CMSC 603 Notes, 3/19/99

Discussed Exam 1
 

Generating Functions

Sequence
    F = ( 0, 1, 1, 2, 3, 5, 8, ...)

Generating Function
    G(x) = 0x⁰ + 1x¹ + 2x² + 2x³ + ...

                   _¥
          = å f_ix^{i
                          i=0}
The advantage to Generating Functions is that once the sequence is transformed
we can use the power of Calculas to manipulate the function.

Ideas:

• Treat infinite sequence as a single object
• Techniques in calculas
• Generating Function techniques

1. Choose general form of generaing function G(x)
2. Using recurrence, derive formula for G(x) in terms of x
3. Deive a closed-form expression for the coefficients of G(x)

              0 ,             n = 0
    f_n =    1 ,              n = 1
              f_n-1 + f_n-2 ,   n > 1

    1)  Choose general from
                          _¥
            G(x) = å f_ix^{i
                          i=0}

    2) Solve for G(x) in terms of x

          f_n = f_n-1 + f_n-2 ,   n > 1
            _{¥              ¥                 ¥}
            å f_ixⁱ = ( å f_i-1xⁱ ) + ( å f_i-2xⁱ )
            ^{i=2                    i=2                        i=2}
            _{¥                       ¥                         ¥}
            å f_ixⁱ        = ( xå f_i-1x^i-1 )  +  ( x²å f_i-2x^i-2 )
            ^{i=2                               i=2                                   i=2}
         G(x) - 0 - x    x[G(x) - 0]            x²G(x)

         G(x) = x + xG(x) + x²G(x)