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A difference equation of order k is an equation that involves the k first differences of a sequence or a function, in the same way as a differential equation of order k relates the k first derivatives of a function. The two above relations allow transforming a recurrence relation of order k into a difference equation of order k, and, conversely ...
A linear recurrence denotes the evolution of some variable over time, with the current time period or discrete moment in time denoted as t, one period earlier denoted as t − 1, one period later as t + 1, etc. The solution of such an equation is a function of t, and not of any
The order of the sequence is the smallest positive integer such that the sequence satisfies a recurrence of order d, or = for the everywhere-zero sequence. [ citation needed ] The definition above allows eventually- periodic sequences such as 1 , 0 , 0 , 0 , … {\displaystyle 1,0,0,0,\ldots } and 0 , 1 , 0 , 0 , … {\displaystyle 0,1,0,0 ...
To compute the terms of a recurrence through according to Miller's algorithm, one first chooses a value much larger than and computes a trial solution taking initial condition to an arbitrary non-zero value (such as 1) and taking + and later terms to be zero.
A sequence () is called hypergeometric if the ratio of two consecutive terms is a rational function in , i.e. (+) / (). This is the case if and only if the sequence is the solution of a first-order recurrence equation with polynomial coefficients.
Find recurrence relations for sequences—the form of a generating function may suggest a recurrence formula. Find relationships between sequences—if the generating functions of two sequences have a similar form, then the sequences themselves may be related. Explore the asymptotic behaviour of sequences. Prove identities involving sequences.
The Eulerian number of the second order, denoted , counts the number of all such permutations that have exactly m ascents. For instance, for n = 3 there are 15 such permutations, 1 with no ascents, 8 with a single ascent, and 6 with two ascents:
In mathematics, the Lucas sequences (,) and (,) are certain constant-recursive integer sequences that satisfy the recurrence relation = where and are fixed integers.Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences (,) and (,).