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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 [Pg.95]

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 [Pg.95]


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... [Pg.540]

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. [Pg.233]

Knowing one C, one can derive a host of others by certain recurrence relations, which we shall now develop. In accordance with Eqs. [Pg.406]

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 ... [Pg.9]

On the strength of the preceding decomposition we establish the recurrence relation for the coefficients... [Pg.351]

Other iterative methods apply equally well to problem (2). Among them the method with the recurrence relation... [Pg.510]

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. [Pg.117]

We next derive some recurrence relations for the Hermite polynomials. If we differentiate equation (D.l) with respect to s, we obtain... [Pg.297]

This recurrence relation may be used to obtain a Hermite polynomial when the two preceding polynomials are Imown. [Pg.297]

The relations (D.5) and (D.6) may be eombined to give a third recurrence relation. Addition of the two equations gives... [Pg.298]

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 ( )... [Pg.298]

The recurrence relation (E.4) is useful for evaluating Pi(p) when the two preceding polynomials are known. [Pg.302]

Equating coefficients of s on each side of this equation yields a second recurrence relation... [Pg.303]

A third recurrence relation may he obtained by differentiating equation (E.4) to give... [Pg.303]

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... [Pg.307]

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. [Pg.171]

Calculation of Mean First Passage Times from Differential Recurrence Relations... [Pg.357]


See other pages where Recurrence relation is mentioned: [Pg.56]    [Pg.234]    [Pg.234]    [Pg.235]    [Pg.236]    [Pg.120]    [Pg.120]    [Pg.120]    [Pg.35]    [Pg.75]    [Pg.165]    [Pg.208]    [Pg.308]    [Pg.390]    [Pg.652]    [Pg.690]    [Pg.142]    [Pg.297]    [Pg.298]    [Pg.302]    [Pg.329]    [Pg.329]    [Pg.330]    [Pg.63]    [Pg.383]    [Pg.383]    [Pg.171]    [Pg.298]    [Pg.357]    [Pg.357]   


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