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Recurrence Relation and it Applications

Dr. Tarkeshwar Singh

Department of Mathematics

October 27, 2021

Types of Recurrence Relations Solution of Recurrence Relations Generating Functions

Outline

Recurrence Relations Types of recurrence relations Solution of Recurrence Relations Generating Functions Non-linear Recurrences.

Types of Recurrence Relations Solution of Recurrence Relations Generating Functions

Recursive Functions

Let a ∈ W and X = {a, a + 1, a + 2,... }. The recursive definition of a function f with domain X consists of three parts, where k > 1. Basis clause: A few initial values of the function f (a), f (a + 1),... , f (a + k − 1) are specified. An equation that specifies such initial values is an initial condition. Recursive clause: A formula to compute f (n) from the k preceding functional values f (n − 1), f (n − 2),... , f (n − k) is made. Such a formula is a recurrence relation (or recursion formula). Terminal clause: Only values thus obtained are valid functional values. (For convenience, we drop this clause from our recursive definition.)

Types of Recurrence Relations Solution of Recurrence Relations Generating Functions

Recurrence equations

Definition A recurrence equation (or recurrence relations) is any equation that can be used to specify an infinite sequence < Xn > n ∈ N by expressing Xn in terms of Xn− 1 , Xn− 2 ,... , X 1 , X 0 and n.

Types of Recurrence Relations Solution of Recurrence Relations Generating Functions

The handshake problem

Example There are n guests at a sesquicentennial ball. Each person shakes hands with everybody else exactly once. Define recursively the number of handshakes h(n) that occur.

Types of Recurrence Relations Solution of Recurrence Relations Generating Functions

Fibonacci Sequence

Example Leonardo Fibonacci, the most outstanding Italian mathematician of the Middle Ages, proposed the following problem around 1202: Suppose there are two newborn rabbits, one male and the other female. Find the number of rabbits produced in a year if: (a). Each pair takes one month to become mature. (b) Each pair produces a mixed pair every month, from the second month. (c) No rabbits die.

Types of Recurrence Relations Solution of Recurrence Relations Generating Functions

Pizza Problem

Example Let fn denote the maximum number of places into which a pizza can be divided with n cuts. Find a formula for fn.

Types of Recurrence Relations Solution of Recurrence Relations Generating Functions

Triangulation of convex polygons

Example The nth^ Catalan number Cn denotes the number of ways to divide a convex (n + 2)-gon into triangles by drawing nonintersecting diagonals.

Types of Recurrence Relations Solution of Recurrence Relations Generating Functions

Linear Recurrence Relation with Constt. Coeff.

Definition A linear recurrence equation of order k is one which can be written as Xn + p 1 (n)Xn− 1 + p 2 (n)Xn− 2 + · · · + pk (n)Xn−k = q(n). Where, p 1 (n), p 2 (n),... , pk (n), &q(n) are functions of n only.

If we replace n by n + k, we can write this linear recurrence equation in the alternative form:

Xn+k + ak− 1 (n)Xn+k− 1 + · · · + a 0 (n)Xn = q′(n).

If all pi (n) are constant ∀i, 1 ≤ i ≤ n, then the above recurrence relation is known as Linear recurrence relation with constant coefficients(LRRWCC).

Types of Recurrence Relations Solution of Recurrence Relations Generating Functions

LRRWCC

The LRRWCC is written as Xn + a 1 Xn− 1 + a 2 Xn− 2 + · · · + ak Xn−k = q(n). If q(n) = 0, then above LRRWCC is known and Linear Homogeneous Recurrence Realtaion with Constant Coefficients (LHRRWCC). If q(n) 6 = 0, then above LRRWCC is known and Linear Non-homogeneous Recurrence Realtaion with Constant Coefficients (LNHRRWCC). Solution of the above recurrence relation is nothing but as sequence < an > which satisfy the above recurrence relation.

Types of Recurrence Relations Solution of Recurrence Relations Generating Functions

Characteristic Root Method

Solution of LHRRWCC

Theorem Let < an > and < bn > be the solutions of the LHRRWCC L(E )(Xn) = 0, then an + bn and kan are also solutions of L(E )(Xn) = 0.

Types of Recurrence Relations Solution of Recurrence Relations Generating Functions

Characteristic Root Method

Solution of LHRRWCC

Example Find the solution of Xn+1 − 3 Xn = 0 for all n ≥ 0 with X 0 = 4.

Types of Recurrence Relations Solution of Recurrence Relations Generating Functions

Characteristic Root Method

Characteristic Root Method

Let L(Xn) = 0 be a linear homogeneous recurrence relation with constant coefficients (LHRRWCC). Then we must have L(E ) = E k^ + ak E k−^1 + · · · + a 1 E + a 0 I = 0. Theorem Let L(Xn) = 0 be a linear homogeneous recurrence relation with constant coefficients (LHRRWCC). If L(E ) = (E − α 1 I )(E − α 2 I )... (E − αk I ) and all αi are distinct, then the general solution of L(E ) = 0 is given by Xn =

∑k i=1 Ai^ αi n. Where Ai are known as arbitrary constants.

Types of Recurrence Relations Solution of Recurrence Relations Generating Functions

Characteristic Root Method

Characteristic Root Method

Suppose we have LHRRWCC is written as Xn + a 1 Xn− 1 + a 2 Xn− 2 + · · · + ak Xn−k = 0. To find the solution we have to follow the following steps: We assume the solution Xn = αn^ and substitute it in above LHRRWCC. We have then αn^ + a 1 αn−^1 + · · · + ak = 0. The above equation is known as characteristic equation of LHRRWCC. If all the roots of the above equations are distinct then the solution is of the type Xn =

∑k i=1 Ai^ αi n. Where Ai are known as arbitrary constants.