Abstract: In the earlier chapters, we have encountered a number of instances where a sequence (H(n) : n = 0, 1, 2,…) is given through a function that describes the n-th term H(n) of the sequence in terms of H(n − 1), H (n − 2),…. One of the early instances of this is the Pascal identity of the binomial coefficients. There are also many other examples such as the derangement number D n , and the Stirling and Catalan numbers encountered in chapter 10 where recurrence relations were used. There are essentially two equivalent definitions of the number n!. We define n! = 1 × 2 × ⋯ × n. In this case, n! is directly computed using a formula as a function of n. On the other hand, we may also define n! = n × (n − 1)!. This obtains n! in terms of (n − 1)! and we say that n! is now recursively defined. There is a conceptual difference between the two approaches. We do not need to know the entire sequence when it is recursively defined and hence are not interested in an explicit formula. It is the knowledge of the previous H(r)’s that helps us find H(n). We give a number of examples.
Publication Year: 2013
Publication Date: 2013-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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