Recursion coefficients

The recursive relation is the most important property for constructive and computational use of orthogonal polynomials. In fact, as will be shown below, knowledge of the recursion coefficients allows the zeros of the orthogonal polynomials to be computed, and with them the quadrature rule. Therefore the calculation of the coefficients of this three-term recurrence relation is of paramount importance. The recursive relationship in Eq. (3.5) generates a sequence of monic polynomials that are orthogonal with respect to the weight function... 

This relation generates the continued fraction that is, in general, infinite. Explicit expressions were derived for the continued recursion coefficients in terms of the elementary reaction rates. The method has also been applied for cases when two-step jumps occur. 

These statements are a consequence of the recursion relations obtained by identifying the coefficients of the power series expansion on the right- and left-hand side of the equation. For example, in (4.6), the coefficient of x" is (n > 1) on the left-hand side, and on the right-hand side a polynomial in R, . [cf. (2.56)], which implies the uniqueness. The coefficients of the polynomial mentioned are non-negative the term occurs, coming from x/, thus Rj > n-i statements that the coefficients are... 

Equations (7-46) and (7-47) connect coefficients with the same j but different m. Other useful recursion relations for C with equal 3 and m result from application of in the form (7-45) to Eq. (7-35), viz. ... 

Campbell s Theorem, 174 Cartwright, M. L., 388 Caywood, T. E., 313 C-coefficients, 404 formulas for, 406 recursion relations, 406 relation to spherical harmonics, 408 tabulations of, 408 Wigner s formula, 408 Central field Dirac equation in, 629 Central force law... 

A table of binomial coefficients is given below (table IV). The numbers resulting from the successive partial differentiations, are given in table V, which can easily be extended with the use of the recursion formula... 

For some differential equations, the two roots and S2 of the indicial equation differ by an integer. Under this circumstance, there are two possible outcomes (a) steps 1 to 6 lead to two independent solutions, or (b) for the larger root 5i, steps 1 to 6 give a solution mi, but for the root S2 the recursion relation gives infinite values for the coefficients a beyond some specific value of k and therefore these steps fail to provide a second solution. For some other differential equations, the two roots of... 

Setting the coefficient of each power of equal to zero gives the recursion formida... 

This result is the recursion formula which allows the coefficient an+2 to he calculated from the coefficient a . Starting with either ao or a an infinite series can be constructed which is even or odd respectively. These two coefficients are of course the two arbitrary constants in the general solution of a second-order differential equation. If one of them, say ao is set equal to zero, the remaining series will contain the constant at and be composed only of odd powers of On the other hand if a 0, the even series will result. It can be shown, however that neither of these infinite series can be accepted as... 

As this relation is correct for all values of s, the coefficients in brackets must vanish The result yields the important recursion formula for the Hermite polynomials,... 

Here again the indices n are independent in each summation, so that n can be replaced by n + 2 in the first term. Then, by posing the coefficient of z" equal to zero, the recursion formula becomes... 

The function y(x) can now be developed in a power series following the method presented in Section 5.2.1. The recursion formula for the coefficients is then of the form... 

The last term on the right in Equation 1.22 represents a doubly reduced matrix element, which can be calculated by recursive formula in terms of the coefficients of fractional parentage [4, 14], tabulated in the work of Nielson and Koster [27]. Finally, Equation 1.18 is rewritten as... 

In this way, all the coefficients Ar of the Frobenius series can be determined step by step. The recursion formula generates two independent series for odd and even values of r. For... 

Once the coefficients for the expansion of the exchange-correlation term have been evaluated, all matrix elements can be calculated analytically. The Obara and Saika [47] recursive scheme has been used for the evaluation of the one and the two electron integrals. The total energy is therefore expressed in terms of the fitting coefficients for the electronic density and the exchange-correlation potential. 

The recursive formulas that have been presented exhibit some advantages over classical batch processing. First, they avoid the inversion of the normal coefficient matrix, since we would usually process a few equations at a time. Obviously, when only one equation is involved each time, the inversion degenerates into computing the reciprocal of a scalar. Furthermore, these sequential relationships can also be used to isolate systematic errors that may be present in the data set, as will be shown in the next chapter. 

Because of a different normalization , the coefficients of the parentheses are not identical for the Gram-Schmidt orthogonalization and for the recursion formula. 

Within this approach (called (SC) because it is size-consistent if localized orbitals are used and the coefficients are, in practice, yielded by a recursive self-consistent procedure) the diagonal elements An of the dressing operator become ... 

An automated and optimal subtraction procedure was previously developed for Eq. (8). The algorithm takes one spectrum after another and optimally subtracts previous signals. The criterion used is minimization of the signal entropy (see Section 4.5.2). The recursive equation used appears in Eq. (9), where is an experimental spectrum, is a set of reference spectra, and x are scalar coefficients. The values of x are easily determined. 

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