Generating FunctionsGenerating function is a method to solve the recurrence relations. Let us consider, the sequence a_{0}, a_{1}, a_{2}....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 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+t^{2} +t^{3}+t^{4}+⋯[By binomial expansion] Comparing, this with equation (i), we get a_{0}=1,a_{1}=1,a_{2}=1 and so on. For, the constant sequence 1,2,3,4,5,..the generating function is Comparing, this with equation (i), we get The generating function of Z^{r},(Z≠0 and Z is a constant)is given by Also,If a^{(1)}_{r} has the generating function G_{1}(t) and a^{(2)}_{r} has the generating function G_{2}(t), then λ_{1} a^{(1)}_{r}+λ_{2} a^{(2)}_{r} has the generating function λ_{1} G_{1}(t)+ λ_{2} G_{2}(t). Here λ_{1} and λ_{2} are constants. Application Areas:Generating functions can be used for the following purposes -
Example: Solve the recurrence relation a_{r+2}-3a_{r+1}+2a_{r}=0 By the method of generating functions with the initial conditions a_{0}=2 and a_{1}=3. Solution: Let us assume that Multiply equation (i) by t^{r} and summing from r = 0 to ∞, we have (a_{2}+a_{3} t+a_{4} t^{2}+⋯)-3(a_{1}+a_{2} t+a_{3} t^{2}+⋯)+2(a_{0}+a_{1} t+a_{2} t^{2}+⋯)=0 +2G(t)=0............equation (ii) Now, put a_{0}=2 and a_{1}=3 in equation (ii) and solving, we get Put t=1 on both sides of equation (iii) to find A. Hence Put t= on both sides of equation (iii) to find B. Hence Thus G (t) = .Hence,a_{r}=1+2^{r}. Next TopicProbability |