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A recurrence relation is an equation that expresses each element of a sequence as a function of the preceding ones. More precisely, in the case where only the immediately preceding element is involved, a recurrence relation has the form
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
A D-finite or holonomic sequence is a natural generalization where the coefficients of the recurrence are allowed to be polynomial functions of rather than constants. [ 38 ] A k {\displaystyle k} -regular sequence satisfies a linear recurrences with constant coefficients, but the recurrences take a different form.
This equation has nonzero solutions that are nonsingular on [−1, 1] only if ℓ and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values. When in addition m is even, the function is a polynomial. When m is zero and ℓ integer, these functions are identical to the Legendre polynomials.
Computational Aspects of Three-Term Recurrence Relations. SIAM Review, 9:24–80 (1967). Walter Gautschi. Minimal Solutions of Three-Term Recurrence Relation and Orthogonal Polynomials. Mathematics of Computation, 36:547–554 (1981). Amparo Gil, Javier Segura, and Nico M. Temme. Numerical Methods for Special Functions. siam (2007)
The main idea is to express an integral involving an integer parameter (e.g. power) of a function, represented by I n, in terms of an integral that involves a lower value of the parameter (lower power) of that function, for example I n-1 or I n-2. This makes the reduction formula a type of recurrence relation. In other words, the reduction ...
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 (,).
In mathematics a P-recursive equation is a linear equation of sequences where the coefficient sequences can be represented as polynomials.P-recursive equations are linear recurrence equations (or linear recurrence relations or linear difference equations) with polynomial coefficients.