Here is an offer of 30 EXTRA POINTS for those who e-mail me their complete answers by Monday, Oct 20, 11 am. Important: The answer must be in electronic form, as this is an internet special. Scanning handwritten notes is not allowed. There won't be any partial credit. Each of the three questions is either completely correct (10 points) or not completely correct (0 points). Here are the questions:
1. For a sequence y_k, we define the forward difference operator Dy_k by y_{k+1}-y_k. Now prove a product and a quotient rule for the operator D. The Casoratian of two sequences is defined similarly as the Wronskian of two functions, except the derivative is now replaced by the forward difference operator. Find the Casoratian of 2^k and 3^k. Find also the Casoratian of 4^k and k4^k. Finally, find two solutions of DDy_k=0 such that their Casoratian is equal to one.
2. Consider the equation y_{n+1}=2y_n+1. Rewrite this equation in the form of Dy_k+ay_{k+1}=b. Then find an "integrating factor" and use methods similar as we did for first order differential equations to solve this first order difference equation subject to the initial condition y_0=0.
3. Consider the equation y_{n+2}=y_{n+1}+y_n. Use methods similar as we did for second order differential equations to solve this second order difference equation, subject to the initial conditions y_0=0 and y_1=1. Hint: Try y_n=r^n.