scholarly journals On Fractional Symmetric Hahn Calculus

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 873 ◽  
Author(s):  
Nichaphat Patanarapeelert ◽  
Thanin Sitthiwirattham
Keyword(s):  
Difference Operator ◽  
Fundamental Properties ◽  
Hahn Difference Operator

In this paper, we study fractional symmetric Hahn difference calculus. The new idea of the symmetric Hahn difference operator, the fractional symmetric Hahn integral, and the fractional symmetric Hahn operators of Riemann–Liouville and Caputo types are presented. In addition, we formulate some fundamental properties based on these fractional symmetric Hahn operators.

scholarly journals Explicit Formulas for the First Form (q,r)-Dowling Numbers and (q,r)-Whitney-Lah Numbers

2021 ◽  
Vol 14 (1) ◽  
pp. 65-81
Author(s):  
Roberto Bagsarsa Corcino ◽  
Jay Ontolan ◽  
Maria Rowena Lobrigas
Keyword(s):  
Recurrence Relation ◽  
Explicit Formula ◽  
Taylor Series ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Difference Operator ◽  
Interpolation Formula ◽  
Explicit Formulas ◽  
Fundamental Properties ◽  
Whitney Numbers

In this paper, a q-analogue of r-Whitney-Lah numbers, also known as (q,r)-Whitney-Lah number, denoted by $L_{m,r} [n, k]_q$ is defined using the triangular recurrence relation. Several fundamental properties for the q-analogue are established such as vertical and horizontal recurrence relations, horizontal and exponential generating functions. Moreover, an explicit formula for (q, r)-Whitney-Lah number is derived using the concept of q-difference operator, particularly, the q-analogue of Newton’s Interpolation Formula (the umbral version of Taylor series). Furthermore, an explicit formula for the first form (q, r)-Dowling numbers is obtained which is expressed in terms of (q,r)-Whitney-Lah numbers and (q,r)-Whitney numbers of the second kind.

Hahn Difference Operator and Associated Jackson–Nörlund Integrals

2012 ◽  
Vol 154 (1) ◽  
pp. 133-153 ◽  
Author(s):  
M. H. Annaby ◽  
A. E. Hamza ◽  
K. A. Aldwoah
Keyword(s):  
Difference Operator ◽  
Hahn Difference Operator

scholarly journals Теория Титчмарша - Вейля сингулярного уравнения Хана - Штурма - Лиувилля

2021 ◽  
pp. 16-26
Author(s):  
B.P. Allahverdiev ◽  
H. Tuna
Keyword(s):  
Hilbert Space ◽  
Continuous Function ◽  
Difference Equation ◽  
Difference Operator ◽  
Liouville Problem ◽  
Limit Circle ◽  
Weyl Theory ◽  
Hahn Difference Operator ◽  
Sturm Liouville ◽  
Ordinary Derivative

In this work, we will consider the singular Hahn--Sturm--Liouville difference equation defined by $-q^{-1}D_{-\omega q^{-1},q^{-1}}D_{\omega ,q}y( x) +v(x) y( x) =\lambda y(x)$, $x\in (\omega _{0},\infty),$ where $\lambda$ is a complex parameter, $v$ is a real-valued continuous function at $\omega _{0}$ defined on $[\omega _{0},\infty)$. These type equations are obtained when the ordinary derivative in the classical Sturm--Liouville problem is replaced by the $\omega,q$-Hahn difference operator $D_{\omega,q}$. We develop the $\omega,q$-analogue of the classical Titchmarsh--Weyl theory for such equations. In other words, we study the existence of square-integrable solutions of the singular Hahn--Sturm--Liouville equation. Accordingly, first we define an appropriate Hilbert space in terms of Jackson--N\"{o}rlund integral and then we study families of regular Hahn--Sturm--Liouville problems on $[\omega_{0},q^{-n}]$, $n\in \mathbb{N}$. Then we define a family of circles that converge either to a point or a circle. Thus, we will define the limit-point, limit-circle cases in the Hahn calculus setting by using Titchmarsh's technique.

scholarly journals Leibnizs rule and Fubinis theorem associated with Hahn difference operators

2016 ◽  
Vol 12 (6) ◽  
pp. 6335-6346 ◽  
Author(s):  
Samer Derham Makarash
Keyword(s):  
Difference Operator ◽  
Difference Operators ◽  
Fubini’S Theorem ◽  
Hahn Difference Operator ◽  
Fubini's Theorem

In $1945$, Wolfgang Hahn introduced his difference operator $D_{q,\omega}$, which is defined by where $\displaystyle{\omega_0=\frac {\omega}{1-q}}$ with $0<q<1, \omega>0.$ In this paper, we establish Leibniz's rule and Fubini's theorem associated with this Hahn difference operator.

Metastable Defects in the Amorphous Silicon-Germanium Alloys

2003 ◽  
Vol 762 ◽  
Author(s):  
J. David Cohen
Keyword(s):  
Amorphous Silicon ◽  
Silicon Germanium ◽  
Dangling Bonds ◽  
Annealing Behavior ◽  
Fundamental Properties ◽  
Silicon Materials ◽  
Salient Features ◽  
The Individual ◽  
Amorphous Silicon Germanium ◽  
Silicon Germanium Alloys

AbstractThis paper first briefly reviews a few of the early studies that established some of the salient features of light-induced degradation in a-Si,Ge:H. In particular, I discuss the fact that both Si and Ge metastable dangling bonds are involved. I then review some of the recent studies carried out by members of my laboratory concerning the details of degradation in the low Ge fraction alloys utilizing the modulated photocurrent method to monitor the individual changes in the Si and Ge deep defects. By relating the metastable creation and annealing behavior of these two types of defects, new insights into the fundamental properties of metastable defects have been obtained for amorphous silicon materials in general. I will conclude with a brief discussion of the microscopic mechanisms that may be responsible.

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