Polynomial generating function

A convenient method of extracting higher moments of the distribution directly from Eq. (8) (to be fully exploited in following sections) is to define the polynomial generating function [10] containing concentrations of species, C =C (0> and the dummy variable x with no physical significance... 

But since x — 1) contains a constant term, if A x 1) is a conventional polynomial, then A x-, )/ x — 1) must also be a conventional polynomial. Thus, all reachable configurations represented by the generating function 1) have the form... 

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 Hermite polynomials // ( ) are defined by means of an infinite series expansion of the generating function g( , ),... 

To obtain the orthogonality and normalization relations for the Hermite polynomials, we multiply together the generating functions g(, 5) and g( , t), both obtained from equation (D.l), and the factor e and then integrate over ... 

The Legendre polynomials Piipi) may be defined as the coefficients of in an infinite series expansion of a generating function g pi, 5)... 

We next derive some recurrence relations for the Legendre polynomials. Differentiation of the generating function g p, s) with respect to s gives... 

The generating functions g " p, s) for the associated Legendre polynomials may be found from equation (E.l) by letting... 

Another relationship for the polynomials Lkip) can be obtained by expanding the generating function g(p, s) in equation (F.l) using (A.l)... 

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... 

At this point, we may proceed in one of two ways, which are mathematically equivalent. In the first procedure, we note that from the generating function (E.l) for Legendre polynomials Pi, equation (J.3) may be written as... 

It is analogous to die generating function for the Hermite polynomials % fEq. (94)], although somewhat mote complicated. It can foe used to obtain die useful recursion relations... 

An alternative definition involves the use of a generating function. This method is especially convenient for the evaluation of certain integrals of the Hermite polynomials and can be applied to other polynomials as well. For the Hermite polynomials the generating function can be written as... 

The associated Legendre polynomials can be defined by the generating function... 

In the last equation Hi(x) is the th Hermite polynomial. The reader may readily recognize that the functions look familiar. Indeed, these functions are identical to the wave functions for the different excitation levels of the quantum harmonic oscillator. Using the expansion (2.56), it is possible to express AA as a series, as has been done before for the cumulant expansion. To do so, one takes advantage of the linearization theorem for Hermite polynomials [42] and the fact that exp(-t2 + 2tx) is the generating function for these polynomials. In practice, however, it is easier to carry out the integration in (2.12) numerically, using the representation of Po(AU) given by expressions (2.56) and (2.57). 

The solutions of differential equations often define series of related functions that can be obtained from simple generating functions or formulae. As an example consider the Legendre polynomials... 

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... 

Theorem 6.1. The generating function of the Poincare polynomials of the Hilbert scheme parameterizing n-points in X, is given by... 

Exercise. Find a differential equation for the generating function F(z, t) supposing that rn and gn are polynomials in n. 

The connection between these associated Legendre polynomials and the angular shape of certain real atomic orbitals px, pz, dxz, dyz, dxz, dxi yZ, and dz2 is explicitly made between "quotes " the connection is exact for m 0, but inexact for m O, where the ( -dependence of exp( imq>) gets involved. From the generating function... 

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 /... 

---