Laguerre generating function

G is then a generating function for these integrals, which occur as coefficients in its expansion in powers of u and and it can he evaluated with the use of the generating function for the associated Laguerre polynomials, given in equation (19). Thus we have... 

The Laguerre polynomials Lkip) are defined by means of the generating function g(p, s)... 

Since the Laguerre polynomial Lk p) divided by k is the coefficient of 5 in the expansion (F.l) of the generating function, we have... 

To demonstrate that Laguerre polynomials in the interval 0 < x < oo may be obtained from the generating function... 

Since LsT x) — ( )s Lr(x) the generating function for the associated Laguerre polynomials follows as... 

It is readily shown from equation (42.1) which defines the generating function for Laguerre polynomials that the associated Laguerre polynomials may be defined by the equation... 

From the generating function for associated Laguerre polynomials... 

In order to obtain the orthogonality and normalization relations of the associate Laguerre polynomials, we make use of the generating function (F. 10). We multiply together g(p, s j), g(p, t j), and the factor p2+ve /> and then integrate over p to give an integral that we abbreviate with the symbol /... 

Laguerre and associated Laguerre polynomials can be found from the following generating functions ... 

An analogous procedure works for Laguerre polynomials. From the generating function (12.150)... 

Therefore, in the light of the Laurent theorem of residues there can be inferred that the complex integral solution of the Hermite equation may produce the Hermite generating function (with the same recipe as was previously done for generating Laguerre function) ... 

This can be done in the same way as proceed with the Laguerre s polynomials, i.e., by considering the product of two Hermite s generating functions ... 

In order to apply the representation theory of so(2,1) to physical problems we need to obtain realizations of the so(2, 1) generators in either coordinate or momentum space. For our purposes the realizations in three-dimensional coordinate space are more suitable so we shall only consider them (for N-dimensional realizations, see Cizek and Paldus, 1977, and references therein). First we shall show how to build realizations in terms of the radial distance and momentum operators, r, pr. These realizations are sufficiently general to express the radial parts of the Hamiltonians we shall consider linearly in the so(2,1) generators. Then we shall obtain the corresponding realizations of the so(2,1) unirreps which are bounded from below. The basis functions of the representation space are simply related to associated Laguerre polynomials. For finding the eigenvalue spectra it is not essential to obtain these explicit realizations of the basis functions, since all matrix elements can... 

It is important to be able to efficiently generate values for the Laguerre functions. There are several ways to do so and each way requires a different amount of computational effort. 

Method A. For low model orders, the Laguerre functions can be generated using Equations (2.4) directly. 

Method B Generating Laguerre Functions Using Polynomials... 

We can now generate the Laguerre functions in Equations (2.4) by setting X = 2pt in the Laguerre polynomials... 

Method D Generating Laguerre Functions Using Difference Equations... 

The radial factor R for the hydrogen atom consists of an exponential factor and a polynomial denoted by G(p). These polynomials are related to the associatedLaguerre functions. Appendix F describes these functions and the Laguerre polynomials of which they are derivatives and gives formulas for generating the polynomials. To express R in terms of r, we use Eqs. (17.3-4), (17.3-8), and (17.3-11) to write... 

Thus we have demonstrated how the L j /(p) polynomials can be generated and that they do satisfy the general associated Laguerre polynomial equation. Schrodinger worked out the Hydrogen orbitals from these functions in his third revolutionary paper [7] and perhaps we can appreciate the patience required to carry the derivation to useful results ... 

The radial functions, R i(r), that are the eigenfunctions of Equation 10.2 can be constructed from a set of polynomials called the Laguerre pol3momials. The Laguerre polynomial of order k can be generated by... 

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