systems.

UNCERTAINTY AND IMPRECISION IN ANALYTICAL CONTEXT: FUZZY LIMITS AND FUZZY DERIVATIVES

2001 ◽  
Vol 09 (05) ◽  
pp. 563-585
Author(s):  
MARK BURGIN
Keyword(s):  
Incomplete Information ◽  
Differential Calculus ◽  
Classical Case ◽  
Open Problems ◽  
Weak Derivative ◽  
The Third ◽  
Differentiable Functions ◽  
Fourth Part ◽  
Real Functions ◽  
Derivatives Of

The main goal of the present paper is to extend such classical constructions as limits and derivatives making them appropriate for management of imprecise, vague, uncertain, and incomplete information. In the second part of the paper, going after introduction, elements of the theory of fuzzy limits are presented. The third part is devoted to the construction of fuzzy derivatives of real functions. Two kinds of fuzzy derivatives are introduced: weak and strong ones. It is necessary to remark that the strong fuzzy derivatives are similar to ordinary derivatives of real functions being their fuzzy extensions. The weak fuzzy derivatives generate a new concept of a weak derivative even in a classical case of exact limits. In the fourth part fuzzy differentiable functions are studied. Different properties of such functions are obtained. Some of them are the same or at least similar to the properties of the differentiable functions while other properties differ in many aspects from those of the standard differentiable functions. Many classical results are obtained as direct corollaries of propositions for fuzzy derivatives, which are proved in this paper. Some of the classical results are extended and completed. The fifth part of the paper contains several interpretations of fuzzy derivatives aiming at application of fuzzy differential calculus to solving practical problems. At the end, some open problems are formulated.

scholarly journals Necessary conditions of optimality of quadrature formulas with weight on some classes of differentiable functions

1987 ◽  
pp. 47
Author(s):  
Ye.Ye. Dunaichuk
Keyword(s):  
Weight Function ◽  
Quadrature Formula ◽  
Necessary Conditions ◽  
Quadrature Formulas ◽  
Necessary Conditions Of Optimality ◽  
Differentiable Functions ◽  
Derivatives Of ◽  
Continuous Weight

For the quadrature formula (with positive, continuous weight function) that is defined by the values of the function and its derivatives of up to and including $(r-1)$-th order, we find necessary conditions of optimality on the classes $W^r_p$, $r = 1, 2, \ldots$, $p = 2$, $p = \infty$.

scholarly journals On a Functional Equation Appearing on the Margins of a Mean Invariance Problem

2020 ◽  
Vol 34 (1) ◽  
pp. 96-103
Author(s):  
Justyna Jarczyk ◽  
Witold Jarczyk
Keyword(s):  
Functional Equation ◽  
Integral Formula ◽  
Differentiable Functions ◽  
Bajraktarević Mean ◽  
Invariance Problem ◽  
Derivatives Of

AbstractGiven a continuous strictly monotonic real-valued function α, defined on an interval I, and a function ω : I → (0, +∞) we denote by Bαω the Bajraktarević mean generated by α and weighted by ω:B_\omega ^\alpha \left({x,y} \right) = {\alpha ^{- 1}}\left({{{\omega \left(x \right)} \over {\omega \left(x \right) + \omega \left(y \right)}}\alpha \left(x \right) + {{\omega \left(y \right)} \over {\omega \left(x \right) + \omega \left(y \right)}}\alpha \left(y \right)} \right),\,\,\,x,y \in I.We find a necessary integral formula for all possible three times differentiable solutions (φ, ψ) of the functional equationr\left(x \right)B_s^\varphi \left({x,y} \right) + r\left(y \right)B_t^\psi \left({x,y} \right) = r\left(x \right)x + r\left(y \right)y,where r, s, t : I → (0, +∞) are three times differentiable functions and the first derivatives of φ, ψ and r do not vanish. However, we show that not every pair (φ, ψ) given by the found formula actually satisfies the above equation.

scholarly journals Some Identities for a Sequence of Unnamed Polynomials Connected with the Bell Polynomials

2017 ◽  
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Bai-Ni Guo
Keyword(s):  
Stirling Numbers ◽  
Higher Order ◽  
Recursive Relation ◽  
Order Derivative ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Derivative Formula ◽  
Higher Order Derivative ◽  
Determinantal Expression

In the paper, using two inversion theorems for the Stirling numbers and binomial coecients, employing properties of the Bell polynomials of the second kind, and utilizing a higher order derivative formula for the ratio of two dierentiable functions, the authors present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, derive two identities connecting the sequence of unnamed polynomials with the Bell polynomials, and recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials.

Differentiable functions and their Fourier coefficients

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Vakhtang Tsagareishvili
Keyword(s):  
Bounded Variation ◽  
Fourier Coefficients ◽  
Special Series ◽  
Differentiable Functions ◽  
Derivatives Of

Abstract In the paper we consider the properties of Fourier coefficients of functions that possess derivatives of bounded variation. We investigate the convergence of the special series of Fourier coefficients with respect to general orthonormal systems (ONS). The obtained results are the best possible. We also describe the behavior of subsequences of general ONS.

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