scholarly journals ON A FAMILY OF (p, q)-HYBRID POLYNOMIALS

2021 ◽  
Vol 45 (03) ◽  
pp. 409-426
Author(s):  
GHAZALA YASMIN ◽  
ABDULGHANI MUHYI
Keyword(s):  
Generating Function ◽  
Graphical Representations ◽  
Appell Polynomials ◽  
Function Series ◽  
Definition Of

In this paper, the class of (p,q)-Bessel-Appell polynomials is introduced. The generating function, series definition and determinant definition of this class are established. Certain members of (p,q)-Bessel-Appell polynomials are considered and some properties of these members are also derived. Further, the class of 2D (p,q)-Bessel-Appell polynomials is introduced by means of the generating function and series definition. In addition, the graphical representations of some members of (p,q)-Bessel-Appell polynomials and 2D (p,q)-Bessel-Appell polynomials are plotted with the help of Matlab.

scholarly journals Certain Results of q -Sheffer–Appell Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 159 ◽  
Author(s):  
Ghazala Yasmin ◽  
Abdulghani Muhyi ◽  
Serkan Araci
Keyword(s):  
Generating Function ◽  
Appell Polynomials ◽  
Function Series

In this paper, the class of q -Sheffer–Appell polynomials is introduced. The generating function, series definition, determinant definition and some other identities of this class are established. Certain members of q -Sheffer–Appell polynomials are investigated and some properties of these members are derived. In addition, the class of 2D q -Sheffer–Appell polynomials is introduced. Further, the graphs of some members of q -Sheffer–Appell polynomials and 2D q -Sheffer–Appell polynomials are plotted for different values of indices by using Matlab.

scholarly journals Certain Results for the Twice-Iterated 2D q-Appell Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1307 ◽  
Author(s):  
Hari M. Srivastava ◽  
Ghazala Yasmin ◽  
Abdulghani Muhyi ◽  
Serkan Araci
Keyword(s):  
Generating Function ◽  
Difference Equations ◽  
Subject Matter ◽  
Mixed Type ◽  
Recurrence Relations ◽  
Appell Polynomials ◽  
Function Series ◽  
Potential Applications ◽  
The Subject ◽  
Polynomial Class

In this paper, the class of the twice-iterated 2D q-Appell polynomials is introduced. The generating function, series definition and some relations including the recurrence relations and partial q-difference equations of this polynomial class are established. The determinant expression for the twice-iterated 2D q-Appell polynomials is also derived. Further, certain twice-iterated 2D q-Appell and mixed type special q-polynomials are considered as members of this polynomial class. The determinant expressions and some other properties of these associated members are also obtained. The graphs and surface plots of some twice-iterated 2D q-Appell and mixed type 2D q-Appell polynomials are presented for different values of indices by using Matlab. Moreover, some areas of potential applications of the subject matter of, and the results derived in, this paper are indicated.

System of thermodynamic quantities in the McMillan-Mayer solution theory

1985 ◽  
Vol 50 (4) ◽  
pp. 791-798 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Generating Function ◽  
Simple Form ◽  
Unit Volume ◽  
Thermodynamic Quantities ◽  
Solution Theory ◽  
Pure Solvent ◽  
Second Derivatives ◽  
Definition Of ◽  
Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

On the symplectically invariant variational principle and generating functions

1990 ◽  
Vol 431 (1883) ◽  
pp. 403-417 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Generating Function ◽  
Generating Functions ◽  
Geometric Approach ◽  
Canonical Transformations ◽  
Weyl Transform ◽  
Analogous Relation ◽  
Definition Of ◽  
Symplectic Invariance

The generating function for canonical transformations derived by Marinov has the important property of symplectic invariance (i. e. under linear canonical transformations). However, a more geometric approach to the rederivation of this function from the variational principle reveals that it is not free from caustic singularities after all. These singularities can be avoided without breaking the symplectic invariance by the definition of a complementary generating function bearing an analogous relation to the Woodward ambiguity function in telecommunications theory as that tying Marinov’s function to the Wigner function and the Weyl transform in quantum mechanics. Marinov’s function is specially apt to describe canonical transformations close to the identity, but breaks down for reflections through a point in phase space, easily described by the new generating function.

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 THE MOMENTS OF MINKOWSKI QUESTION MARK FUNCTION: THE DYADIC PERIOD FUNCTION

2009 ◽  
Vol 52 (1) ◽  
pp. 41-64 ◽  
Author(s):  
GIEDRIUS ALKAUSKAS
Keyword(s):  
Integral Equation ◽  
Generating Function ◽  
Period Function ◽  
Question Mark ◽  
Real Distribution ◽  
Exponential Generating Function ◽  
Wave Forms ◽  
Schmidt Operator ◽  
Definition Of ◽  
Minkowski Question Mark Function

AbstractThe Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane \ (1, ∞). The exponential generating function satisfies an integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert–Schmidt operator. Finally, we describe p-adic distribution of rationals in the Stern–Brocot tree. Surprisingly, the Eisenstein series G2(z) does manifest in both real and p-adic cases.

scholarly journals On certain generalized q-Appell polynomial expansions

2014 ◽  
Vol 68 (2) ◽  
Author(s):  
Thomas Ernst
Keyword(s):  
Generating Function ◽  
Difference Operators ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Recurrence Formulas ◽  
Symmetry Relations ◽  
Polynomial Expansions

AbstractWe study q-analogues of three Appell polynomials, the H-polynomials, the Apostol-Bernoulli and Apostol-Euler polynomials, whereby two new q-difference operators and the NOVA q-addition play key roles. The definitions of the new polynomials are by the generating function; like in our book, two forms, NWA and JHC are always given together with tables, symmetry relations and recurrence formulas. It is shown that the complementary argument theorems can be extended to the new polynomials as well as to some related polynomials. In order to find a certain formula, we introduce a q-logarithm. We conclude with a brief discussion of multiple q-Appell polynomials.

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.

scholarly journals Sample Stratification in Verification of Ensemble Forecasts of Continuous Scalar Variables: Potential Benefits and Pitfalls

2017 ◽  
Vol 145 (9) ◽  
pp. 3529-3544 ◽  
Author(s):  
Joseph Bellier ◽  
Isabella Zin ◽  
Guillaume Bontron
Keyword(s):  
General Framework ◽  
Graphical Representations ◽  
Ensemble Forecasts ◽  
Probability Score ◽  
Numerical Example ◽  
Continuous Ranked Probability Score ◽  
Potential Benefits ◽  
Rank Histogram ◽  
Areal Precipitation ◽  
Definition Of

In the verification field, stratification is the process of dividing the sample of forecast–observation pairs into quasi-homogeneous subsets, in order to learn more on how forecasts behave under specific conditions. A general framework for stratification is presented for the case of ensemble forecasts of continuous scalar variables. Distinction is made between forecast-based, observation-based, and external-based stratification, depending on the criterion on which the sample is stratified. The formalism is applied to two widely used verification measures: the continuous ranked probability score (CRPS) and the rank histogram. For both, new graphical representations that synthesize the added information are proposed. Based on the definition of calibration, it is shown that the rank histogram should be used within a forecast-based stratification, while an observation-based stratification leads to significantly nonflat histograms for calibrated forecasts. Nevertheless, as previous studies have warned, statistical artifacts created by a forecast-based stratification may still occur, thus a graphical test to detect them is suggested. To illustrate potential insights about forecast behavior that can be gained from stratification, a numerical example with two different datasets of mean areal precipitation forecasts is presented.

scholarly journals Construction of the Type 2 Poly-Frobenius-Genocchi Polynomials with Their Certain Applications

2020 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Genocchi Polynomials ◽  
Definition Of ◽  
Special Case

Motivated by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim, in the present paper, we consider a class of new generating function for the Frobenius-Genocchi polynomials, called the type 2 poly-Frobenius-Genocchi polynomials, by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Frobenius-Genocchi polynomias equal a linear combination of the classical Frobenius-Genocchi polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Frobenius-Genocchi polynomials and Bernoulli polynomials of order k. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim, we introduce the unipoly-Frobenius-Genocchi polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Frobenius-Genocchi polynomials and the classical Frobenius-Genocchi polynomials.

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