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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
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)
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.
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.
The values (), …, of the partition function (1, 2, 3, 5, 7, 11, 15, and 22) can be determined by counting the Young diagrams for the partitions of the numbers from 1 to 8. In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n.
Many families of special functions satisfy a recurrence relation that relates the values of the functions of different orders with common argument . The modified Bessel functions of the first kind I n ( x ) {\displaystyle I_{n}(x)} satisfy the recurrence relation
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 (,).