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:
∞
i=0
ai = lim
n→∞
n
ai
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,
an+1
= r.
an
All geometric series are of the form
∞
a · r n = a + ar + ar 2 + ar 3 + · · ·
n=0
and converge to
a
1−r
if and only if |r| < 1.
3
A Real Geometric Series
1
r
···
a
ar
ar 2
ar 3
···
a
1−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 < 1, it converges by the monotonic sequence theorem.
4
A Complex Geometric Series
2
1 + z + z2 + z3 + z4
1 + z + z2 + z3
1 + z + z2
1
z3
z2
z
1+z
z4
−1
1
2
5
Formal Power Series
Given a sequence c0 , c1 , c2 , . . ., the corresponding formal power series is
given by
∞
cn q n
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!q n = 1 + q + 2q 2 + 6q 3 + 24q 4 + 120q 5 + · · ·
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 q n can be computed in a finite number of
operations.
Example
F (q) =
∞
1
1
1
1
=
+
+ ···
+
n
2
3
1
−
q
1
−
q
1
−
q
1
−
q
n=1
has no formal power series expansion since the constant term 1 appears in
every term. In other words, the coefficient of q 0 cannot be computed in a
finite number of operations.
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Example
The following function has a well-defined formal power series.
∞
n=1
q2
q
q3
qn
+
=
+
+ ···
1 − qn
1 − q 1 − q2 1 − q3
The coefficient of q 4 can be computed in the following manner
∞
qn
q2
q
q3
q4
+
=
+
+
1 − q n q4
1 − q 1 − q 2 1 − q 3 1 − q 4 q4
n=1
q(1 + q + q 2 + q 3 + q 4 + · · · )
=
+ q 2 (1 + q 2 + q 4 + q 6 + q 8 + · · · )
+ q 3 (1 + q 3 + q 6 + q 9 + q 12 + · · · )
4
4
8
+ q (1 + q + q + q
12
+q
16
+ · · · )
q4
= (q + q + q + q ) + (q + q ) + (0) + (q )
2
= 3
3
4
2
4
4
q4
8
Convergence of a Formal Power Series
In general, the coefficient of q N in
∞
n=1
qn
1 − qn
is the number of divisors of N .
∞
qn
1 − qn
n=1
= q + 2q 2 + 2q 3 + 3q 4 + 2q 5 + 4q 6 + 2q 7 + · · ·
9
Algebra of Formal Power Series
∞
an q ±
n=0
∞
r=0
n
ar q r
∞
bn q
n=0
∞
s=0
bs q s
n
=
=
∞
(an ± bn ) q n
n=0
n
∞
n=0
ar bn−r
qn
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
∞
cn q n
n=0
Example
(1 + q)n is the generating function for
n n
n
,
,
·
·
·
,
0
1
n .
1
is the generating function for 1, 1, 1, 1, . . ..
1−q
k−1 k k+1
1
is
the
generating
function
for
k−1 , k−1 , k−1 , . . ..
(1 − q)k
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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.
A(q)B(q) is the generating function for
n
r=0
ar bn−r
number of ways to select n objects from A ∪ B.
, the
n≥0
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 + · · ·
2
1−q
is the G.F. for the number of ways to write n as a sum of twos.
1
= 1 + q + 2q 2 + 2q 3 + 3q 4 + 3q 5 + 4q 6 + · · ·
2
(1 − q)(1 − q )
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 − q 2 ) · · · (1 − 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=1
1
= 1 + 1q + 2q 2 + 3q 3 + 5q 4 + 7q 5 + 11q 6 + · · ·
i
1−q
is the G.F. for the number of integer partitions.
14
Hypergeometric Series
The series
∞
cn
n=0
is said to be hypergeometric if c0 = 1 and for all integers n ≥ 0,
a rational function of n.
cn+1
is
cn
Example
x
e =
tan−1 (x) =
∞
xn
n=0
∞
∞
x2n
cos(x) =
(−1)
(2n)!
n=0
n!
2n+1
n x
(−1)
(2n + 1)
ln(1 − x) = −
n=0
n
∞
xn
n=1
n
15
Suppose
(a + n)z
cn+1
=
cn
b+n
Then
cn+1 =
(a + n)z
· cn =
b+n
..
.
=
=
where
(a + n)z (a + n − 1)z
·
· cn−1
b+n
b+n−1
az
(a + n)z (a + n − 1)z
·
···
· c0
b+n
b+n−1
b
(a)n+1 z n+1
a(a + 1)(a + 2) · · · (a + n)z n+1
=
b(b + 1)(b + 2) · · · (b + n)
(b)n+1
1
(z)n =
z(z + 1)(z + 2) · · · (z + n − 1)
if n = 0
otherwise
is called a shifted factorial.
16
Generalized Hypergeometric Series
Ratio of consecutive terms is a rational function of n:
∞
(a1 )n (a2 )n · · · (ar )n z n
a1 , a2 , . . . , ar
F
;
z
=
r s
b1 , . . . , bs
(b1 )n · · · (bs )n
n!
n=0
(1 + z)a =
ln(1 + z) =
sin−1 (z) =
tan−1 (z) =
ez =
−a
; −z
1 F0
−
1, 1
; −z
z 2 F1
2
1/2, 1/2 2
;z
z 2 F1
3/2
1/2, 1
2
z 2 F1
; −z
3/2
−
;z
0 F0
−
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Basic Hypergeometric Series
Ratio of consecutive terms is a rational function of q n :
r φs
a1 , a2 , . . . , ar
; q, z
b1 , . . . , bs
=
∞
(a1 ; q)n (a2 ; q)n · · · (ar ; q)n
n=0
(b1 ; q)n · · · (bs ; q)n
zn
(q; q)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 − zq 2 ) · · · (1 − zq n−1 ) otherwise
18
q-analog of the binomial series
Theorem (Cauchy)
∞
(−a/z; q)n z n
n=0
(q; q)n
∞
(z + a)(z + aq) · · · (z + aq n−1 )
n=0
(1 − q)(1 − q 2 ) · · · (1 − q n )
∞
1 + aq n
=
1 − zq n
n=0
=
(1 + a)(1 + aq)(1 + aq 2 ) · · ·
(1 − z)(1 − zq)(1 − 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.
19
Weighted Tilings
Definition
A tiling is a covering of an infinitely long board:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 · · ·
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.
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q-analog of the binomial series
Weight tiles in the following manner:

i
with i

zq if t is a
w(t) = aq i if t is a
with i


1
if t is a
or
or
to its left
to its left
Theorem (Cauchy)
∞
(−a/z; q)n z n
n=0
=
(q; q)n
∞
1 + aq n
=
1 − zq n
n=0
z + a (z + a)(z + aq) (z + a)(z + aq)(z + aq 2 )
+
+
+ ···
1+
1−q
(1 − q)(1 − q 2 )
(1 − q)(1 − q 2 )(1 − q 3 )
1 + a 1 + aq 1 + aq 2 1 + aq 3
·
·
·
···
1 − z 1 − zq 1 − zq 2 1 − zq 3
21
Theorem (Cauchy)
∞
(−a/z; q)n z n
n=0
(q; q)n
∞
1 + aq n
=
1 − zq 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.
in position i accounts for a weight of aq n−i .
A
This process accounts for a weight of
n
(z + aq n−i ) = (−a/z; q)n z n
i=1
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Theorem (Cauchy)
∞
(−a/z; q)n z n
(q; q)n
n=0
∞
1 + aq n
=
1 − zq 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 q 3j
∞
1
Accounting for all values of j:
(q 3 )j =
1 − q3
j=0
Accounting for all positions:
n
i=1
1
1
=
i
1−q
(q; q)n
23
Theorem (Cauchy)
∞
(−a/z; q)n an
n=0
(q; q)n
∞
1 + aq n
=
1 − zq 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
n
(1 + aq )
∞
j=0
1 + aq n
(zq ) =
1 − zq n
n j
Multiplying over n ≥ 0 completes the construction.
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Specializations
z = q, a = 0
No gray squares, black squares weighted by q j if it has j − 1 white squares
to its left:
∞
n=0
∞
qn
1
=
(q; q)n
1 − qn
n=1
Generating function for partitions.
z = 0, a = q
No black squares, gray squares weighted by q j if it is in position j:
∞
q n(n−1)/2
n=0
(q; q)n
=
∞
(1 + q n )
n=1
Generating function for partitions into distinct parts.
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Specializations
z = q, a = −q N +1
∞ 1
N +n−1 n
q =
(1 − q)(1 − q 2 ) · · · (1 − q N )
n
n=0
Generating function for partitions using the numbers 1 through N .
n
(q; q)n
=
(q; q)k (q; q)n−k
k
where
is a q-analog of
n
k
.
n
n
lim
=
q→1 k
k
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Other Identities
Heine
(cq; q)∞
∞
(−c/a; q)n (−q/b; q)n an bn
n=0
(q; q)n (cq; q)n
∞
(1 + bcq n )(1 + aq n+1 )
=
1 − abq n
n=0
Lebesgue:
∞
(−z; q)n
n=0
(q; q)n
Cauchy:
n+1
q( 2 ) =
∞
(1 + q n )(1 + zq 2n−1 )
n=1
∞
2
znqn
=1
(zq; q)∞
(q;
q)
(zq;
q)
n
n
n=0
27
Sylvester:
2
∞
∞
znqn
=
(1 + zq 2n−1 )
2
2
(q ; q )n
n=0
n=1
Rogers:
2
2
∞
∞
znqn
zn qn
2 2
= (−zq ; q )∞
(q; q)n
(q 2 ; q 2 )n (−zq 2 ; q 2 )n
n=0
n=0
and many many many more....
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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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