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Sama Arjika

Publications and source records attributed to Sama Arjika.

At least 19 recordsLinked to original sources
arXiv2022

On certain properties of perturbed Freud-type weight: a revisit

In this paper, monic polynomials orthogonal with deformation of the Freud-type weight function are considered. These polynomials fullfill linear differential equation with some polynomial coefficients in their holonomic form. The aim of this work is explore certain characterizing properties of perturbed Freud type polynomials such as nonlinear recursion relations, finite moments, differential-recurrence and differential relations satisfied by the recurrence coefficients as well as the corresponding semiclassical orthogonal polynomials. We note that the obtained differential equation fulfilled by the considered semiclassical polynomials are used to study an electrostatic interpretation for the distribution of zeros based on the original ideas of Stieltjes.

math.CA↗
arXiv2021

Generalized q-difference equations for general q-polynomials with double q-binomial coefficients

In this paper, we use the generalized q-polynomials with double q-binomial coefficients and homogeneous q-operators [J. Difference Equ. Appl. 20 (2014), 837--851.] to construct q-difference equations with seven variables, which generalize recent works of Jia et al [Symmetry 2021, 13, 1222.]. In addition, we derive Rogers formulas, extended Rogers formulas and Srivastava--Agarwal type bilinear generating functions for generalized q-polynomials, which generalize generating functions for Cigler's polynomials [J. Difference Equ. Appl. 24 (2018), 479--502.]. Finally, we also derive mixed generating functions using q-difference equations.

math.CO↗
arXiv2021

A General Family of $q$-Hypergeometric Polynomials and Associated Generating Functions

In this paper, we introduce a general family of $q$-hypergeometric polynomials and investigate several $q$-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of $q$-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized $q$-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various $q$-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called $(p,q)$-variations of the $q$-results, which we have investigated here, because the additional parameter $p$ is obviously redundant.

math.CO↗
arXiv2021

$q$-difference equation for generalized trivariate $q$-Hahn polynomials

In this paper, we introduce a family of trivariate $q$-Hahn polynomials $Ψ_n^{(a)}(x,y,z|q)$ as a general form of Hahn polynomials $ψ_n^{(a)}(x|q),$ $ψ_n^{(a)}(x,y|q)$ and $F_n(x,y,z;q)$. We represent $Ψ_n^{(a)}(x,y,z|q)$ by the homogeneous $q$-difference operator $\widetilde{L}(a,b; θ_{xy})$ introduced by Srivastava {\it et al} [H. M. Srivastava, S. Arjika and A. Sherif Kelil, {\it Some homogeneous $q$-difference operators and the associated generalized Hahn polynomials}, Appl. Set-Valued Anal. Optim. {\bf 1} (2019), pp. 187--201.] to derive: extended generating, Rogers formula, extended Rogers formula and Srivastava-Agarwal type generating functions involving $Ψ_n^{(a)}(x,y,z|q)$ by the $q$-difference equation.

math.CA↗
arXiv2021

Certain generating functions for Cigler's polynomials

In this paper, we use the homogeneous $q$-operators [J. Difference Equ. Appl. {\bf20 } (2014), 837--851.] to derive Rogers formulas, extended Rogers formulas and Srivastava-Agarwal type bilinear generating functions for Cigler's polynomials [J. Difference Equ. Appl. {\bf 24} (2018), 479--502.]. Finally, we also derive two interesting transformation formulas between ${}_2Φ_1, \, {}_2Φ_2$ and ${}_3Φ_2$.

math.CO↗
arXiv2021

A note on fractional Askey--Wilson integrals

In this paper, we generalize fractional $q$-integrals by the method of $q$-difference equation. In addition, we deduce fractional Askey--Wilson integral, reversal type fractional Askey--Wilson integral and Ramanujan type fractional Askey--Wilson integral.

math.CA↗
arXiv2020

A note on generalized $q$-difference equations for general Al-Salam--Carlitz polynomials

In this paper, we deduce the generalized $q$-difference equations for general Al-Salam--Carlitz polynomials and generalize Arjika's recently results [$q$-difference equation for homogeneous $q$-difference operators and their applications, J. Differ. Equ. Appl. {\bf 26}, 987--999 (2020)]. In addition, we obtain transformational identities by the method of $q$-difference equation. Moreover, we deduce $U(n+1)$ type generating functions and Ramanujan's integrals involving general Al-Salam--Carlitz polynomials by $q$-difference equation.

math.CO↗
arXiv2020

A Note on Generalized $q$-Difference Equations and Their Applications Involving $q$-Hypergeometric Functions

In this paper, we use two $q$-operators $\mathbb{T}(a,b,c,d,e,yD_x)$ and $\mathbb{E}(a,b,c,d,e,yθ_x)$ to derive two potentially useful generalizations of the $q$-binomial theorem, a set of two extensions of the $q$-Chu-Vandermonde summation formula and two new generalizations of the Andrews-Askey integral by means of the $q$-difference equations. We also briefly describe relevant connections of various special cases and consequences of our main results with a number of known results.

math.CO↗
arXiv2020

Generating Functions for Some Families of the Generalized Al-Salam-Carlitz $q$-Polynomials

In this paper, by making use of the familiar $q$-difference operators $D_q$ and $D_{q^{-1}}$, we first introduce two homogeneous $q$-difference operators $\mathbb{T}({\bf a},{\bf b},cD_q)$ and $\mathbb{E}({\bf a},{\bf b}, cD_{q^{-1}})$, which turn out to be suitable for dealing with the families of the generalized Al-Salam-Carlitz $q$-polynomials $ϕ_n^{({\bf a},{\bf b})}(x,y|q)$ and $ψ_n^{({\bf a},{\bf b})}(x,y|q)$. We then apply each of these two homogeneous $q$-difference operators in order to derive generating functions, Rogers type formulas, the extended Rogers type formulas and the Srivastava-Agarwal type linear as well as bilinear generating functions involving each of these families of the generalized Al-Salam-Carlitz $q$-polynomials. We also show how the various results presented here are related to those in many earlier works on the topics which we study in this paper.

math.CA↗
arXiv2019

On $q^2$-trigonometric functions and their $q^2$-Fourier transform

In this paper, we first construct generalized $q^2$-cosine, $q^2$-sine and $q^2$-exponential functions. We then use $q^2$-exponential function in order to define and investigate a $q^2$-Fourier transform. We establish $q$-analogues of inversion and Plancherel theorems.

math-ph↗
arXiv2019

Quantum statistical properties of multiphoton hypergeometric coherent states and the discrete circle representation

We review the definition of hypergeometric coherent states, discussing some representative examples. Then we study mathematical and statistical properties of hypergeometric Schrödinger cat states, defined as orthonormalized eigenstates of $k$-th powers of nonlinear $f$-oscillator annihilation operators, with $f$ of hypergeometric type. These "$k$-hypercats" can be written as an equally weighted superposition of hypergeometric coherent states $|z_l\rangle, l=0,1,\dots,k-1$, with $z_l=z e^{2πi l/k}$ a $k$-th root of $z^k$, and they interpolate between number and coherent states. This fact motivates a continuous circle representation for high $k$. We also extend our study to truncated hypergeometric functions (finite dimensional Hilbert spaces) and a discrete exact circle representation is provided. We also show how to generate $k$-hypercats by amplitude dispersion in a Kerr medium and analyze their generalized Husimi $Q$-function in the super- and sub-Poissonian cases at different fractions of the revival time.

math-ph↗
arXiv2019

Some homogeneous $q$-difference operators and the associated generalized Hahn polynomials

In this paper, we first construct the homogeneous $q$-shift operator $\widetilde{E}(a,b;D_{q})$ and the homogeneous $q$-difference operator $\widetilde{L}(a,b; θ_{xy})$. We then apply these operators in order to represent and investigate generalized Cauchy and a general form of Hahn polynomials. We derive some $q$-identities such as: generating functions, extended generating functions, Mehler's formula and Roger's formula for these $q$-polynomials.

math.CA↗
arXiv2019

Summation formula for generalized discrete $q$-Hermite II polynomials

In this paper, we provide a family of generalized discrete $q$-Hermite II polynomials denoted by $\tilde{h}_{n,α}(x,y|q)$. An explicit relations connecting them with the $q$-Laguerre and Stieltjes-Wigert polynomials are obtained. Summation formula is derived by using different analytical means on their generating functions.

math-ph↗
arXiv2014

Even and odd generalized hypergeometric coherent states

In this paper, we investigate a large class of generalized hypergeometric states $|p,q,z\rangle$, depending on a complex variable $z$ and two sets of parameters, $(a_1,\cdots,a_p)$ and $(b_1,\cdots,b_q)$. Even and odd generalized hypergeometric states $|p,q,z\rangle_e$ and $|p,q,z\rangle_o$ are also defined and analyzed. The moment problem is solved by the Mellin transform techniques. For particular values of $p$ and $q$, the photon-counting statistics, quantum optical properties and geometry of these states are discussed.

math-ph↗