Polynomial Families with Interesting Zeros

Robert BoyerDrexel University

April 30, 2010

Taylor Expansions with Finite Radius of Convergence

Work of Robert Jentzsch, 1917

Partial Sums of Geometric Series

pn(z) =n

!

k=0

zk = 1 + z + z2 + · · · + zn =zn+1 ! 1

z ! 1

pn(z) has zeros at all the (n + 1)-st roots of unity except at z = 1

Zeros all lie on the unit circle and fill it up as n " #

Taylor Polynomials for f (z)

pn(z) =n

!

k=0

f (k)(0)

k!zk

No simple formula to get the zeros of pn(z)

Assume that the Taylor series!!

k=0

f (k)(0)

k!zk converges for |z | < R

Question: What happens to the zeros of pn(z) as n " #?

Figure: Taylor polynomial for sec(x) – Degree 100

Figure: Taylor polynomial for arcsin(x) – Degree 100

Figure: !(x) – Degree 100

2

2

1

10

-1

0

-2

-1-2

Figure:"

xk/sin(k) – Degree 400

-1 1.5-0.5

1

0.5

-0.5

0.50

-1

01

Figure:"

xk/sin(k) – Degree 7,000

Taylor Polynomials for Exponential and Cosine

Work of Szego

Figure:"

(70x)k/k! – Degree 70

Figure: Szego Curve |ze1!z | = 1

Figure:"

(70x)k/k! – Degree 70 – with limiting curve

Figure: Exponential with zero attractor - degree 1000

Figure: Scaled Cosine Polynomial with limiting curve – Degree 70

Figure: Cosine with zero attractor - degree 1000

Taylor Polynomials for Linear Combinations of Exponentials

Work of Bleher and Mallison, 2006

3 e(8+2 i)z ! (9 ! 12 i) e(4+7 i)z + (2 + i) e("7+4 i)z

+ (30 + 30 i) e(6"7 i)z + (8 ! 5 i) e(6"4 i)z

+ (3 ! 9 i) e(4+4 i)z + 2 ie("2"4 i)z

Figure: Linear Combination of exp’s - degree 250

Figure: Linear Combination of exp’s - degree 1000

Bernoulli Polynomials

Work of Boyer and Goh, 2007

Bernoulli Polynomials

Generating Function

text

et ! 1=

!!

n=0

Bn(x)tn

Formula for Sums of Powers of Integers

m!

k=1

kn =Bn+1(m) ! Bn+1(1)

m + 1

Table of Bernoulli Polynomials

B1(x) = x ! 1/2

B2(x) = x2 ! x + 1/6

B3(x) = x3 ! 3/2 x2 + 1/2 x

B4(x) = x4 ! 2 x3 + x2 ! 1/30

B5(x) = x5 ! 5/2 x4 + 5/3 x3 ! 1/6 x

B6(x) = x6 ! 3 x5 + 5/2 x4 ! 1/2 x2 + 1/42

B7(x) = x7 ! 7/2 x6 + 7/2 x5 ! 7/6 x3 + 1/6 x

B8(x) = x8 ! 4 x7 + 14/3 x6 ! 7/3 x4 + 2/3 x2 ! 1/30

Figure: Bernoulli Polynomial Degree 70

Figure: Bernoulli Polynomial Degree 70 with limiting curve

Figure: Bernoulli Polynomial Degree 500

Figure: Bernoulli Polynomial Degree 1000

Figure: Bernoulli Polynomial Degree 500 with limiting curve

Figure: Bernoulli Polynomial Degree 1,000 with limiting curve

Appell Polynomials

Work of Boyer and Goh, 2010

Appell Polynomials

ext

g(t)=

!!

n=0

Pn(x)tn, P #

n(x) = Pn"1(x)

Basic Example: Pn(x) =xn

n!with g(t) = 1

Example:

n!

k=0

xn

n!, with generating function g(t) = 1 ! t

Another Method: Given any sequence {an}, the polynomialfamily below is Appell:

Pn(x) =n

!

k=0

an"k

k!xk

Figure: Appell Polynomial – Degree 100 – g(t) = (t ! 1)(t2 + 2)

Figure: Two Szego Curves

Figure: Degree 100, g(t) = (t ! 1)(t2 + 2)

Figure: Degree 400, g(t) = (t ! 1)(t2 + 2)

Figure: Appell Polynomial – Degree 400 –g(t) = (t ! 1/(1.2e i3!/16))(t ! 1/(1.3e i7!/16))(t ! 1/1.5)

Figure: Appell Polynomial –g(t) = (t ! 1/(1.2e i3!/16))(t ! 1/(1.3e i7!/16))(t ! 1/1.5)

Figure: Appell Polynomials

Figure: Appell Polynomials

Polynomials Satisfying a Linear Recurrence

pn+1(z) =k

!

j=1

qj(z)pn"j(z)

Example

pn+1(z) = (z + 1 ! i) pn + (z + 1) (z ! i) pn"1 +#

z3 + 10$

pn"2

Example

pn+1(z) = [(z + 1 ! i) + (z + 1) (z ! i)]pn"1 +#

z3 + 10$

pn"2

Fibonacci Type Polynomials

Fibonacci numbers: Fn+2 = Fn+1 + Fn have polynomial version:

Fn+1(x) = xFn(x) + Fn"1(x), F1(x) = 1,F2(x) = x .

Their zeros are all purely imaginary.

More general versions “Tribonacci”:

Tn+3(x) = x2Tn+2(x) + xTn+1 + Tn(x),

T0(x) = 0,T1(x) = 1,T2(x) = x2

The following two examples have the following recurrences.Both have the same initial conditions:

p0(z) = z6!z4+i , p1(z) = z!i+2, p2(z) = (2+i)2 (z2!8)

Example One

pn+1(z) = [(z +1+ i) + (z +1)] (z ! i) pn"1(z)+ (z3 +10) pn"2(z)

Example Two

pn+1(z) = (z+1+i) pn(x)+(z+1) (z!i) pn"1(z)+(z3+10) pn"2(z)

Figure: Tribonacci Polynomial - Degree 238

Example One

Figure: Generalized Fibonacci - Degree 76

Figure: Generalized Fibonacci - Degree 506

Figure: Generalized Fiboncaci - Degree 1006

Example Two

Figure: Another Example: Generalized Fibonaaci - Degree 506

Jacobi Polynomials

K Driver and P. Duren, 1999

P(!,")n (z) =

1

n!

!(" + n + 1)

!(" + # + n + 1)

n!

m=0

%

n

m

&

!(" + # + n + m + 1)

!(" + m + 1)

%

z ! 1

2

&m

" = kn + 1, # = !n ! 1, with k = 2

Lemniscate: |z ! 1|k |z + 1| =

%

2

k + 1

&k+1

kk

Figure: Jacobi Polynomial - Degree 50

Figure: Jacobi Polynomial - Degree 500

Figure: Jacobi Polynomial - Degree 700

Mandelbrot Polynomials with Fractal Zeros

pn+1(x) = xpn(x)2 + 1, p0(x) = 1

Figure: Mandelbot Polynomial - Degree 210 ! 1 = 1023

Figure: Mandelbot Polynomial - Degree 211 ! 1 = 2, 048

Figure: Mandelbot Polynomial - Degree 212 ! 1 = 4, 095

Polynomials Associated with Painleve Equations

Peter A Clarkson and Elizabeth L Mansfield, 2003

Painleve Di!erential Equations and VorobevYablonskii

Polynomials

Suppose that Qn(z) satisfies the recursion relation

Qn+1Qn"1 = zQ2n ! 4[QnQ

##

n ! (Q #

n)2], Q0(z) = 1,Q1(z) = z .

Then the rational function

w(z ; n) =d

dzln

Qn"1(z)

Qn(z)

satisfies PII

w ## = 2w3 + zw + ", " = n $ Z+.

Further, w(z ; 0) = 0 and w(z ;!n) = !w(z ; n).The VorobevYablonskii polynomials are monic with degreen(n + 1)/2.

Figure: Painleve Polynomial - Degree 325

Figure: Painleve Polynomial - Degree 1, 275

Richard Stanley Examples from Combinatorics, 2001

Chromatic Polynomial of a Graph

A complete bipartite graph G has its vertices broken into twodisjoint subsets A and B so that every vertex in A is connected byan edge with every vertex in B .

A coloring of a graph with r colors is an assignment that uses allthe possible colors so that if vertices v and w are connected by anedge they must have di"erent colors.

The chromatic polynomial pG (x) of a graph G is determined by itsvalues on the positive integers:

pG (n) = # all colorings of G using n colors

Figure: Chromatic Polynomial for a Complete Bipartite Graph – Degree1,000

q-Catalan Numbers

Catalan numbers: Cn = 1n+1

#2nn

$

with recurrence

Cn+1 =n

!

i=0

CiCn"1

Counts properly parathenized expressions or nondecreasing binarypaths

q-Catalan Polynomials Cn(q)

Cn+1(q) =n

!

i=0

Ci (q)Cn"i (q)q(i+1)(n"i), C0(q) = 1,

deg(Cn) =

%

n

2

&

, Cn(1) = Cn

Geometric-Combinatorial Meaning

Cn(q) =!

P:path

qarea(P)

where P is any lattice path from (0, 0) to (n, n) with step either(1, 0) or (0, 1) satisfies the additional condition that the path P

never rises above the line y = x .area(P) means the area underneath the path.

Figure: q-catalan Polynomial - Degree 190

Figure: q-catalan Polynomial - Degree 11,175

Partition Polynomials

Work of Boyer and Goh, 2007

Polynomial Partition Polynomials

Partition Numbers

4 = 4, 4 = 3+1, 4 = 2+2, 4 = 2+1+1, 4 = 1+1+1+1

p1(4) = 1, p2(4) = 2, p3(4) = 1, p4(4) = 1

Hardy-Ramanujan Asymptotics

p(n) %1

4n&

3e#&

2n/3.

Partition Polynomials

Fn(x) =n

!

k=1

pk(n)xk

1

1

0.5

0.50

-0.5

0

-1

-0.5-1

Figure: Partition Polynomial - degree 200

1000 400

16

300200

4

500

12

0

8

Figure: Digits of Partition Polynomial - degree 500

250

200

150

50

0

300

800000 20000

100

6000040000

Figure: Digits of Partition Polynomial - degree 80,000

10

1

0.5-0.5

-1

-0.5

0.5

0-1

Figure: All Zeros of Partition Polynomial - degree 10,000

1

1

0.5

0.50

-0.5

0

-1

-0.5-1

Figure: Partition Polynomial Attractor

1

0.8

0.6

0.4

0.2

010.50-0.5-1

Figure: Partition Polynomial Attractor in Upper Half Plane

0.4

-0.4

0.2

0-0.6-0.8-1

1

0

0.8

-0.2

0.6

Figure: Partition Polynomial Attractor in Second Quadrant

0.7

-0.6

0.65

0.6

-0.640.55

-0.68-0.72

0.8

-0.56

0.75

Figure: Triple Point for Partition Polynomial - degree 400

0.76

-0.6

0.72

0.68

-0.64

0.64

-0.68-0.72

0.8

-0.56

Figure: Triple Point for Partition Polynomial - degree 5,000

0.76

-0.6

0.72

0.68

-0.64

0.64

-0.68-0.72

0.8

-0.56

Figure: Triple Point for Partition Polynomial - degree 50,000

1

0.8

0.6

0.4

0.2

0

10.5-0.5 0-1

Figure: Region I for Partition Polynomial

0.5

0.1

0.6

0.4

0

10.50-0.5-1

0.3

0.7

0.2

Figure: Region II for Partition Polynomial

-0.5-0.6-0.7-0.8

0.9

0.8

0.7

0.6

-0.3

0.5

0.4

-0.4

Figure: Region III for Partition Polynomial

0.4

0.2

1

0.8

0

10 0.5

-0.5

0.6

-1