Etant donne un nombre quelconque de quantites numeriques α1, α2, ..., α n , on sail qu’on peut en approcher simultanement par des fractions de memo denominatcur, de telle sortc qu’on ait 

$$\begin{array}{l} {\alpha _1} = \frac{{{A_1}}}{A} + \frac{{{\partial _1}}}{{A\sqrt A }}, \\{\alpha _2} = \frac{{{A_2}}}{A} + \frac{{{\partial _2}}}{{A\sqrt[n]{A}}}, \\ , \\ {\alpha _n} = \frac{{{A_n}}}{A} + \frac{{{\partial _n}}}{{A\sqrt[n]{A}}}, \\ \end{array}$$

∂1, ∂2, …, ∂ n ne pouvant depasser une limite qui depend seulement de n. C’est, comme on voit, une extension du mode d’approximation resultant de la theorie; des fractions continues, qui correspondrait au cas le plus simple de n = 1. Or, on peut se proposer une generalisation semblable de la theorie des fractions continues algebriques, en cherchant les expressions approchees de n fonctions φ1 (x), φ2(x), ..., φ n (x) par des fractions ralionnelles $ \frac{{{\Phi _1}\left( x \right)}}{{\Phi \left( x \right)}},\frac{{{\Phi _2}\left( x \right)}}{{\Phi \left( x \right)}},....,\frac{{{\Phi _n}\left( x \right)}}{{\Phi \left( x \right)}}$, de maniere crue les developpemcnts en serie suivant les puissances croissantcs de la variable coincident jusqu’a une puissance determinec x M. Voici d’abord, a cet egard, un premier resultat qui s’oflre immediatcment.

Sur la Fonction Exponentielle

The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the variable scaled by a factor depending on the degree. This note presents an elementary proof of the orthogonality of wave functions with differing energy levels. It is also shown that this is the only other natural orthogonality for Laguerre polynomials. By expanding in terms of the usual Laguerre polynomial basis an analogous strange orthogonality is obtained for Meixner polynomials.

A Laguerre Polynomial Orthogonality and the Hydrogen Atom

In this paper, we perform a further investigation for the Fubini polynomials. By making use of the generating function methods and Pade approximation techniques, we establish some new identities for the two variable Fubini polynomials. Some special cases as well as immediate consequences of the main results presented here are also considered.

https://www.mdpi.com/2227-7390/7/2/115/pdf

Some Identities for the Two Variable Fubini Polynomials

The fertile marriage between the two Glauber parity and displacement operators

Abstract In this study, a collocation approach is presented to solve Bagley–Torvik equation, which is a class of fractional differential equations. As most fractional differential equations do not have exact analytical solutions, it is needed numerical methods. This study is important because it presents a numerical method for fractional differential equations. The main purpose of this method is to obtain the approximate solution based on Laguerre polynomials of the Bagley–Torvik equation. To date, a collocation method based on the Laguerre polynomials has not been studied for the solutions of the Bagley–Torvik equation. This reveals the novelty of the study. The approximate solution is sought in form of the fractional powers of the Laguerre polynomials. By using the Caputo derivative, the matrix relation is created for term with fractional derivative in the equation. Similarly, the matrix relation of second derivative is computed in equation. Then, by using these matrix relations and the collocation points, the Bagley–Torvik problem is converted into a system of the linear algebraic equations. The solution of this system gives the coefficients of the assumed solution. Secondly, an error estimation method is given with the help of the residual function and also the Laguerre polynomial solution is improved by using the estimated error function. Then, the method is applied to four examples and the obtained numerical results are shown in tables and graphs. Also, the comparisons are made with other methods in the literature and so the presented method gives better results than other methods.

Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials

In this paper, we perform a further investigation for the generalized Laguerre polynomials. By applying the generating function methods and Padé approximation techniques, we establish some new identities for the generalized Laguerre polynomials, and give some illustrative special cases as well as immediate consequences of the main results. c ©2016 All rights reserved.

/pdf/some-identities-for-the-generalized-laguerre-polynomials-krgowmorxn.pdf

Some identities for the generalized Laguerre polynomials

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Some identities for the generalized Laguerre polynomials