The Recurrence Formula

As already mentioned, the optimum strategy for most simply finding the values of li(0,n a,b) is given by applying the recurrence (4.8). The same pertains to the 

When b = 0, a. special case of the formulas of the preceding section exists. In this case, the problem simplifies considerably, since 

As with the result (4.42), an explicit representation of f (m, n a, b) is obtained for the general case b 0. The result is by similar arguments leading to Equation 4.42 

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Thus, we see that this problem has a unique solution if the value y is given for some i. For the sake of simplicity let be known in advance for i = 0. With this, one can determine all the values y-, y, . . by the recurrence formula just established. In the case qi z= q = const and ipi = 0 this provides support for the view that the whole collection of yi constitutes a geometric progression. If qi = q and (pi qb 0 then... 

The requirement a << 1 necessitates making some modifications for stability of this or that difference scheme. As a final result of minor changes, the recurrence formulae have the representations... 

The evaluation of more complicated integrals can be effected by combining Ibis result with the recurrence formulae wc have already established for the 1 Termite functions. For instance, it follows from equation (40.3) that... 

After Smith and Ewart ( ), we define the kinetics of emulsion polymerization in terms of the recurrence formula,... 

Theorem. For any three consecutive orthonormal polynomials defined by Eq. (7.4), we have the recurrence formula... 

We wish to add now another important property of orthogonal poly-nomials. " From the recurrence formula (7.10) the Christoffel-Darboux relation can be easily deduced ... 

Use of the recurrence formula at Eq. (A9) requires the explicit evaluation of the following integral ... 

If a polymer chain is built monomer by monomer, from time zero onwards, the probability of obtaining a chain with N monomers after a time t is a function PN(t) given by the recurrence formulas... 

The development of the recurrence formulas is outlined in Prob. 2-3. An improved form of these expressions was recently proposed by Boston and Sullivan.1 For the special case of a conventional distillation column in which model 2 (see Fig. 2-2) for the feed plate is assumed, the procedure proposed by Boston and Sullivan (see Prob. 2-3) may be used.to reduce the above formulas to the following form... 

After the recurrence formulas have been applied for each component i and the set of component vapor rates (Vji)ca have been found, the corresponding set of liquid rates (/,) , are then found by use of Eq. (2-10). These sets of calculated... 

Application of the recurrence formulas for tridiagonal matrix equations follows ... 

The recurrence formulas given by Eqs. (2-20) and (2-21) for solving equations which are tridiagonal in form may be developed as outlined below by use of the gaussian elimination. The system of linear equations given by Eq. (2-19) is represented by the following matrix equation... 

Begin with the set of algebraic equations given by Eq. (10-11) and develop the recurrence formulas given by Eq. (10-12). 

Making use of the recurrence formulae for modified Bessel s functions ... 

If is to be invariant, AJ — A m must be an even number since the requirement on A m for the x or y component is A m = +1, it follows that AJ must be odd. Detailed calculation using the recurrence formulas shows that AJ = +1 only. 

Again this relation requires A J to be odd if the integral is to be invariant under reflection. Detailed calculation using the recurrence formula shows that AJ = 1 only. 

The molar density of reactant A in the inlet stream of the first CSTR is Cao/s in the Laplace domain. Hence, when the recurrence formula (2-30) is applied to the first reactor (i.e., k = 1),... 

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