scholarly journals -Pascal and -Wronskian Matrices with Implications to -Appell Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Thomas Ernst
Keyword(s):  
Difference Equation ◽  
Recent Article ◽  
Matrix Multiplication ◽  
Euler Polynomials ◽  
Matrix Approach ◽  
Appell Polynomials ◽  
Sum Of Products ◽  
Functional Matrix

We introduce a -deformation of the Yang and Youn matrix approach for Appell polynomials. This will lead to a powerful machinery for producing new and old formulas for -Appell polynomials, and in particular for -Bernoulli and -Euler polynomials. Furthermore, the --polynomial, anticipated by Ward, can be expressed as a sum of products of -Bernoulli and -Euler polynomials. The pseudo -Appell polynomials, which are first presented in this paper, enable multiple -analogues of the Yang and Youn formulas. The generalized -Pascal functional matrix, the -Wronskian vector of a function, and the vector of -Appell polynomials together with the -deformed matrix multiplication from the authors recent article are the main ingredients in the process. Beyond these results, we give a characterization of -Appell numbers, improving on Al-Salam 1967. Finally, we find a -difference equation for the -Appell polynomial of degree .

scholarly journals A class of big (p,q)-Appell polynomials and their associated difference equations

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3085-3121
Author(s):  
H.M. Srivastava ◽  
B.Y. Yaşar ◽  
M.A. Özarslan
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Recurrence Relation ◽  
Difference Equations ◽  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Equivalence Theorem ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Special Case

In the present paper, we introduce and investigate the big (p,q)-Appell polynomials. We prove an equivalance theorem satisfied by the big (p, q)-Appell polynomials. As a special case of the big (p,q)- Appell polynomials, we present the corresponding equivalence theorem, recurrence relation and difference equation for the big q-Appell polynomials. We also present the equivalence theorem, recurrence relation and differential equation for the usual Appell polynomials. Moreover, for the big (p; q)-Bernoulli polynomials and the big (p; q)-Euler polynomials, we obtain recurrence relations and difference equations. In the special case when p = 1, we obtain recurrence relations and difference equations which are satisfied by the big q-Bernoulli polynomials and the big q-Euler polynomials. In the case when p = 1 and q ? 1-, the big (p,q)-Appell polynomials reduce to the usual Appell polynomials. Therefore, the recurrence relation and the difference equation obtained for the big (p; q)-Appell polynomials coincide with the recurrence relation and differential equation satisfied by the usual Appell polynomials. In the last section, we have chosen to also point out some obvious connections between the (p; q)-analysis and the classical q-analysis, which would show rather clearly that, in most cases, the transition from a known q-result to the corresponding (p,q)-result is fairly straightforward.

scholarly journals Matrix approach to hypercomplex Appell polynomials

2017 ◽  
Vol 116 ◽  
pp. 2-9 ◽  
Author(s):  
Lidia Aceto ◽  
Helmut Robert Malonek ◽  
Graça Tomaz
Keyword(s):  
Matrix Approach ◽  
Appell Polynomials

scholarly journals Two examples related to conical energies

2022 ◽  
Vol 47 (1) ◽  
pp. 261-281
Author(s):  
Damian Dąbrowski
Keyword(s):  
Singular Integral ◽  
Recent Article ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
The Other ◽  
Independent Interest ◽  
Sufficient Condition ◽  
Lipschitz Graphs ◽  
Big Pieces

In a recent article (2021) we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and Mörters (2000), the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.

scholarly journals On the Polynomial Solution of Divided-Difference Equations of the Hypergeometric Type on Nonuniform Lattices

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 47 ◽  
Author(s):  
Mama Foupouagnigni ◽  
Salifou Mboutngam
Keyword(s):  
Difference Equation ◽  
Divided Difference ◽  
Polynomial Solution ◽  
Formal Proof ◽  
Second Order ◽  
Fixed Degree ◽  
Hypergeometric Type ◽  
The Mean ◽  
The Right

In this paper, we provide a formal proof of the existence of a polynomial solution of fixed degree for a second-order divided-difference equation of the hypergeometric type on non-uniform lattices, generalizing therefore previous work proving existence of the polynomial solution for second-order differential, difference or q-difference equation of hypergeometric type. This is achieved by studying the properties of the mean operator and the divided-difference operator as well as by defining explicitly, the right and the “left” inverse for the second operator. The method constructed to provide this formal proof is likely to play an important role in the characterization of orthogonal polynomials on non-uniform lattices and might also be used to provide hypergeometric representation (when it does exist) of the second solution—non polynomial solution—of a second-order divided-difference equation of hypergeometric type.

scholarly journals New Generalized Apostol-Frobenius-Euler polynomials and their Matrix Approach

2021 ◽  
Vol 45 (03) ◽  
pp. 393-407
Author(s):  
MARÍA JOSÉ ORTEGA ◽  
WILLIAM RAMÍREZ ◽  
ALEJANDRO URIELES
Keyword(s):  
Polynomial Matrix ◽  
Product Formula ◽  
Euler Polynomial ◽  
Euler Polynomials ◽  
Matrix Approach ◽  
Pascal Matrix ◽  
Differential Properties ◽  
Euler Matrix

In this paper, we introduce a new extension of the generalized Apostol-Frobenius-Euler polynomials ℋn[m−1,α](x; c,a; λ; u). We give some algebraic and differential properties, as well as, relationships between this polynomials class with other polynomials and numbers. We also, introduce the generalized Apostol-Frobenius-Euler polynomials matrix ????[m−1,α](x; c,a; λ; u) and the new generalized Apostol-Frobenius-Euler matrix ????[m−1,α](c,a; λ; u), we deduce a product formula for ????[m−1,α](x; c,a; λ; u) and provide some factorizations of the Apostol-Frobenius-Euler polynomial matrix ????[m−1,α](x; c,a; λ; u), which involving the generalized Pascal matrix.

scholarly journals Reliability Characterization of Binary-Imaged Multi-State Coherent Threshold Systems

2020 ◽  
Vol 6 (1) ◽  
pp. 309-321
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Fares Ahmad Muhammad Ghaleb
Keyword(s):  
System Structure ◽  
System Failure ◽  
Weighted Sum ◽  
Zero Level ◽  
Algebraic Expressions ◽  
Sum Of Products ◽  
Expected Values ◽  
System Success ◽  
Karnaugh Map

A notable reliability model is the binary threshold system (also called the weighted-k-out-of-n system), which is a dichotomous system that is successful if and only if the weighted sum of its component successes exceeds or equals a particular threshold. The aim of this paper is to extend the utility of this model to the reliability analysis of a homogeneous binary-imaged multi-state coherent threshold system of (m+1) states, which is a non-repairable system with independent non-identical components. The paper characterizes such a system via switching-algebraic expressions of either system success or system failure at each non-zero level. These expressions are given either (a) as minimal sum-of-products formulas, or (b) as probability–ready expressions, which can be immediately converted, on a one-to-one basis, into probabilities or expected values. The various algebraic characterizations can be supplemented by a multitude of map representations, including a single multi-value Karnaugh map (MVKM) (giving a superfluous representation of the system structure function S), (m+1) maps of binary entries and multi-valued inputs representing the binary instances of S, or m maps, again of binary entries and multi-valued inputs, but now representing the success/failure at every non-zero level of the system. We demonstrate how to reduce these latter maps to conventional Karnaugh maps (CKMs) of much smaller sizes. Various characterizations are inter-related, and also related to pertinent concepts such as shellability of threshold systems, and also to characterizations via minimal upper vectors or via maximal lower vectors.

scholarly journals Associated Consistency Characterization of Two Linear Values for Tu Games by Matrix Approach

2012 ◽  
Author(s):  
Genjiu Xu ◽  
René van den Brink ◽  
Gerard van der Laan ◽  
Hao Sun
Keyword(s):  
Matrix Approach ◽  
Tu Games ◽  
Associated Consistency

scholarly journals About the Stabilization of a Nonlinear Perturbed Difference Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
M. De la Sen
Keyword(s):  
Fixed Point ◽  
Difference Equation ◽  
Slight Modification ◽  
Unmodeled Dynamics ◽  
Nonlinear Difference Equations ◽  
Oscillatory Solutions ◽  
Fixed Point Principle ◽  
Single Controller ◽  
Different Order

This paper investigates the local asymptotic stabilization of a very general class of instable autonomous nonlinear difference equations which are subject to perturbed dynamics which can have a different order than that of the nominal difference equation. In the general case, the controller consists of two combined parts, namely, the feedback nominal controller which stabilizes the nominal (i.e., perturbation-free) difference equation plus an incremental controller which completes the stabilization in the presence of perturbed or unmodeled dynamics in the uncontrolled difference equation. A stabilization variant consists of using a single controller to stabilize both the nominal difference equation and also the perturbed one under a small-type characterization of the perturbed dynamics. The study is based on Banach fixed point principle, and it is also valid with slight modification for the stabilization of unstable oscillatory solutions.

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