Generating Functions

Generating function is a method to solve the recurrence relations.

Let us consider, the sequence a₀, a₁, a₂....a_r of real numbers. For some interval of real numbers containing zero values at t is given, the function G(t) is defined by the series
            G(t)= a₀, a₁t+a₂ t²+⋯+a_r t_r+............equation (i)

This function G(t) is called the generating function of the sequence a_r.

Now, for the constant sequence 1, 1, 1, 1.....the generating function is

It can be expressed as

            G(t) =(1-t)^-1=1+t+t² +t³+t⁴+⋯[By binomial expansion]

Comparing, this with equation (i), we get

            a₀=1,a₁=1,a₂=1 and so on.

For, the constant sequence 1,2,3,4,5,..the generating function is
            G(t) = because it can be expressed as
            G(t) =(1-t)^-2=1+2t+3t² +4t³+⋯+(r+1) t^r

Comparing, this with equation (i), we get
a₀=1,a₁=2,a₂=3,a₃=4 and so on.

The generating function of Z^r,(Z≠0 and Z is a constant)is given by
            G(t)= 1+Zt+Z² t²+Z³ t³+⋯+Z^r t^r
            G(t)=       [Assume |Zt|<1]
So,       G(t)= generates Z^r,Z≠0

Also,If a⁽¹⁾_r has the generating function G₁(t) and a⁽²⁾_r has the generating function G₂(t), then λ₁ a⁽¹⁾_r+λ₂ a⁽²⁾_r has the generating function λ₁ G₁(t)+ λ₂ G₂(t). Here λ₁ and λ₂ are constants.

Application Areas:

Generating functions can be used for the following purposes -

• For solving recurrence relations
• For proving some of the combinatorial identities
• For finding asymptotic formulae for terms of sequences

Example: Solve the recurrence relation a_r+2-3a_r+1+2a_r=0

By the method of generating functions with the initial conditions a₀=2 and a₁=3.

Solution: Let us assume that

Multiply equation (i) by t^r and summing from r = 0 to ∞, we have

(a₂+a₃ t+a₄ t²+⋯)-3(a₁+a₂ t+a₃ t²+⋯)+2(a₀+a₁ t+a₂ t²+⋯)=0
     [∴ G(t)=a₀+a₁ t+a₂ t²+⋯]

 +2G(t)=0............equation (ii)

Now, put a₀=2 and a₁=3 in equation (ii) and solving, we get

Put t=1 on both sides of equation (iii) to find A. Hence
            -1=- A       ∴ A = 1

Put t= on both sides of equation (iii) to find B. Hence
            = B       ∴ B = 1

Thus G (t) = .Hence,a_r=1+2^r.