Recurrence relation example

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. 

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). 

Calculation of the viscoelastic functions proceeds as above where, for example, Eq. (T 7) is the reduced relaxation modulus for the cubic array. The incomplete gamma function of order 5/2 may be obtained in simpler form through a recurrence relation and ... 

Further, regarding ijrn), it is also possible to have the explicit Lanczos algorithm by deriving the expression that holds the whole result with no recourse to recurrence relations. For example, applying the explicit Lanczos polynomial operator Q (U) from Eq. (177 to tq) = 0) will generate the wave packet IV n) according to ijrn) = Q (U) V o) as in Eq. (91). Therefore, the final result is the following expression for the explicit Lanczos states Vr ) ... 

Here y = x/r[ = =C, /2/IkT is chosen as the inertial effects parameter (y = /2/y is effectively the inverse square root of the parameter y used earlier in Section I). Noting e initial condition, Eq. (134), all the < (()) in Eq. (136) will vanish with the exception of n = 0. Furthermore, Eq. (136) is an example of hoyv, using the Laplace integration theorem above, all recurrence relations associated with the Brownian motion may be generalized to fractional dynamics. The normalized complex susceptibility /(m) = x ( ) z"( ) is given by linear response theory as... 

Here, the formula is just 2 raised to a power, the value of which is defined by each element of the domain. Notice that the use of r as a counting index is arbitrary any other appropriate letter (with the exception of u which we have used already) would do. A counting index such as r is often termed a dummy index.. An alternative way of generating this sequence is accomplished using a recurrence relation as the prescription, where each successive term is obtained from the previous term. For example, the sequence given in equation (1.8) can alternatively be expressed as ... 

Perhaps the most common example of recursion in programming and in mathematics (where recursion is called a recurrence relation) is the factorial function ... 

Another common example of a recurrence relation is one that defines the Fibonacci numbers. This sequence of numbers, called the Fibonacci sequence after the wealthy Italian merchant Leonardo Fibonacci of Pisa, plays important roles in mathematics and nature. These numbers are such that, after the first two, every number in the sequence equals the sum of the two previous numbers ... 

This program computes the first 30 Fibonacci numbers, using an array size of 30. This method of using arrays to store previous results is usually the preferred method for evaluating recurrence relations, because it allows even complex expressions to be processed in a uniform and efficient manner. Of course, in this example, the programmer can even avoid the array by retaining the last two values in two variables. 

The first stage of the analysis is to find suitable values for c through the indicial relation. The second stage is to find the relations for a from the recurrence relation. This second stage has many twists and turns and can best be learned by example. We consider these relations in the next section. 

The recurrence relation is obviously the bracketed terms set to zero. In most practical problems, uncovering the recurrence relation is not so straightforward, and the coefficients = f a X U = /(< 2X so on, must be found one at a time, as we show later in example Ex. 3.2. However, in the present case, we have the general result for any value c... 

The diagram representing three-term recurrence relations may also illustrate the termination of the pertinent expansions. For example, let aj) 0 for n<3, but nij = tiaj+ P2 2 = If to construct a solution for which =... 

We here discuss the two best-developed approaches to this idea, namely Biermann s function merging mechanism, and Summers recurrence relation detection mechanism. Both reflect passive, non-incremental, two-step, consistent synthesis from positive, ground, pre-synthesis I/0-example(s) that are selected by an agent who knows the intended function. 

This recurrence relation detection mechanism requires multiple positive examples, which overcomes the potential ambiguity problems of single-example approaches. It is more efficient than Biermann s, but less robust as the examples must be carefully chosen. A striking difference with Biermann s approach is that recursion is here detected by folding parts of several traces obtained from different examples, whereas Biermann s mechanism detects recursion by folding a single trace onto itself. 

The mentioned Basic Synthesis Theorem constitutes a major breakthrough, as it provides a firm theoretical foundation to synthesis from examples. Summers also describes a technique that automatically introduces accumulator parameters when no recurrence relations can be found this amounts to descending generalization [Deville 90]. The results of Summers have spawned considerable efforts for generalization and improvement, especially by [Jouannaud and Kodratoff 83] [Kodratoff and Jouannaud 84]. Their achievements are very encouraging as the developed sequence matching algorithms are very efficient. 

The PRISM method modifies the McMurchie-Davidson recurrence relations to process contracted rather than primitive integrals. The first step in this procedure is to introduce simultaneous contraction and scaling. Examples of this are... 

Subsequently the transformation from the two-center integrals p q] in the Cartesian Hermite presentation to the Cartesian presentation (see equation 66) should be considered. The recurrence relation of this transformation, equation (68), is especially suitable for the required manipulations. Bearing this in mind the new recurrence relation for the contracted intermediate integrals is, for example,... 

We have now succeeded in setting up a set of recurrence relations by means of which the two-electron Cartesian integrals may be obtained from the Boys function. The resulting expressions are rather complicated, however, involving as many as eight distinct contributions. Unlike the McMurchie-Davidson scheme, the Obara-Saika scheme does not treat the two electrons separately since the recurrences (9.10.24) and (9.10.25), for example, affect the indices of all four orbitals. In Section 9.10.3, we shall see how the Obara-Saika recurrences may be simplified considerably when used in conjunction with two other types of recurrence relations the electron-transfer recurrences and the horizontal recurrences. 

Stored in reservoirs, and a growing social perception of problems related to water quantity and quality in reservoirs is expected. Just as an example, the recurrent drought episodes suffered by the Barcelona metropolitan area (Spain) in recent years placed reservoirs at the center-stage since then people are regularly informed about water resources stored in nearby reservoirs in newspapers and TV, and an iconic view from a reservoir has become a nonofficial monitor of the water supply volume available for the city (Fig. 1). 

For example, protection of the milk supply from contamination with antibiotics can be done at a relatively low cost at the farm, where the main costs are recurrent costs related to ensuring that the milk from treated cows... 

For multifactorial diseases, the recurrence risk decreases rapidly for more remotely related relatives. For example, one study of autism reported a sibling risk of 4.5%, an uncle-niece risk of 0.1%, and a first-cousin risk of 0.05%. In contrast, the risk of carrying a single-gene mutation decreases by only 1/2 with each successive degree of relationship (i.e., 50% chance for siblings, 25% for unde-niece relationships, and 12.5% for first cousins). 

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