On the ω-multiple Charlier polynomials

The main aim of this paper is to define and investigate more general multiple Charlier polynomials on the linear lattice $\omega \mathbb{N} = \{ 0,\omega ,2\omega ,\ldots \} $ ω N = { 0 , ω , 2 ω , … } , $\omega \in \mathbb{R}$ ω ∈ R . We call these polynomials ω-multiple Charlier polynomials. Some of their properties, such as the raising operator, the Rodrigues formula, an explicit representation and a generating function are obtained. Also an $( r+1 )$ ( r + 1 ) th order difference equation is given. As an example we consider the case $\omega =\frac{3}{2}$ ω = 3 2 and define $\frac{3}{2}$ 3 2 -multiple Charlier polynomials. It is also mentioned that, in the case $\omega =1$ ω = 1 , the obtained results coincide with the existing results of multiple Charlier polynomials.

HAHN'S PROBLEM WITH RESPECT TO SOME PERTURBATIONS OF THE RAISING OPERATOR \(X-c\)

In this paper, we study the Hahn's problem with respect to some raising operators perturbed of the operator \(X-c\), where \(c\) is an arbitrary complex number. More precisely, the two following characterizations hold: up to a normalization, the \(q\)-Hermite (resp. Charlier) polynomial is the only \(H_{\alpha,q}\)-classical (resp. \(\mathcal{S}_{\lambda}\)-classical) orthogonal polynomial, where \(H_{\alpha, q}:=X+\alpha H_q\) and \(\mathcal{S}_{\lambda}:=(X+1)-\lambda\tau_{-1}.\)

The water industry and decarbonisation of cities: A review

: The urban water industry is a very energy intensive industry. Higher water quality standards are driving a level of energy growth that is threatening to move it to the top rank. Climate change is further exacerbating this situation: Growing aridity is variously imposing an enhanced carbon burden through water recycling, trans-regional pipelines and desalination plants. Natural disasters too can often affect water quality requiring energy hungry mitigations. There’s clear evidence that a failure to appropriately weight energy considerations in water infrastructure is commonplace and that this is an unsustainable position for the industry and is prejudicial to working towards zero carbon cities. Real time tracking of CO2e emissions is an important starting point in raising operator consciousness and introducing rivalry between utilities in attaining abatement. So too is reaching out to the resource and manufacturing sectors to form strategic alliances as well as seeking to enter into closer relationships with the energy sector.

The cosmological bootstrap: weight-shifting operators and scalar seeds

A key insight of the bootstrap approach to cosmological correlations is the fact that all correlators of slow-roll inflation can be reduced to a unique building block — the four-point function of conformally coupled scalars, arising from the exchange of a massive scalar. Correlators corresponding to the exchange of particles with spin are then obtained by applying a spin-raising operator to the scalar-exchange solution. Similarly, the correlators of massless external fields can be derived by acting with a suitable weight-raising operator. In this paper, we present a systematic and highly streamlined derivation of these operators (and their generalizations) using tools of conformal field theory. Our results greatly simplify the theoretical foundations of the cosmological bootstrap program.

MENENTUKAN PERSAMAAN GELOMBANG SCHR�DINGER POTENSIAL NON-SENTRAL SCARF HIPERBOLIK PLUS ROSEN-MORSE TRIGONOMETRIK MENGGUNAKANMETODE SUPERSIMETRI MEKANIKA KUANTUM

Penelitian ini bertujuan untuk menentukan persamaan gelombang Schrdinger potensial non-sentral Scarf hiperbolik plus Rosen-Morse trigonometrik menggunakan metode Supersimetri Mekanika Kuantum (SUSI MK). Persamaangelombang radial diperoleh dari persamaan Schrdinger bagian radial, sedangkan persamaan gelombang sudut diperoleh dari persamaan Schrdinger bagian sudut polar. Penentuan persamaan gelombang tingkat dasar ditentukan dengan sifat lowering operator dan persamaan gelombang tereksitasi ditentukan dengan sifat raising operator. Jadi, baik untuk bagian radial maupun bagian polar ditentukan dengan menggunakan metode operator supersimetri. Adapun tampilan gambar dari fungsi gelombang bagian polar menggunakan aplikasi program maple 12.

Solution of The Schrödinger Equation for Trigonometric Scarf Plus Poschl-Teller Non-Central Potential Using Supersymmetry Quantum Mechanics

<span>In this paper, we show that the exact energy eigenvalues and eigen functions of the Schrödinger <span>equation for charged particles moving in certain class of noncentral potentials can be easily <span>calculated analytically in a simple and elegant manner by using Supersymmetric method <span>(SUSYQM). We discuss the trigonometric Scarf plus Poschl-Teller systems. Then, by operating <span>the lowering operator we get the ground state wave function, and the excited state wave functions <span>are obtained by operating raising operator repeatedly. The energy eigenvalue is expressed in the <span>closed form obtained using the shape invariant properties. The results are in exact agreement with <span>other methods.</span></span></span></span></span></span></span><br /></span>

Solution Of Schrödinger Equation For Poschl-Teller Plus Scarf Non-Central Potential Using Supersymmetry Quantum Mechanics Aproach

In this paper, we show that the exact energy eigenvalues and eigenfunctions of the Schrödinger equation for charged particles moving in a certain class of noncentral potentials can be easily calculated analytically in a simple and elegant manner by using Supersymmetric method (SUSYQM). We discuss the Poschl-Teller plus Scarf non-central potential systems. Then, by operating the lowering operator we get the ground state wave function, and the excited state wave functions are obtained by operating raising operator repeatedly. The energy eigenvalue is expressed in the closed form obtained using the shape invariant properties. The results are in exact agreement with other methods.Keyword: supersymmetry, non-central potentials, poschl teller plus scarf.

A Note on Young's Raising Operator

Consider the following formula due to Young [7] for the calculation of the homogeneous product sum, hλ, in terms of Schur functions;where the operation Srs is defined as follows:Y1: Srs, where r < s, “represents the operation of moving one letter from the s-th row up to the r-th row; and the resulting term is regarded as zero, when any row becomes less than a row below it, or when letters from the same row overlap; as, for instance, happens when λ1 = λ2 in the case of S13S23.“The following example of the above is given by Robinson [4].Calculation by other means shows that the above analysis of h(3,2,1) is correct; however, it will be noticed that the operator S123S23 does not appear in the above yet it is not specifically excluded by the rule Y1.