CSL105: Discrete Mathematical Structures. Ragesh Jaiswal, CSE, IIT Delhi

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3 Definition (Linear homogeneous recurrence) A linear homogeneous recurrence relation of degree k with constant coefficients is a recurrence relation of the form a n = c 1 a n 1 + c 2 a n c k a n k, where c 1, c 2,..., c k are real numbers, and c k 0. Linear means that that RHS is a sum of linear terms of the previous elements of the sequence. a n = a n 1 + a n 2 is a linear recurrence relation whereas a n = a n 1 + an 2 2 is not.

4 Definition (Linear homogeneous recurrence) A linear homogeneous recurrence relation of degree k with constant coefficients is a recurrence relation of the form a n = c 1 a n 1 + c 2 a n c k a n k, where c 1, c 2,..., c k are real numbers, and c k 0. Linear means that that RHS is a sum of linear terms of the previous elements of the sequence. Homogeneous means that there are no terms in the RHS that are not multiples of a j s. a n = a n 1 + a n 2 is homogeneous whereas a n = a n 1 + a n is not.

5 Definition (Linear homogeneous recurrence) A linear homogeneous recurrence relation of degree k with constant coefficients is a recurrence relation of the form a n = c 1 a n 1 + c 2 a n c k a n k, where c 1, c 2,..., c k are real numbers, and c k 0. Linear means that that RHS is a sum of linear terms of the previous elements of the sequence. Homogeneous means that there are no terms in the RHS that are not multiples of a j s. The coefficients of all the terms on the RHS are constants. The degree is k since a n is expressed as the previous k terms of the sequence.

6 Definition (Linear homogeneous recurrence) A linear homogeneous recurrence relation of degree k with constant coefficients is a recurrence relation of the form a n = c 1 a n 1 + c 2 a n c k a n k, where c 1, c 2,..., c k are real numbers, and c k 0. a n = r n is a solution of the recurrence if and only if r k c 1 r k 1... c k = 0. (1) (1) is called the characteristic equation of the recurrence relation. The solutions of the characteristic equation are called the characteristic roots of the recurrence relation.

7 Let c 1 and c 2 be real numbers. Suppose r 2 c 1 r c 2 = 0 has two distinct roots r 1 and r 2. Then the sequence {a n } is a solution of the linear homogeneous recurrence relation a n = c 1 a n 1 + c 2 a n 2 if and only if a n = α 1 r n 1 + α 2r n 2 for all n = 0, 1, 2,..., where α 1 and α 2 are constants.

8 Let c 1 and c 2 be real numbers. Suppose r 2 c 1 r c 2 = 0 has two distinct roots r 1 and r 2. Then the sequence {a n } is a solution of the linear homogeneous recurrence relation a n = c 1 a n 1 + c 2 a n 2 if and only if a n = α 1 r n 1 + α 2r n 2 for all n = 0, 1, 2,..., where α 1 and α 2 are constants. What is the solution of the recurrence relation a n = a n a n 2 with a 0 = 2 and a 1 = 7?

9 Let c 1 and c 2 be real numbers. Suppose r 2 c 1 r c 2 = 0 has two distinct roots r 1 and r 2. Then the sequence {a n } is a solution of the linear homogeneous recurrence relation a n = c 1 a n 1 + c 2 a n 2 if and only if a n = α 1 r n 1 + α 2r n 2 for all n = 0, 1, 2,..., where α 1 and α 2 are constants. Let c 1 and c 2 be real numbers with c 2 0. Suppose that r 2 c 1 r c 2 = 0 has only one root r 0. A sequence {a n } is a solution of the recurrence relation a n = c 1 a n 1 + c 2 a n 2 if and only if a n = α 1 r n 0 + α 2nr n 0, for n = 0, 1, 2,..., where α 1 and α 2 are constants. What is the solution of the recurrence relation a n = 6a n 1 9 a n 2 with a 0 = 1 and a 1 = 6?

10 Let c 1, c 2,..., c k be real numbers. Consider the linear homogeneous recurrence relation a n = c 1 a n 1 + c 2 a n c k a n k. Suppose the characteristic equation of the recurrence relation has k distinct characteristic roots r 1, r 2,..., r k. Then {a n } is a solution of the recurrence relation if and only if a n = α 1 r n 1 + α 2r n α kr n k for n = 0, 1, 2,..., where α 1, α 2,..., α k are constants. What is the solution of the recurrence relation a n = 6a n 1 11 a n 2 + 6a n 3 with a 0 = 2, a 1 = 5, and a 2 = 15?

11 Let c 1, c 2,..., c k be real numbers. Consider the linear homogeneous recurrence relation a n = c 1 a n 1 + c 2 a n c k a n k. Suppose the characteristic equation of the recurrence relation has t k distinct characteristic roots r 1, r 2,..., r t with multiplicities m 1, m 2,..., m t, respectively, so that m i 1 for i = 1, 2,..., t and m 1 + m m t = k. Then {a n } is a solution of the recurrence relation if and only if a n = (α 1,0 + α 1,1 n α 1,m1 1n m 1 1 )r n 1 +(α 2,0 + α 2,1 n α 2,m2 1n m 2 1 )r n (α t,0 + α t,1 n α t,mt 1n mt 1 )r n t for n = 0, 1, 2,..., where α i,j are constants for 1 i t and 0 j m i 1. What is the solution of the recurrence relation a n = 3a n 1 3 a n 2 a n 3 with a 0 = 1, a 1 = 2, and a 2 = 1?

12 A linear non-homogeneous recurrence relation with constant coefficients is a recurrence of the form: a n = c 1 a n 1 + c 2 a n c k a n k + F (n), where F (n) is a function not identically equal to zero and depending only on n. The recurrence relation a n = c 1 a n 1 + c 2 a n c k a n k is called the associated homogeneous recurrence relation. If {a n (p) } is a particular solution of the non-homogeneous linear recurrence relation with constant coefficients a n = c 1 a n 1 + c 2 a n c k a n k + F (n), then every solution is of the form {a n (p) + a n (h) }, where {a n (h) } is a solution of the associated homogeneous recurrence relation a n = c 1 a n 1 + c 2 a n c k a n k.

13 If {a n (p) } is a particular solution of the non-homogeneous linear recurrence relation with constant coefficients a n = c 1 a n 1 + c 2 a n c k a n k + F (n), then every solution is of the form {a n (p) + a n (h) }, where {a n (h) } is a solution of the associated homogeneous recurrence relation a n = c 1 a n 1 + c 2 a n c k a n k. Find all solutions of the recurrence relation a n = 3a n 1 + 2n. What is the solution with a 1 = 3?

14 If {a n (p) } is a particular solution of the non-homogeneous linear recurrence relation with constant coefficients a n = c 1 a n 1 + c 2 a n c k a n k + F (n), then every solution is of the form {a n (p) + a n (h) }, where {a n (h) } is a solution of the associated homogeneous recurrence relation a n = c 1 a n 1 + c 2 a n c k a n k. Find all solutions of the recurrence relation a n = 3a n 1 + 2n. What is the solution with a 1 = 3? Findall solutions if the recurrence relation a n = 5a n 1 6a n n.

15 Suppose {a n } satisfies the linear non-homogeneous recurrence relation a n = c 1 a n 1 + c 2 a n c k a n k + F (n), where c 1, c 2,..., c k are real numbers, and F (n) = (b t n t + b t 1 n t b 1 n + b 0 )s n, where b 0, b 1,..., b t are s real numbers. When s is not a root of the characteristic equation of the associated linear homogeneous recurrence relation, there is a particular solution of the form (p t n t + p t 1 n t p 1 n + p 0 )s n. When s is a root of this characteristic equation and its multiplicity is m, there is a particular solution of the form n m (p t n t + p t 1 n t p 1 n + p 0 )s n.

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