Combinatorial Analysis of the Geometric Series

David P. Little

April 7, 2015

www.math.psu.edu/dlittle

1

Analytic Convergence of a Series

The series ∞ ai i=0 converges analytically if and only if the sequence of partial sums,

sn = a0 + a1 + ···+ an converges.

In other words, an infinite sum is defined to be the limit of a finite sum: ∞ n ai = lim ai n→∞ i=0 i=0

2 The Geometric Series

The series ∞ an n=0 is geometric if there exists r ∈ C such that for all integers n ≥ 0, a n+1 = r. an

All geometric series are of the form ∞ a · rn = a + ar + ar2 + ar3 + ··· n=0 and converge to a 1 − r if and only if |r| < 1.

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A Real Geometric Series

1

r ···

2 3 a a ar ar ar ··· 1−r

The sum of the widths of the rectangles is given by the geometric series

a + ar + ar2 + ar3 + ar4 + ···

And if 0

4 A Complex Geometric Series

2 1+z + z2 + z3 + z4 1+z + z2 + z3

1+z + z2 1 z2 z3 z 1+z

z4

−1 12

5

Formal Power Series

Given a sequence c0,c1,c2,..., the corresponding formal power series is given by ∞ n cnq n=0

For a formal power series, convergence comes down to the computability of the coefficients cn, and not the values of q that result in a convergent series.

The formal power series ∞ n!qn =1+q +2q2 +6q3 +24q4 + 120q5 + ··· n=0 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 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 qn can be computed in a finite number of operations.

Example ∞ 1 1 1 1 F (q)= = + + + ··· 1 − qn 1 − q 1 − q2 1 − q3 n=1 has no formal power series expansion since the constant term 1 appears in every term. In other words, the coefficient of q0 cannot be computed in a finite number of operations.

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Example The following function has a well-defined formal power series.

∞ qn q q2 q3 = + + + ··· 1 − qn 1 − q 1 − q2 1 − q3 n=1

The coefficient of q4 can be computed in the following manner ∞ qn q q2 q3 q4 = + + + 1 − qn q4 1 − q 1 − q2 1 − q3 1 − q4 q4 n=1 = q(1 + q + q2 + q3 + q4 + ···) + q2(1 + q2 + q4 + q6 + q8 + ···) + q3(1 + q3 + q6 + q9 + q12 + ···) + q4(1 + q4 + q8 + q12 + q16 + ···) q4 =(q + q2 + q3 + q4)+(q2 + q4)+(0)+(q4) q4 =3

8 Convergence of a Formal Power Series

In general, the coefficient of qN in

∞ qn 1 − qn n=1 is the number of divisors of N.

∞ qn = q +2q2 +2q3 +3q4 +2q5 +4q6 +2q7 + ··· 1 − qn n=1

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Algebra of Formal Power Series

∞ ∞ ∞ n n n anq ± bnq = (an ± bn) q n=0 n=0 n=0 ∞ ∞ ∞ n r s n arq bsq = arbn−r q r=0 s=0 n=0 r=0

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.

10 Generating Functions

The function F (q) is the generating function of the sequence {cn} if its power series representation is given by ∞ n cnq n=0

Example n n n n (1 + q) is the generating function for 0 , 1 , ··· , n . 1 1, 1, 1, 1,... 1 − q is the generating function for . 1 k−1 , k , k+1 ,... (1 − q)k is the generating function for k−1 k−1 k−1 .

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Combinatorial Interpretations

Let A and B be disjoint multisets and let an be the number of ways to select n objects from A and let bn 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 {an + bn}n≥0, the number of ways to select n things from A or n things from B but not both. n A(q)B(q) is the generating function for arbn−r ,the r=0 n≥0 number of ways to select n objects from A ∪ B.

12 An Example

1 =1+q + q2 + q3 + ··· 1 − q is the G.F. for the number of ways to write n as a sum of ones.

1 =1+q2 + q4 + q6 + ··· 1 − q2 is the G.F. for the number of ways to write n as a sum of twos.

1 =1+q +2q2 +2q3 +3q4 +3q5 +4q6 + ··· (1 − q)(1 − q2) is the G.F. for the number of ways to write n as an unordered sum of ones and twos.

13

Partitions

1 (1 − q)(1 − q2) ···(1 − qN ) 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.

∞ 1 =1+1q +2q2 +3q3 +5q4 +7q5 +11q6 + ··· 1 − qi i=1 is the G.F. for the number of integer partitions.

14 Hypergeometric Series

The series ∞ cn n=0 cn+1 is said to be hypergeometric if c0 =1and for all integers n ≥ 0, is cn a rational function of n.

Example

∞ xn ∞ x2n ex = cos(x)= (−1)n n! (2n)! n=0 n=0 ∞ x2n+1 ∞ xn tan−1(x)= (−1)n ln(1 − x)=− (2n +1) n n=0 n=1

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Suppose c (a + n)z n+1 = cn b + n Then (a + n)z (a + n)z (a + n − 1)z c = · c = · · c n+1 b + n n b + n b + n − 1 n−1 . . (a + n)z (a + n − 1)z az = · ··· · c b + n b + n − 1 b 0 a(a +1)(a +2)···(a + n)zn+1 (a) zn+1 = = n+1 b(b +1)(b +2)···(b + n) (b)n+1 where 1 if n =0 (z)n = z(z +1)(z +2)···(z + n − 1) otherwise is called a shifted factorial.

16 Generalized Hypergeometric Series Ratio of consecutive terms is a rational function of n: ∞ n a1,a2,...,ar (a1)n(a2)n ···(ar)n z rFs ; z = b1,...,bs (b ) ···(b ) n! n=0 1 n s n

−a (1 + z)a = F ; −z 1 0 − 1, 1 ln(1 + z)=z F ; −z 2 1 2 1/2, 1/2 sin−1(z)=z F ; z2 2 1 3/2 1/2, 1 tan−1(z)=z F ; −z2 2 1 3/2 − ez = F ; z 0 0 −

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Basic Hypergeometric Series Ratio of consecutive terms is a rational function of qn: ∞ n a1,a2,...,ar (a1; q)n(a2; q)n ···(ar; q)n z rφs ; q,z = b1,...,bs (b ; q) ···(b ; q) (q; q) n=0 1 n s n n

where the symbol (z; q)n is called a q-shifted factorial and defined by 1 if n =1 (z; q)n = (1 − z)(1 − zq)(1 − zq2) ···(1 − zqn−1) otherwise

18 q-analog of the binomial series

Theorem (Cauchy) ∞ (−a/z; q) zn ∞ 1+aqn n = (q; q) 1 − zqn n=0 n n=0

∞ (z + a)(z + aq) ···(z + aqn−1) (1 + a)(1 + aq)(1 + aq2) ··· = (1 − q)(1 − q2) ···(1 − qn) (1 − z)(1 − zq)(1 − zq2) ··· n=0

Combinatorial Proof Show that both sides have the same formal power series expansion. Specifically, we will show that the coefficient of qn on both sides of the equation counts the same set of combinatorial objects.

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Weighted Tilings

Definition A tiling is a covering of an infinitely long board:

123456789101112131415··· using different types of tiles: 3 5

The weight of a tiling T is given by w(T )= w(t) t∈T where w(t) is the weight of the tile t. The weight of a white square will always be 1. Each tiling will have a finite number of non-white square tiles.

20 q-analog of the binomial series Weight tiles in the following manner:  i zq if t is a with i or to its left w(t)= aqi if t is a with i or to its left  1 if t is a

Theorem (Cauchy) ∞ (−a/z; q) zn ∞ 1+aqn n = (q; q) 1 − zqn n=0 n n=0

z + a (z + a)(z + aq) (z + a)(z + aq)(z + aq2) 1+ + + + ··· 1 − q (1 − q)(1 − q2) (1 − q)(1 − q2)(1 − q3) 1+a 1+aq 1+aq2 1+aq3 = · · · ··· 1 − z 1 − zq 1 − zq2 1 − zq3

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Theorem (Cauchy) ∞ (−a/z; q) zn ∞ 1+aqn n = (q; q) 1 − zqn n=0 n n=0

Proof. PART I: Interpret infinite series STEP 1: Place n black or gray squares in positions 1, 2, 3,...,n.

···

A in position i accounts for a weight of z. A in position i accounts for a weight of aqn−i. This process accounts for a weight of n n−i n (z + aq )=(−a/z; q)nz i=1

22 Theorem (Cauchy) ∞ (−a/z; q) zn ∞ 1+aqn n = (q; q) 1 − zqn n=0 n n=0

Proof. PART I: Interpret infinite series 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 q3j ∞ 1 j (q3)j = Accounting for all values of : 1 − q3 j=0 n 1 1 = Accounting for all positions: 1 − qi (q; q) i=1 n

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Theorem (Cauchy) ∞ (−a/z; q) an ∞ 1+aqn n = (q; q) 1 − zqn n=0 n n=0

Proof. PART II: Interpret infinite product Each tiling can be broken up into segments: j ≥ 0 black squares ··· ···

The weight of the nth segment for n ≥ 0 is given by

∞ 1+aqn (1 + aqn) (zqn)j = 1 − zqn j=0

Multiplying over n ≥ 0 completes the construction.

24 Specializations z = q, a =0 No gray squares, black squares weighted by qj if it has j − 1 white squares to its left:

∞ qn ∞ 1 = (q; q) 1 − qn n=0 n n=1 Generating function for partitions. z =0, a = q No black squares, gray squares weighted by qj if it is in position j:

∞ qn(n−1)/2 ∞ = (1 + qn) (q; q) n=0 n n=1 Generating function for partitions into distinct parts.

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Specializations z = q, a = −qN+1 ∞ N + n − 1 1 qn = n (1 − q)(1 − q2) ···(1 − qN ) n=0 Generating function for partitions using the numbers 1 through N. where n (q; q) = n k (q; q)k(q; q)n−k n is a q-analog of k . n n lim = q→1 k k

26 Other Identities

Heine ∞ (−c/a; q) (−q/b; q) anbn ∞ (1 + bcqn)(1 + aqn+1) (cq; q) n n = ∞ (q; q) (cq; q) 1 − abqn n=0 n n n=0

Lebesgue: ∞ ∞ (−z; q)n (n+1) n 2n−1 q 2 = (1 + q )(1 + zq ) (q; q) n=0 n n=1

Cauchy: ∞ znqn2 (zq; q) =1 ∞ (q; q) (zq; q) n=0 n n

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Sylvester: ∞ znqn2 ∞ = (1 + zq2n−1) (q2; q2) n=0 n n=1 Rogers:

∞ znqn2 ∞ znqn2 =(−zq2; q2) (q; q) ∞ (q2; q2) (−zq2; q2) n=0 n n=0 n n and many many many more....

28 References

Free Books: “generatingfunctionology” by H. Wilf “A=B” 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 (1952), 147-167 www.math.psu.edu/dlittle

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