"In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms."
Relationships between successive terms of a sequence, which can be used to define sequences or solve problems involving them.
Introduction to Sequences: A sequence is a set of numbers that are in a specific order. Different types of sequences include arithmetic, geometric, and recursive sequences.
Linear Recurrence Relations: Linear recurrence relations involve finding a formula for the nth term of a sequence in terms of previous terms. They can be solved by using characteristic equations.
Nonlinear Recurrence Relations: Nonlinear recurrence relations involve finding a formula for the nth term of a sequence in terms of both previous terms and the value of n. They can be more difficult to solve than linear recurrence relations.
Generating Functions: Generating functions are power series that can be used to represent and manipulate sequences. They can be used to solve recurrence relations and to derive formulas for the nth term of a sequence.
Stirling Numbers: Stirling numbers are a type of combinatorial number that count the number of ways to partition a set into subsets of a specific size. They can be used to solve recurrence relations related to combinatorics.
Eulerian Numbers: Eulerian numbers are a type of combinatorial number that count the number of permutations with a specific number of ascents or descents. They can also be used to solve recurrence relations related to combinatorics.
Continued Fractions: A continued fraction is a type of expression for a real number that involves an infinite series of fractions. Continued fractions can be used to approximate irrational numbers and to solve recurrence relations.
Solving Recurrence Relations with Matrix Methods: Matrix methods involve using matrices to solve recurrence relations. This can be a more efficient method for solving large and complex recurrence relations.
Applications of Recurrence Relations: Recurrence relations have many real-world applications, including in finance, physics, biology, and computer science. Understanding recurrence relations can help in modeling and analyzing these complex systems.
Linear homogeneous recurrence relation with constant coefficients: This is a recurrence relation where each term is a linear combination of the previous k terms, where k is a fixed positive integer, and the coefficients are constants.
Linear non-homogeneous recurrence relation with constant coefficients: This is a recurrence relation where the same condition as above holds, but with a non-homogeneous term present.
Non-linear recurrence relation: A recurrence relation that is not linear.
Homogeneous recurrence relation: A recurrence relation containing 0 in the non-homogeneous term.
Non-homogeneous recurrence relation: A recurrence relation containing a non-zero non-homogeneous term.
First order recurrence relation: A recurrence relation where each term depends only on the previous term.
Second order recurrence relation: A recurrence relation where each term depends on the two previous terms.
Third order recurrence relation: A recurrence relation where each term depends on the three previous terms.
Geometric recurrence relation: A recurrence relation where each term is a constant multiple of the previous term.
Arithmetic recurrence relation: A recurrence relation where each term is a constant added to the previous term.
Fibonacci recurrence relation: A recurrence relation where each term is the sum of the two previous terms.
Second order, linear, homogeneous recurrence relation with constant coefficients: This is a specific type of recurrence relation where each term depends on the two previous terms, and the relation is linear and homogeneous.
Second order, linear, non-homogeneous recurrence relation with constant coefficients: This is a specific type of recurrence relation where each term depends on the two previous terms, and the relation is linear and non-homogeneous.
Higher order, linear, homogeneous recurrence relation with constant coefficients: A recurrence relation where each term depends on the previous k terms, where k is a fixed positive integer, and the relation is linear and homogeneous.
Higher order, linear, non-homogeneous recurrence relation with constant coefficients: A recurrence relation where the same conditions as above hold but with a non-homogeneous term present.
Diophantine recurrence relation: A recurrence relation involving the solution of Diophantine equations.
Linear recurrence relation with polynomial coefficients: A recurrence relation where the coefficients are polynomials rather than constants.
Multivariate recurrence relation: A recurrence relation involving multiple variables.
Bernoulli recurrence relation: A recurrence relation involving Bernoulli numbers.
Chebyshev recurrence relation: A recurrence relation involving Chebyshev polynomials.
"This number k is called the order of the relation."
"The rest of the sequence can be calculated by repeatedly applying the equation."
"In linear recurrences, the nth term is equated to a linear function of the k previous terms."
"A famous example is the recurrence for the Fibonacci numbers."
"where the order k is two and the linear function merely adds the two previous terms."
"For these recurrences, one can express the general term of the sequence as a closed-form expression of n."
"Linear recurrences with polynomial coefficients depending on n are also important."
"Because many common elementary and special functions have a Taylor series whose coefficients satisfy such a recurrence relation."
"Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of n."
"Yes, the concept of a recurrence relation can be extended to multidimensional arrays."
"That is, indexed families that are indexed by tuples of natural numbers."
"For these recurrences, one can express the general term of the sequence as a closed-form expression of n."
"Many common elementary and special functions have a Taylor series whose coefficients satisfy such a recurrence relation."
"Obtaining a closed-form solution: a non-recursive function of n."
"This number k is called the order of the relation."
"The rest of the sequence can be calculated by repeatedly applying the equation."
"The nth term is equated to a linear function of the k previous terms."
"One can express the general term of the sequence as a closed-form expression of n."
"That is, indexed families that are indexed by tuples of natural numbers."