Loading....
Coupon Accepted Successfully!

 

Generating Function

 

The generating function for the sequence a1, a2, ...., ar, .... of real numbers of a numeric functions (a0, a1, a2, ...ar,...) is the infinite series.

A(z) = a0 + a1z + a2z2 + ...+ arzr + ...

619.png 

Some Results

  1. Let a and b are any two discrete numeric functions and b = αa, then B(z) = α A(z) where B(z) is the generating function corresponding to numeric b and A(z) is that of a.
    e.g., The generating function of the numeric function
    624.png
    is 629.png  
  2. For any three discrete numeric functions a, b and c, if c = a + b, then C(z) = A(z) + B(z) is the generating function representation.
    e.g., The generating function of the numeric function ar = 1r + 2r (r  0)
    is 634.png 
  3. Let a be a numeric function and A(z) its generating function. Let b be a numeric function such that br = αrar
    Then, the generating function of b is
    639.png
    644.png 
    B(z) = A(αz)
    e.g., The generating function of the numeric function ar = 1, r  0 is 649.png 
  4. Let A(z) be the generating function of a. Then, z1 A(z) is the generating function of Sia for any positive integer i.
  5. Let A(z) be the generating function of a. Then, z–i[A(z) – a0 – a1z – a2z2 – ...–ai–1 zi–1is the generating function of S–i a.
    e.g., The generating function of ar = 3r+2, z  0 is
    654.png 
  6. For b = a, the generating function is given by
    659.pngand for b =  a B(z) = A(z) – z A(z) 
  7. Let C = a*b, i.e., C is the convolution of two numeric functions and its generating function C (z) = A(z) · B(z)665.png is the coefficient of zr in the product of670.png  





Test Your Skills Now!
Take a Quiz now
Reviewer Name