and the compositional inverse when it exists is A.

Transcription

1 Lecture B jacques@ucsd.edu Notation: R denotes a ring, N denotes the set of sequences of natural numbers with finite support, is a generic element of N, is the infinite zero sequence, n 0 R[[ X]] denotes the ring of formal power series over R where = (X, X 2,... ). The X notation [ X n]a extracts the coefficient of in the formal power series A. Let X n j and j denote the formal derivative and integral with respect to a variable X j. The reciprocal of a formal power series A when it exists is A and the compositional inverse when it exists is A. 2 Introduction to combinatorial calculus We introduce the combinatorial calculus of formal power series here, but with sparse emphasis on algebraic properties of rings of formal power series. We are more interested in the combinatorial properties of the coefficients of a formal power series, and in interpreting the combinatorial meaning of the usual operations from algebra and analysis and their effect on coefficients. The main idea is to identify a sequence of elements of a ring R with a formal power series which is treated as an algebraic object in the ring of formal power series R[[ X]]. Generally but not always, the ring R will be commutative with unity. Fibonacci numbers. We start with a famous illustrative example: the Fibonacci numbers. The Fibonacci numbers are integers defined by the recurrence a n = a n + a n 2 for n 2 with a = a 2 = and a 0 = 0. If we associate (a n ) n N with the series A(X) = a n X n, then summing the recurrence times X n on both sides from n = 2 leads to A(X) = n 0 X X X 2. Now of course for this to mae sense, we have to mae sense of this as a formal power series in C[[X]] in particular the reciprocal of X X 2 must be defined. Assuming this maes sense, we might now use the geometric series formula and then binomial theorem to say j ( ) j A(X) = X (X + X 2 ) 2 = X X j+. j=0 j+=n j=0 Now we apply the coefficient operator: defining a n = [X n ]A(X) we have for n N a n+ = [X n+ ]A(X) = ( ) j ( ) =. n

2 This is an explicit formula for Fibonacci numbers as well as a combinatorial identity for the sum. Another approach is to write X X X = ( 2 5 Xφ ) Xφ where φ = ( + 5)/2 and φ = ( 5)/2. Now using the geometric series formula we see for n 0: a n = (φ n φ n ). 5 This in some sense is more explicit than the former expression for a n, and proves the identity ( ) = (φ n+ φ n+ ). n 5 If we go a bit further, we note that the sum of the first n Fibonacci numbers is exactly the coefficient of X n in the formal power series Consider the formal power series A(X) X. (a n+2 )X n = A(X) X X 2. Now observe this is exactly A(X)/( X). Therefore we immediately obtain a 0 + a + + a n = a n+2. The ideas in this example generalize greatly to many other algebraic operations in rings of formal power series, and we develop many connections between algebraic identities and combinatorial identities, as well as methods for extracting coefficients from formal power series. Derangements. Let s consider one more example, involving derangements of [n]. If a n is the number of derangements of [n], we saw that a n = n! ( ).! It maes sense to divide by n!, multiply by X n and then sum: A(X) = a n n! Xn = 2 ( ) X n.!

3 Now the right hand side is recognizable as the product of two power series, similarly to the end of the discussion of Fibonacci numbers, namely ( ) X! X. Since the first series is exp( X) again we have to mae sure this is sensible notation ( X)A(X) = exp( X). Applying the coefficient operator,we find for n : a n+ (n + )a n = ( ) n+. Now this has a combinatorial interpretation: it says that the number of derangements of [n + ] and the number of permutations of [n + ] with exactly one fixed point differ by or by. Thus again an algebraic equation has provided combinatorial meaning. 2. Definitions Let R be a ring with unity R. Let N be the set of sequences of non-negative integers with finite support each indexed by N. For N, associate an element a n R. Let n X = (X, X 2,..., X n ) and define to be the monomial i= X n Xn i i for indeterminates or variables X i : i N. Here we define = which is the multiplicative unity in R. Consider X 0 the set R[[ X]] of objects A( X) = n a n X n. Define addition of A( X) and B( X) = b n X n as A( X) + B( X) = n 0 (a n + b n) X n and multiplication A( X) B( X) = n ( m+ l= n The set R[[ X]] together with multiplication and addition is the ring of formal power series over R. In some texts, given a sequence ā indexed by vectors in N, the notation ā A( X) is used to denote that ā is the sequence of coefficients in A( X) with ā n as the coefficient of X n. Traditionally, R[ X] is the ring of polynomials over R, and topologically R[[ X]] can be viewed as the completion of R[ X] in the metric derived from R N as follows: let f : N N be a bijection. For A, B R[[ X]], a mb l ) X n. d(a, B) = 2 A ā B b = min{f( n) : ā n b n}. For us the most useful case is when R is an integral domain, for then so too are R[ X] and R[[ X]]. The multiplicative identity in R[[ X]] is the formal power series in which the 3

4 coefficient of every monomial is zero except the monomial X 0 appears with coefficient. The latter is called the constant term of the formal power series A, and denoted A( 0). More generally, the coefficient operator is defined for each n by [ X n]a( X) = a n. We give a differential formula for extracting this coefficient below. 2.2 Reciprocal The multiplicative inverse or reciprocal of a formal power series A( X), when it exists, is a formal power series B( X) such that A( X) B( X) = B( X) A( X) =. Not all elements of R[[ X]] have reciprocals. One of the fundamental topics in combinatorial calculus is to determine inversion theorems theorems which allow us to effectively compute coefficients in inverses. Here is a necessary and sufficient condition for a formal power series to have an inverse: Theorem Let R be a commutative ring with unity and let A R[[ X]]. Then A has a reciprocal if and only if A( 0) is invertible in R, and the reciprocal is unique. Proof Suppose A( 0) is invertible with inverse a. We claim that the reciprocal is B( X) = a ( aa( X)). One checs quicly by definition of multiplication that A( X)B( X) =. The only question is to show B R[[ X]], and the only way that can fail is if the coefficient of some in X n B is not a finite sum of elements of R. Note that C( X) = aa( X) has a zero constant term and therefore the lowest power of X j in any monomial in ( aa( X)) for N is at least. This means that the coefficient of in B( X) is the same as the coefficient of X n X n in a K C( X) which is a finite sum of elements of R. Now suppose A( 0) is not invertible. Then for any power series B( X) R[[ X]], by definition of multiplication the coefficient of X 0 is A( 0)B( 0), and this is not since A( 0) is not invertible. So in this case A( X) has no inverse. Uniqueness is clear. The reciprocal of a formal power series A is usually denoted /A. For instance, the geometric series identity familiar from calculus for x <, namely x = x 4

5 can now be proved more generally in the ring R[[X]] of formal power series in one variable X: by the rules of multiplication: ( X)( + X + X ) = + 0X + 0X 2 + = so + X + X is indeed the reciprocal of X. Most important in our wor will be methods for computing coefficients of reciprocal series and so-called inversion theorems. 2.3 Infinite sums and products We have defined the sum and product of two formal power series, and these definitions extend to sums and products of finitely many formal power series. Now if A, A 2,... are formal power series and the coefficient of is zero in all but finitely many of the power X n series, we can define = to be the formal power series in which the coefficient of is the sum of the coefficients X n of in the A. We call the family {A : N} summable. If {A : N} is summable, X n then we can define the infinite product ( + A ) N as follows: the coefficient of in this product is determined by taing the product over X n N of the coefficients of monomial X n in A such that = = For instance, the n n. Euler products discussed for partitions are now valid formal power series: A ( + X i ) R[[ X]] n= is well-defined since {X : N} is summable, and if X is replaced with X for N then we recover the formal power series whose coefficients are partition numbers (see Lecture A). 2.4 Formal derivative and integral For this section, R is a commutative ring with unity R and zero characteristic. Let n and fix j N. For a monomial denote j = 0 if n j = 0 and X n, X n j = n j X X n n j j X n i i. The formal derivative of A R[[ X]] is i j j A = n a n j X n. 5

6 Similarly, let = j X n n j + Xn j+ j i j X n i i and, provided R has characteristic zero, define the formal integral j A = n a n j X n. Formal integration and differentiation follow the usual rules of calculus, for instance j (A+ B) = j A + j B and j (A B) = B j A + A j B and similarly for the quotient rule. The proofs are straight from the definitions of addition and multiplication of power series, and also extend to infinite sums and products of a summable set of power series. One of the nice facts in R[[ X]] is that j and are commuting operators (unlie in analysis). A useful result which allows the extraction of coefficients is Taylor s Theorem: Theorem 2 If R has characteristic zero and j A = j B and A( 0) = B( 0) then A = B. Furthermore, let A(X) = n a n X n where n N and let n denote the partial derivative operator with n j derivatives of X j for j N. [ X n]a(x) = B( 0) where B(X) = j= n j! na(x). 2.5 Compositional inverse Another important operation on formal power series is composition. Let A R[[ X]] and suppose that B = (B ) N is a set of formal power series such that {a n B n} is summable. Then the composition of A with B is A B = A( B) = n a n B n. In the special case of the ring R[[X]] of single variable formal power series, we define the compositional inverse of A R[[X]] to be a power series B R[[X]] such that A B(X) = X. This entity exists and is unique when A(0) = 0 and A( X) is invertible, and is denoted A as opposed to the reciprocal which is. The Lagrange inversion formula is a way to A express the coefficients of the compositional inverse of a formal power series, and we prove this later. 2.6 Explicit Examples For simplicity we start off with power series in the ring C[[X]] of one-variable formal power series over the complex numbers, although in applications we will often need two or more variables. 6

7 Reciprocal and geometric series. Recall the geometric series formula X = X which comes from the definition of reciprocal given before. Compositional inverse. The compositional inverse of A(X) = X + X 2 + X 3 + = X/( X) is A (X) = X X 2 +X 3 = X/(+X) and there is no reciprocal. However, /( X) has no compositional inverse. Exponential and logarithmic power series. Define the exponential and logarithmic power series respectively as exp(x) = X! log( X) = = X. The definition of formal derivative implies exp(x) = exp(x) and log( X) = /( X), and we also have the rule (exp(x)) = exp(x) for N and exp(x) = / exp( X), in accordance with the corresponding power series from calculus. The composition log(exp(x)) maes sense and by the chain rule, log(exp(x)) = and log(exp(0)) = 0. However X also satisfies these conditions, and therefore since C has characteristic zero, Taylor s Theorem shows log(exp(x)) = X. Similarly, exp(log( X)) = X, but note that log X is not a formal power series. Another way to derive exp(x) is from a differential equation: one can show that if A C[[X]] and A(X) = A(X) then A is proportional to exp(x): the differential equation shows (n + )a n+ = a n for n which has solution a n = a 0 /n!. The Binomial Series. This is one of the most important series combinatorially. For indeterminates X and Y, defined the binomial series in C[X, Y ] as ( + X) Y = ( ) Y X. If n is a positive integer, then (X + Y ) n = X Y n is a finite sum. The usual rules ( + X) Y ( + X) Z = ( + X) Y +Z, (( + X) Y ) Z = ( + X) Y Z and ( + X) Y = /( + X) Y are satisfied by this formal power series. Taylor s Theorem. Let f : C C be a function that is analytic at the origin. Then the power series provided by a Taylor expansion of f at the origin can be viewed as a formal power series. The methods of complex analysis extend to formal power series, provided 7

8 that each operation does indeed produce a formal power series i.e. the coefficients must be finite sums of complex numbers. So for instance sin(x) = ( ) X 2+ (2 + )! defines a formal power series, and it satisfies the same rules as the ordinary power series for X, for instance the formal power series we get from arcsin(x) is indeed the compositional inverse of sin(x). However, log(x) is not a formal power series since log(z) is not analytic at z = Identities Using operations on formal power series, we can obtain combinatorial identities from algebraic identities. Generally we may pass from an algebraic identity to a combinatorial identity by comparing coefficients, and we may tae a combinatorial identity and sum each side multiplied by X n to get an algebraic identity, taing care to ensure that coefficients are finitely generated. Pascal s Triangle Identity and Binomial Sums This arises by factoring an algebraic quantity: recall the identity states for n, N, ( ) ( ) ( ) n n n = +. We have ( + X) n = ( + X)( + X) n for n N and the coefficient of X n is ( n ) by the binomial theorem. On the other hand, by the definition of multiplication, with a 0 = = a, a 2 = a 3 = = 0 and b = ( ) n, ( + X)( + X) n = ( a l b l )X = l=0 ( ( ) n ) + X and comparing coefficients gives the result. For many identities involving binomial sums, for instance ( ) 2 ( ) n 2n = n using the basic rule of multiplication of formal power series is effective. The left hand side is [X n ]( + X) 2n. Now ( + X) n ( + X) n = l=0 8 ( ) n X l l

9 and reading off the coefficient with = n gives the identity. Another tric is to interchange the order of summation in multiple sums. For instance, while it is not possible to explicitly compute the inner (harmonic) sum in if we interchange the sums we get l= =l = l= 2 + l 2 + l = 2 l l. This sum is computed by letting X = /2 in the following computation: l=0 We conclude that the given sum is log 2. l= X l = log( X). Evaluating sums. We may use operations on formal power series to evaluate explicit sums. Let d. To determine we notice for any polynomial P, = d P (X )A = 0 P ()a X. Applying this to the geometric series A(X) = X = Xn+ X with P (X) = X d we obtain d X = P (X X) Xn+ X. Setting X =, we get n(n + )/2 for d = and n(n + )(2n + )/6 for d = 2, and in principle we can compute the sum for any value of d. As a second example, consider the sum. ( + d ) (d) = 9

10 This can be determined via formal integration: if A(X) is the geometric power series, and P (X) = X d, then d X. For d = 2 we get 2 X = log( X) = X + ( X) log( X). Putting X = we get that the sum is. Of course there is an easier way: it is a telescoping sum in disguise ( + ) = ( ) =. + For d = 3 we get = = X X2 2 log( X) + X log( X) 2 X2 log( X). Again putting X = we get that the sum is. We should note that maing a substitution 4 into a formal power series converts it to a power series where issues of convergence arise. The substitution step is only valid within the radius of convergence of the power series. Formal power series for partial sums. The nth Harmonic sum is H n = n. In the exercises you will be ased to show via formal integration that: H n X n = n= log( X). X This relies on the basic observation that if A(X) is a formal power series for the sequence (a n ) n N, then A(X) X is a formal power series for the partial sums of (a n ) n N. Let N. Suppose we wish to determine Bivariate series. B (Y ) = 0 Y n.

11 If we sum B (Y )X we get A(X, Y ) = X Y n = X Y n = Y n ( + X) n = To determine B (Y ) we require [X ]A(X, Y ). This is straightforward: Y ( + X). [X ]A(X, Y ) = [X ] Y ( + X) = [X ] Y XY Y = ( Y ) Y = Y Y ( Y ). + In particular we see independent of that 2 n = 2. Of course there is a more direct approach: notice using formal derivatives B (Y ) = Y! Y n = Y! Y = Y ( Y ) +. Some other examples of bivariate series will be encountered as formal power series of Stirling numbers of the first and second ind. Inverse pairs. Suppose we are given an identity b n = a c. A useful thing to do is to determine a sequence d such that a n = b d. The sequences a n and b n are referred to as inverse pairs. This may occasionally be done via formal power series. We give an example, which might be referred to as binomial inversion: let (a n ) n N {0} be a sequence of complex numbers and for n N {0} let (b n ) n N be defined by: b n = a. Then we claim for all n N {0}, a n = ( ) n b.

12 To see this, we have b n n! Xn = Now this is the product of the two series a! (n )! Xn. exp(x) = n! Xn and a! X. Therefore a! X = exp( X) b n n! = ( ) b!(n )! Xn. Comparing coefficients of X n gives the result. In other words, we managed to find a formal power series relation B(X) = C(X)A(X) where A(X) and B(X) have the a n and b n as coefficients, or some simple function of a n and b n, and then writing A(X) = B(X)/C(X) gives the inverse relation. Recurrence equations. By a recurrence equation we mean a sequence (a n ) n N defined by a n = f(a n, a n 2,..., a 0 ) for some function f together with some initial conditions on the sequence. For instance, the Fibonacci equation a n = a n + a n 2 for n 2 and a 0 = 0 and a = we solved by creating a formal power series for each side and then determining a closed formula for that power series and reading off coefficients. In general this approach leads to functional and partial differential equations for the unnown formal power series A(X), however often these are difficult or impossible to solve explicitly. This also generalizes to rings R[[X]] however the technical complications often become quite a barrier. We will nevertheless discuss particular well-nown cases where these can be overcome. Here we give one more example: let (a n ) n N satisfy a = and a 2n = 2a n and a 2n+ = a n + a n+ for n N. Summing the recurrence, A(X) = ( + X + X 2 )A(X 2 ) where A(X) = n= a nx n. Repeating this functional equation gives (noting summability): A(X) = ( + X 2 + X 2 ). = We may interpret this Euler product has coefficients representing the number of partitions of an integer into powers of two with some restrictions on the multiplicities of consecutive powers of two. The special topic of linear recurrences with constant coefficients can be effectively solved using formal power series. 2