Recurrence relation equation

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 ... 

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... 

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 ( )... 

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 also remark that Eq. (5.44) may be decomposed into separate sets of equations for the odd and even ap(t) which are decoupled from each other. Essentially similar differential recurrence relations for a variety of relaxation problems may be derived as described in Refs. 4, 36, and 73-76, where the frequency response and correlation times were determined exactly using scalar or matrix continued fraction methods. Our purpose now is to demonstrate how such differential recurrence relations may be used to calculate mean first passage times by referring to the particular case of Eq. (5.44). 

An equation of the kind (2.1) which determines the subsequent coefficients in terms of the first two is called a recurrence relation. 

Equation (3.5) gives the two possible values p, pa of p. If we take one of these values, pt say, anti substitute it in the recurrence relation (3.6) we obtain the corresponding value of the coefficients cT anti hence the solution... 

Inserting the value of y x) from equation (11.7) wc see that these coefficients are determined by the recurrence relation... 

Recurrence Relations for the Legendre Polynomials. If wc differentiate both sides of equation (13.2) with respect to k we Slave... 

Recurrence Relations for the Bessel coefficients. If we differentiate the generating equation (25.0) with respect to x obtain the relation I / 1 f. / 1... 

Similarly, corresponding to the root q = 1 of the indicia equation we have the recurrence relation... 

The recurrence relations for (/ (a ) follow immediately from those fur //N( r). For instance equation (33.4) is equivalent to the relation... 

Equating to zero the coefficient oT f" in the expansion on the left we obtain the recurrence relation... 

This identity can then he used to derive recurrence relations for the associated Laguerre polynomials similar to those of equations (42.8) and (42.9) (cf. Examples C(ii), (iii) below). 

We now ask the system to solve (D2) for Y as a function of X using the ODE command The general solution with the two integration constants, %K1 and %K2 is given in (D3) in about two CPU seconds The program can also find powerseries solutions for some differential equations when it can solve the recurrence relation ... 

A solution to equation (16a) having the form = Tk X is called a harmonic of the attached number k. Clearly, this function satisfies problem (16a) with the initial condition u0(x) = Tk X k x). Let us find out the conditions under which every harmonic y, k = 1,2,..., N — 1, is stable. From the recurrence relations... 

In order to derive an analytic expression for Lstr(Z) from Eqs. (467), we generate the recurrence relations for the dynamical quantity / . Assuming small angular deflections p, we omit the terms comprising sinm p with m > 2. Then using equations of motion for the angle p(f), we derive... 

From a recurrence relation like (73) and a sufficient number of initial conditions an explicit formula for the quantity in question may be obtained by the standard method of difference equations. In the present case K Rj(l, 3) = 4 and K Rj(2, 3) = 50 may serve as the initial conditions. The resulting explicit formula for K Rj(m, 3) reads... 

In general case Eqs. (4.60) and (4.61) present infinite sets of the five-term (pentadiagonal) recurrence relations with respect to the index l. In certain special cases (t - 0 or a - 0), they reduce to three-term (tridiagonal) recurrence relations. In this section the sweep procedure for solving such relations is described. This method, also known as the Thomas algorithm, is widely used for recurrence relations entailed by the finite-difference approximation in the solution of differential equations (e.g., see Ref. 61). In our case, however, the recurrence relation follows from the exact expansion (4.60) of the distribution function in the basis of orthogonal spherical functions and free of any seal of proximity, inherent to finite-difference method. Moreover, in our case, as explained below, the sweep method provides the numerical representation of the exact solution of the recurrence relations. 

Equations (4.60) and (4.61) are also able to yield the vector recurrence relations for the case of a skew bias field, that is, when vectors h and n are not parallel. In this case one should ascribe to each bi as many as 21+1 components, corresponding to different values of the azimuthal index m. Another problem, involving vector recurrence relations, is a steady-state nonlinear oscillations of bi in a high-AC field. To study the harmonic content of the nonlinear response, one has to expand all the moments b t)l in the Fourier series. Then the Fourier coefficients may be treated as components of a... 

Unlike Xi, which in principle cannot be evaluated analytically at arbitrary a [90] for Xjnt an exact solution is possible for arbitrary values of the anisotropy parameter. Two ways were proposed to obtain quadrature formulas for Tjnt. One method [91] implies a direct integration of the Fokker-Planck equation. Another method [58] involves solving three-term recurrence relations for the statistical moments of W. The emerging solution for Tjnt can be expressed in a finite form in terms of hypergeometric (Kummer s) functions. Equivalence of both approaches was proved in Ref. 92. 

The sets of relaxation times x and weight coefficients vc entering Eqs. (4.233) and (4.234) were evaluated numerically. Substitution of expansions (4.228) into the Fokker-Planck equation (4.225) yields a homogeneous tridiagonal recurrence relation... 

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