Recurrence relation

To obtain the second set of equations, we insert ipi = n 2 = w n = 0 into Equations 4.64 and 4.79 and mi = m2 = m = 0 in the latter. We proceed in the above manner with the exception that the GF as well as its expansion must be differentiated with respect to zj. This gives 

In both sets of equations, our notation suppresses the parametric dependence of In on the spectroscopic parameters. The coefficients c v, g i,v, and dp,v appearing in these equations are given in the Appendix C. Notice the upper indices on andg v 

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An analytical solution for / appears possible only in equilibrium setting the left-hand side of Eq. (26) equal to zero, it can be represented as a recurrence relation... 

The key result in this section will be the derivation of linear recurrence relations for 7)v,p in terms of 7j,p, for j < n [jen 88a]. We begin by introducing an invariance matrix, whose powers correspond to the lattice sizes on which f> is defined. 

Knowing one C, one can derive a host of others by certain recurrence relations, which we shall now develop. In accordance with Eqs. 

Other ideas are connected with reduction of the original second-order difference equation (9) to three first-order ones, which may be, generally speaking, nonlinear. First of all, the recurrence relation with indeterminate coefficients a,- and f3i is supposed to be valid ... 

On the strength of the preceding decomposition we establish the recurrence relation for the coefficients... 

Other iterative methods apply equally well to problem (2). Among them the method with the recurrence relation... 

For reference, the Hermite polynomials for = 0 to = 10 are listed in Table 4.1. When needed, higher-order Hermite polynomials are most easily obtained from the recurrence relation (D.5). If only a single Hermite polynomial is wanted and the neighboring polynomials are not available, then equation (D.4) may be used. 

We next derive some recurrence relations for the Hermite polynomials. If we differentiate equation (D.l) with respect to s, we obtain... 

This recurrence relation may be used to obtain a Hermite polynomial when the two preceding polynomials are Imown. 

The relations (D.5) and (D.6) may be eombined to give a third recurrence relation. Addition of the two equations gives... 

To find the differential equation that is satisfied by the Hermite polynomials, we first differentiate the second recurrence relation (D.6) and then substitute (D.6) with n replaeed by n — 1 to eliminate the first derivative of i ( )... 

The recurrence relation (E.4) is useful for evaluating Pi(p) when the two preceding polynomials are known. 

Equating coefficients of s on each side of this equation yields a second recurrence relation... 

A third recurrence relation may he obtained by differentiating equation (E.4) to give... 

We solve the recurrence relation (E.4) for Pi p), multiply both sides by Pi p), integrate with respect to p from —1 to +1, and note that one of the integrals vanishes according to the orthogonality relation (E. 18), so that... 

The interested reader can construct the complete verification conditions and notice what "obviousfacts" are needed to check them. For example, to verify (1) one must know that ( ) = 1, to verify (M-) that k < n implies ( ) = (n ), to verify (5) that k < n implies n( jj ) is divisible by k, and to verify (6) the recurrence relation above. 

Calculation of Mean First Passage Times from Differential Recurrence Relations... 

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