scholarly journals Incomplete generalized Vieta–Pell and Vieta–Pell–Lucas polynomials

2021 ◽  
Vol 27 (4) ◽  
pp. 245-256
Author(s):  
Bahar Kuloğlu ◽  
◽  
Engin Özkan ◽  
Anthony G. Shannon ◽  
◽  
...  
Keyword(s):  
Generating Function ◽  
Recurrence Relations ◽  
Lucas Polynomials

In this paper, we introduce the incomplete Vieta–Pell and Vieta–Pell–Lucas polynomials. We give some properties, the recurrence relations and the generating function of these polynomials with suggestions for further research.

scholarly journals Incomplete Bivariate Fibonacci and Lucas -Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Dursun Tasci ◽  
Mirac Cetin Firengiz ◽  
Naim Tuglu
Keyword(s):  
Generating Function ◽  
Lucas Numbers ◽  
Lucas Polynomials ◽  
Fibonacci And Lucas Numbers

We define the incomplete bivariate Fibonacci and Lucas polynomials. In the case , , we obtain the incomplete Fibonacci and Lucas numbers. If , , we have the incomplete Pell and Pell-Lucas numbers. On choosing , , we get the incomplete generalized Jacobsthal number and besides for the incomplete generalized Jacobsthal-Lucas numbers. In the case , , , we have the incomplete Fibonacci and Lucas numbers. If , , , , we obtain the Fibonacci and Lucas numbers. Also generating function and properties of the incomplete bivariate Fibonacci and Lucas polynomials are given.

scholarly journals On the p,q-Humbert Functions from the View Point of the Generating Function Method

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Ayman Shehata
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Function Method ◽  
Recurrence Relations ◽  
View Point ◽  
Starting Point ◽  
Explicit Representations

The main object of the present paper is to construct new p,q-analogy definitions of various families of p,q-Humbert functions using the generating function method as a starting point. This study shows a class of several results of p,q-Humbert functions with the help of the generating functions such as explicit representations and recurrence relations, especially differential recurrence relations, and prove some of their significant properties of these functions.

scholarly journals THE GENERALIZED Q-BESSEL MATRIX FUNCTION OF TWO VARIABLES

2020 ◽  
Vol 1 (4) ◽  
Author(s):  
Fadhl S. N. Alsarahi
Keyword(s):  
Difference Equation ◽  
Bessel Function ◽  
Generating Function ◽  
Matrix Function ◽  
Recurrence Relations ◽  
Special Function ◽  
Applied Mathematics ◽  
Function Of Two Variables

The Bessel function is probably the best known special function, within pure and applied mathematics. In this paper, we introduce the generalized q-analogue Bessel matrix function of two variables. Some properties of this function, such as generating function, q-difference equation, and recurrence relations are obtained.

scholarly journals Multifarious Results for q-Hermite Based Frobenius Type Eulerian Polynomials

2019 ◽  
Author(s):  
Waseem Khan ◽  
Idrees Ahmad Khan ◽  
Mehmet Acikgoz ◽  
Ugur Duran
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Series Representation ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Eulerian Polynomials ◽  
New Class ◽  
Genocchi Polynomials

In this paper, a new class of q-Hermite based Frobenius type Eulerian polynomials is introduced by means of generating function and series representation. Several fundamental formulas and recurrence relations for these polynomials are derived via different generating methods. Furthermore, diverse correlations including the q-Apostol-Bernoulli polynomials, the q-Apostol-Euler poynoomials, the q-Apostol-Genocchi polynomials and the q-Stirling numbers of the second kind are also established by means of the their generating functions.

scholarly journals Notes on $q$-Hermite Based Unified Apostol Type Polynomials

2019 ◽  
Author(s):  
Waseem Khan
Keyword(s):  
Generating Function ◽  
Recurrence Relations ◽  
Series Representation ◽  
Stirling Numbers ◽  
New Class

In this article, a new class of $q$-Hermite based unified Apostol type polynomials is introduced by means of generating function and series representation. Several important formulas and recurrence relations for these polynomials are derived via different generating methods. We also introduce $q$-analog of Stirling numbers of second kind of order $\nu$ by which we construct a relation including aforementioned polynomials.

scholarly journals Families of prudent self-avoiding walks

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Mireille Bousquet-Mélou
Keyword(s):  
Generating Function ◽  
Recurrence Relations ◽  
Square Lattice ◽  
Random Generation ◽  
Fourth Class ◽  
End Distance ◽  
International Audience ◽  
Self Avoiding Walk ◽  
Natural Classes ◽  
Self Avoiding Walks

International audience A self-avoiding walk on the square lattice is $\textit{prudent}$, if it never takes a step towards a vertex it has already visited. Préa was the first to address the enumeration of these walks, in 1997. For 4 natural classes of prudent walks, he wrote a system of recurrence relations, involving the length of the walks and some additional "catalytic'' parameters. The generating function of the first class is easily seen to be rational. The second class was proved to have an algebraic (quadratic) generating function by Duchi (FPSAC'05). Here, we solve exactly the third class, which turns out to be much more complex: its generating function is not algebraic, nor even $D$-finite. The fourth class ―- general prudent walks ―- still defeats us. However, we design an isotropic family of prudent walks on the triangular lattice, which we count exactly. Again, the generating function is proved to be non-$D$-finite. We also study the end-to-end distance of these walks and provide random generation procedures. Un chemin auto-évitant sur le réseau carré est $\textit{prudent}$, s'il ne fait jamais un pas en direction d'un point qu'il a déjà visité. Préa est le premier à avoir cherché à énumérer ces chemins, en 1997. Pour 4 classes naturelles de chemins prudents, il donne un système de relations de récurrence, impliquant la longueur des chemins et plusieurs paramètres "catalytiques'' supplémentaires. La première classe a une série génératrice simple, rationnelle. La deuxième a une série algébrique (quadratique) (Duchi, FPSAC'05). Nous comptons ici les chemins de la troisième classe, et observons un saut de complexité: la série obtenue n'est ni algébrique, ni même différentiellement finie. La quatrième classe, celle des chemins prudents généraux, résiste encore. Cependant, nous définissons un modèle isotrope de chemins prudents sur réseau triangulaire, que nous résolvons de nouveau, la série obtenue n'est pas différentiellement finie. Nous étudions aussi la vitesse d'éloignement de ces chemins, et proposons des algorithmes de génération aléatoire.

scholarly journals Determinant Forms, Difference Equations and Zeros of the q-Hermite-Appell Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 258 ◽  
Author(s):  
Subuhi Khan ◽  
Tabinda Nahid
Keyword(s):  
Generating Function ◽  
Difference Equations ◽  
Recurrence Relations ◽  
Computer Experiment ◽  
Appell Polynomials ◽  
Distribution Of Zeros ◽  
Hybrid Form ◽  
Special Polynomials

The present paper intends to introduce the hybrid form of q-special polynomials, namely q-Hermite-Appell polynomials by means of generating function and series definition. Some significant properties of q-Hermite-Appell polynomials such as determinant definitions, q-recurrence relations and q-difference equations are established. Examples providing the corresponding results for certain members belonging to this q-Hermite-Appell family are considered. In addition, graphs of certain q-special polynomials are demonstrated using computer experiment. Thereafter, distribution of zeros of these q-special polynomials is displayed.

scholarly journals A Lie-theoretic interpretation of multivariate hypergeometric polynomials

2012 ◽  
Vol 148 (3) ◽  
pp. 991-1002 ◽  
Author(s):  
Plamen Iliev
Keyword(s):  
Lie Algebra ◽  
Generating Function ◽  
Hypergeometric Series ◽  
Recurrence Relations ◽  
Multinomial Distribution ◽  
Discrete Variables ◽  
Hypergeometric Polynomials

AbstractIn 1971, Griffiths used a generating function to define polynomials in d variables orthogonal with respect to the multinomial distribution. The polynomials possess a duality between the discrete variables and the degree indices. In 2004, Mizukawa and Tanaka related these polynomials to character algebras and the Gelfand hypergeometric series. Using this approach, they clarified the duality and obtained a new proof of the orthogonality. In the present paper, we interpret these polynomials within the context of the Lie algebra $\mathfrak {sl}_{d+1}(\mathbb {C})$. Our approach yields yet another proof of the orthogonality. It also shows that the polynomials satisfy d independent recurrence relations each involving d2+d+1 terms. This, combined with the duality, establishes their bispectrality. We illustrate our results with several explicit examples.

ON THE SIGNATURE OF COHERENT SYSTEMS AND APPLICATIONS

2007 ◽  
Vol 22 (1) ◽  
pp. 19-35 ◽  
Author(s):  
Ioannis S. Triantafyllou ◽  
Markos V. Koutras
Keyword(s):  
Present Article ◽  
Generating Function ◽  
Recurrence Relations ◽  
Simple Relation ◽  
Function Approach ◽  
Sufficient Condition ◽  
Coherent Systems ◽  
Simple Sufficient Condition ◽  
Circular System ◽  
Preservation Property

In the present article we provide a formula that facilitates the evaluation of the signature of a reliability structure by a generating function approach. A simple sufficient condition is also derived for proving the nonpreservation of the IFR property for the system's lifetime (when the components are IFR) by exploiting the signature of the system. As an application of the general results, we deduce recurrence relations for the signature of a linear consecutive k-out-of-n: F system. We establish a simple relation between the signature of a linear and a circular system and investigate the IFR preservation property under the formulation of such systems.

scholarly journals General-Appell Polynomials within the Context of Monomiality Principle

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Subuhi Khan ◽  
Nusrat Raza
Keyword(s):  
Differential Equation ◽  
Generating Function ◽  
General Class ◽  
Recurrence Relations ◽  
Appell Polynomials ◽  
Special Polynomials ◽  
New Family ◽  
Special Polynomial

A general class of the 2-variable polynomials is considered, and its properties are derived. Further, these polynomials are used to introduce the 2-variable general-Appell polynomials (2VgAP). The generating function for the 2VgAP is derived, and a correspondence between these polynomials and the Appell polynomials is established. The differential equation, recurrence relations, and other properties for the 2VgAP are obtained within the context of the monomiality principle. This paper is the first attempt in the direction of introducing a new family of special polynomials, which includes many other new special polynomial families as its particular cases.

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