Recursion relations

The first tenn is zero if j due to the orthogonality of the Hemiite polynomials. The recursion relation in equation (B 1,2.4 ) is rearranged... 

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... 

Formulas for C-Goefficients Recursion Relations.— Table 7-1 suggests that there is a unique correspondence between ipjj and iaja (first row). Does this mean that the two are equal If so, the coeffitient... 

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... 

Substituting this form of / into the eq.(5) gives a recursion relation which allows to determine all the ajt s for any arbitrary choice of one of them. Choosing ao=l, one gets... 

Making use of recursive relations between them it is easy to see that all the above-mentioned functions, except Pq, increase unlimitedly with the distance from masses, 8 00 which contradicts the condition at infinity. Therefore, terms with E (/ ) have to be discarded too, and this gives one more simplification of Equation (2.132)... 

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... 

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... 

Derive the recursion relation [Eq. (136)] for the associated Laguerre polynomials. 

The slowest part of the construction of this table is the evaluation of the entries in the first column. The simple trapezoid rule, as given by Eq, (65), is applied with successive sectioning of the slices. It can be seen that by descending the column a limiting value can, in principle, be obtained. However, the convergence is very slow. With the use of the recursion relation... 

The third column of Table 1 is calculated by applying the recursion relation to the values shown in the second column, etc. It corresponds to the method of Milne It is apparent that the convergence becomes much more rapid with each successive column. For this particular example foe same Limiting values is obtained as either n or m becomes very large. 

This evolution of a complex set of numbers from something very simple is rather like a recursion rule. For example, the wave function for a harmonic oscillator contains the Hermite polynomial, Hb(t/), which satisfies the recursion relation ... 

End-to-end association. The simplest model is one in which each association step has the same equilibrium constant K. This is equivalent to the familiar Flory most-probable polymerization scheme. Thus we write the recursion relations... 

Also, following the standard recursion relations of Bessel functions, the recursion relations for both i and K are ... 

See Lead zirconate titanate ceramics Quality number 219 Quantum transmission 59 Reciprocal space 123, 353 Reciprocity principle 88 Reconstruction 14, 327 Au(lll) 327 DAS model 16 Si(lll)-2X1 14 Recursion relations 352 Repulsive atomic force 185, 192 Resonance frequency 234, 241 piezoelectric scanners 234 vibration isolation system 241 Resonance interactions 171, 177 and tunneling 177 Resonance theory of the chemical bond 172... 

Since this recursion relation links every other coefficient, we can choose to solve for the even and odd functions separately. Choosing ao and then determining all of the even % in terms of this ao, followed by rescaling all of these a to make the function normalized generates an even solution. Choosing aj and determining all of the odd a in like manner, generates an odd solution. 

The coefficients e are determined by the three-term recursion relation... 

In a given problem, characterized by a force-function and boundary condition, we look for sets /, of functions for which the surface integral in Eq. (7) vanishes and for which, if f is a member of such a set, V2/, and K f can be written as linear combinations of members of the set. Then Eq. (7) may become a recursion relation which in principle (and sometimes in practice) can be solved for the time-dependent behavior of the quantities in terms of their values at 1 = 0. Let ( denote the expectation of/at time t, and 0 its expectation at t = 0. In the absence of a subscript, the angular brackets will denote the usual ensemble average over equilibrium states. 

Equation (9.155) is a recursion relation for the characters of the symmetric direct product of a representation with itself n times note that it is consistent with (9.147). 

Rather elaborate recursion relations can be found for all these integrals when care is taken to preserve numerical accuracy. Since usually all values of n are needed anyway, the intermediate values of n as well as the largest value n=N and the smallest n=0 are useful. 

We mention that these functions are easily generated from fo = 1 and the recursion relation... 

If the voltage is high enough, the noise of isolated contacts can be considered as white at frequencies at which the distribution function / fluctuates. This allows us to consider the contacts as independent generators of white noise, whose intensity is determined by the instantaneous distribution function of electrons in the cavity. Based on this time-scale separation, we perform a recursive expansion of higher cumulants of current in terms of its lower cumulants. In the low-frequency limit, the expressions for the third and fourth cumulants coincide with those obtained by quantum-mechanical methods for arbitrary ratio of conductances Gl/Gr and transparencies Pl,r [9]. Very recently, the same recursive relations were obtained as a saddle-point expansion of a stochastic path integral [10]. 

From these arguments we can establish that the Dimer-RVB states (2) satisfy a recursion relation (RR) given by,... 

Here, the left and right propagator functions, G q) and G (q), are calculated by the following recursive relations... 

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