Combinatorial Analysis of the Geometric Series

Transcription

1 Combinatorial Analysis of the Geometric Series David P. Little April 7, Analytic Convergence of a Series The series converges analytically if and only if the sequence of partial sums, s n a 0 + a + + a n converges. i0 a i In other words, an infinite sum is defined to be the limit of a finite sum: i0 a i lim n n i0 a i 2

2 The Geometric Series The series is geometric if there exists r C such that for all integers n 0, All geometric series are of the form a n a n+ a n r. a r n a + ar + ar 2 + ar 3 + and converge to if and only if r <. a r 3 A Real Geometric Series r a ar ar 2 ar 3 a r The sum of the widths of the rectangles is given by the geometric series a + ar + ar 2 + ar 3 + ar 4 + And if 0 <r<, it converges by the monotonic sequence theorem. 4

3 A Complex Geometric Series 2 +z + z 2 + z 3 + z 4 +z + z 2 + z 3 z 3 z 2 z +z + z 2 +z z Formal Power Series Given a sequence c 0,c,c 2,..., the corresponding formal power series is given by c n q n For a formal power series, convergence comes down to the computability of the coefficients c n, and not the values of q that result in a convergent series. The formal power series n!q n +q +2q 2 +6q 3 +24q q 5 + is combinatorially significant since the sequence of coefficients is the number of permutations, but yet it has no analytic significance because its radius of convergence is 0. 6

4 Convergence of a Formal Power Series Convergence Computability The formal power series of the function F (q) exists if and only if for every integer n 0, the coefficient of q n can be computed in a finite number of operations. Example F (q) n q n q + q 2 + q 3 + has no formal power series expansion since the constant term appears in every term. In other words, the coefficient of q 0 cannot be computed in a finite number of operations. 7 Example The following function has a well-defined formal power series. n n q n q n q q + q2 q 2 + q3 q 3 + The coefficient of q 4 can be computed in the following manner ( ) q n ( ) q q n q 4 q + q2 q 2 + q3 q 3 + q4 q 4 q( + q + q 2 + q 3 + q 4 + ) + q 2 ( + q 2 + q 4 + q 6 + q 8 + ) + q 3 ( + q 3 + q 6 + q 9 + q 2 + ) + q 4 ( + q 4 + q 8 + q 2 + q 6 + ) q 4 (q + q 2 + q 3 + q 4 )+(q 2 + q 4 )+(0)+(q 4 ) 3 q 4 q 4 8

5 Convergence of a Formal Power Series In general, the coefficient of q N in is the number of divisors of N. n q n q n n q n q n q +2q 2 +2q 3 +3q 4 +2q 5 +4q 6 +2q Algebra of Formal Power Series a n q n ± b n q n ( )( ) a r q r b s q s r0 s0 (a n ± b n ) q n ( n ) a r b n r r0 q n The collection of formal power series with the operations of addition and multiplication defined above forms a ring. Series with nonzero constant term are the elements that have a multiplicative inverse. 0

6 Generating Functions The function F (q) is the generating function of the sequence {c n } if its power series representation is given by c n q n Example ( + q) n is the generating function for ( ( n 0), n ( ),, n n). is the generating function for,,,,... q ( q) k is the generating function for ( k ) ( k, k ) ( k, k+ k ),... Combinatorial Interpretations Let A and B be disjoint multisets and let a n be the number of ways to select n objects from A and let b n be the number of ways to select n objects from B. If A(q) and B(q) are the corresponding generating functions, then A(q)+B(q) is the generating function for {a n + b n } n 0, the number of ways to select n things from A or n things from B but not both. A(q)B(q) is the generating function for { n } a r b n r r0 number of ways to select n objects from A B. n 0,the 2

7 An Example q +q + q2 + q 3 + is the G.F. for the number of ways to write n as a sum of ones. q 2 +q2 + q 4 + q 6 + is the G.F. for the number of ways to write n as a sum of twos. ( q)( q 2 ) +q +2q2 +2q 3 +3q 4 +3q 5 +4q 6 + is the G.F. for the number of ways to write n as an unordered sum of ones and twos. 3 Partitions ( q)( q 2 ) ( q N ) is the G.F. for the number of ways to write n as an unordered sum of positive integers less than or equal to N. Definition An integer partition of n is a weakly decreasing sequence of positive integers that sum to n. i q i +q +2q2 +3q 3 +5q 4 +7q 5 +q 6 + is the G.F. for the number of integer partitions. 4

8 Hypergeometric Series The series is said to be hypergeometric if c 0 and for all integers n 0, c n+ a rational function of n. c n c n is Example e x tan (x) x n n! ( ) n x2n+ (2n +) cos(x) ln( x) ( ) n x2n n x n n (2n)! 5 Suppose Then c n+ (a + n)z b + n c n. c n+ c n (a + n)z b + n (a + n)z b + n (a + n )z b + n c n (a + n)z (a + n )z az b + n b + n b c 0 a(a +)(a +2) (a + n)zn+ (a) n+z n+ b(b +)(b +2) (b + n) (b) n+ where (z) n is called a shifted factorial. { if n 0 z(z +)(z +2) (z + n ) otherwise 6

9 Generalized Hypergeometric Series Ratio of consecutive terms is a rational function of n: [ ] a,a 2,...,a r (a ) n (a 2 ) n (a r ) n z n rf s ; z b,...,b s (b ) n (b s ) n n! [ ] ( + z) a a F 0 ; z ln( + z) z 2 F [, 2 ; z sin (z) z 2 F [ /2, /2 3/2 ; z2 [ tan /2, (z) z 2 F 3/2 ; z2 [ ] e z 0 F 0 ; z ] ] ] 7 Basic Hypergeometric Series Ratio of consecutive terms is a rational function of q n : [ ] a,a 2,...,a r rφ s ; q,z b,...,b s (a ; q) n (a 2 ; q) n (a r ; q) n z n (b ; q) n (b s ; q) n where the symbol (z; q) n is called a q-shifted factorial and defined by { if n (z; q) n ( z)( zq)( zq 2 ) ( zq n ) otherwise 8

10 q-analog of the binomial series Theorem (Cauchy) ( a/z; q) n z n +aq n zq n (z + a)(z + aq) (z + aq n ) ( q)( q 2 ) ( q n ) ( + a)( + aq)( + aq2 ) ( z)( zq)( zq 2 ) Combinatorial Proof Show that both sides have the same formal power series expansion. Specifically, we will show that the coefficient of q n on both sides of the equation counts the same set of combinatorial objects. 9 Weighted Tilings Definition A tiling is a covering of an infinitely long board: using different types of tiles: 3 5 The weight of a tiling T is given by w(t ) t T w(t) where w(t) is the weight of the tile t. The weight of a white square will always be. Each tiling will have a finite number of non-white square tiles. 20

11 q-analog of the binomial series Weight tiles in the following manner: zq i if t is a with i or to its left w(t) aq i if t is a with i or to its left if t is a Theorem (Cauchy) ( a/z; q) n z n +aq n zq n + z + a q + (z + a)(z + aq) ( q)( q 2 ) + (z + a)(z + aq)(z + aq2 ) ( q)( q 2 )( q 3 ) + +a z +aq zq +aq2 zq 2 +aq3 zq 3 2 Theorem (Cauchy) ( a/z; q) n z n Proof. PART I: Interpret infinite series +aq n zq n STEP : Place n black or gray squares in positions, 2, 3,...,n. A in position i accounts for a weight of z. A in position i accounts for a weight of aq n i. This process accounts for a weight of n (z + aq n i )( a/z; q) n z n i 22

12 Theorem (Cauchy) ( a/z; q) n z n Proof. PART I: Interpret infinite series +aq n zq n STEP 2: Insert white squares to the left of each black/gray square } {{ } j Inserting j white squares increases the weight by a factor of q 3j Accounting for all values of j: (q 3 ) j q 3 Accounting for all positions: n i j0 q i 23 Theorem (Cauchy) ( a/z; q) n a n Proof. PART II: Interpret infinite product Each tiling can be broken up into segments: +aq n zq n j 0 black squares {}}{ The weight of the nth segment for n 0 is given by ( + aq n ) j0 (zq n ) j +aqn zq n Multiplying over n 0 completes the construction. 24

13 Specializations z q, a 0 No gray squares, black squares weighted by q j if it has j white squares to its left: q n Generating function for partitions. z 0, a q n q n No black squares, gray squares weighted by q j if it is in position j: q n(n )/2 ( + q n ) n Generating function for partitions into distinct parts. 25 Specializations z q, a q N+ [ N + n n ] q n ( q)( q 2 ) ( q N ) Generating function for partitions using the numbers through N. where is a q-analog of ( n k). [ ] n k lim q (q; q) k k [ ] n k ( ) n k 26

14 Other Identities Heine (cq; q) Lebesgue: ( c/a; q) n ( q/b; q) n a n b n (cq; q) n ( + bcq n )( + aq n+ ) abq n ( z; q) n q (n+ 2 ) ( + q n )( + zq 2n ) n Cauchy: (zq; q) z n q n2 (zq; q) n 27 Sylvester: Rogers: z n q n2 (q 2 ; q 2 ) n ( + zq 2n ) n z n q n2 ( zq 2 ; q 2 ) z n q n2 (q 2 ; q 2 ) n ( zq 2 ; q 2 ) n and many many many more... 28

15 References Free Books: generatingfunctionology by H. Wilf AB by H. Wilf, D. Zeilberger, M. Petkovsek More Texts: Basic Hypergeometric Series by G. Gasper & M. Rahman The Theory of Partitions by G. E. Andrews Special Functions by G. E. Andrews, R. Askey, R. Roy Papers: L. J. Slater, Further Identities of the Rogers-Ramanujan Type, Proc. London Math. Soc. (2) 54 (952),