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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 ...
In the mathematical subfield of numerical analysis, de Boor's algorithm [1] is a polynomial-time and numerically stable algorithm for evaluating spline curves in B-spline form. It is a generalization of de Casteljau's algorithm for Bézier curves.
The evaluation of incomplete exponential Bell polynomial B n,k (x 1,x 2,...) on the sequence of ones equals a Stirling number of the second kind: {} =, (,, …,). Another explicit formula given in the NIST Handbook of Mathematical Functions is
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.
For n = 1 this results in the already known recurrence formula, just arranged differently, and with n = 2 it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest m) which implies the evenness or oddness of these polynomials.
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.
There are exactly two ways in which this can be accomplished. We could do this by forming a singleton cycle, i.e., leaving the extra object alone. This increases the number of cycles by 1 and so accounts for the [] term in the recurrence formula. We could also insert the new object into one of the existing cycles.
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 ...