Generating Functions

Transcription

1 Generating Functions Prachi Pendse January 9, 03 Motivation for Generating Functions Question How many ways can you select 6 cards from a set of 0? ( + ) 0 = ( + )( + )( + )... = ( ) The coefficient of 6 is = 0! 6 6!4! = 0. This is a binomial coefficient. Question How many ways can a voter vote yes 4 times if there are 6 issues to vote on in a ballot and a voter can vote yes, vote no or abstain on each issue? ( + + y) 6 = ( + + y)( + + y)( + + y)... =... + y y 4 + y The sum of the coefficients of the terms with y 4 is = 60. Another way to put it is a+b+c=6 c=4 6! a!b!c! a b y c = This is a sum of multinomial coefficients. a+b= 6! a!b!4! b y 4. Question 3 How many ways can you select a 4-letter combination from the set {A, B, C} if A can be included at most once, B at most twice, and C at most thrice? ( + a)( + b + b )( + c + c + c 3 ) = +a+b+c+ab+ac+bc+b +c +abc+...+ab c+abc +ac 3 +b c +bc The answer is the sum of the coefficients of the terms where the total degree of the variables is 4. But the question only asks how many ways there are, not what the ways are. We do not need to differentiate between a s, b s, and c s.

2 ( + )( + + )( ) = We can determine the ways to select an n-letter combination satisfying the given conditions by looking at the coefficient of the n term. There are ways to select a 4-letter combination. Question 4 How many ways can you order a combination of soups and sandwiches for r dollars if soup costs $ and sandwiches $3? ( )( ) = The coefficient of the r term in this polynomial gives the number of ways to spend r dollars. Generating Functions Definition (Formal Power Series) A formal power series over a field F is an infinite sequence α = a 0, a, a,... It is a function from the set of natural numbers to F. Definition ( Ordinary Power Series Generating Function) Given a sequence α = a 0, a, a,... the function f() = a n n is called the ordinary power series generating function of α. Definition 3 (Eponential Generating Function) Given a sequence α = a n a 0, a, a,... the function g() = n! n is called the eponential generating function of α. The first type of series is useful when the sequence α grows linearly with n, while the second is used when α grows eponentially with n. m ( ) m Eample (Binomial Epansion) (+) m = n is the generating n {( ) ( ) ( ) ( ) } m m m m function for α =,,,...,, 0, 0, 0, m Eample (Sequence of s) When α = {,,,...}, f() = = n. This has the closed form. Eample 3 (Geometric Series) The generating function of a geometric sequence with a n = Ar n is A + Ar + Ar + Ar = Ar n n. This has the closed form A r.

3 It is possible to do many operations on generating functions, such as adding, subtracting, multiplying, dividing, or raising to a power by constants, polynomials, or generating functions (ecluding dividing by 0), multiplying by, and differentiating. These operations apply to generating functions in both series form and closed form. In an ordinary power series generating function (opsgf), adding and subtracting is useful to shift indices of the summation, while multiplying and dividing by is useful to get the power of at n = i to be i. The effect of dividing an opsgf by (-) is to replace the sequence that is generated by the sequence of its partial sums. [Wilf] In an eponential generating function (egf), differentiation serves the same function as addition and subtraction of terms, both operations shift the indices of the summation. Since they can be easier to construct, generating functions are useful in finding relations between formal power series. Question Find the closed form of the generating function of the sequence of integers α = {,, 3,...}. A: We realize we get the coefficients,, 3,... when we differentiate the generating function for a sequence of s. f() = = n = f () = = n n = ( ) Multiplying by results in a generating function of the form n n. The closed form of this generating function is ( ). 3 Closed Form of the Fibonacci Numbers The Fibonacci sequence is F = f n where f 0 = 0, f =, and f n = f n + f n for n >. The f n terms are defined in the form of a recurrence relation of length. We can find a closed form for f n using generating functions. Define the generating function of the Fibonacci sequence as f() = f n n = f 0 + f + = f n n (f n + f n ) n = + f n n + f n n We can rewrite this in terms of f() and solve for a closed form of the generaing function of the Fibonacci sequence. 3

4 f() = + f() + f() f() f() f() = f()( ) = f() = We want to factor into ( α )( α ) = +( α α )+(α α ) where α α = and α α =. Solving: α α = α + α = α α = α = α α α = α = α α α = 0 The equation is the characteristic polynomial of the Fibonacci recurrence. It s solutions are α = + and α =. Incidentally α equals φ, the golden ratio. In partial fractions, f() = = A α + A α = (A + A ) (A α + A α ) ( α )( α ), where A + A = 0 and A α + A α =. Solving: A + A = 0 A = A A α + A α = A α A α = A (α α ) = A = A = α α = α α = ), easily rewritten as the sum of two Thus f() = ( α α generating functions. ( f() = ) α n n α n n = (α n α n ) n. Since the original definition of our generating function was f() = have found a closed form for the Fibonacci numbers f n = αn α n. f n n we Generating functions provide a method to solve recurrence relations of finite length and provide closed forms for their terms. 4

5 4 References H.S. Wilf, 994, generatingfunctionology, University of Pennsylvania, Philadelphia, USA, http : // wilf/gf ologylinked.pdf Lerma, A.M., 003, Generating functions, http : // mlerma/problem solving/results/gen f unc.pdf Day, Roger, 003, Combinatorics Topics for K-8 Teachers: Generating Functions, http : //math.illinoisstate.edu/day/courses/old/30/contentgeneratingf unctions.html