Taylor Expansions with Finite Radius of Convergence Work of Robert Jentzsch, 1917

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1 Polynomial Families with Interesting Zeros Robert Boyer Drexel University April 30, 2010

2 Taylor Expansions with Finite Radius of Convergence Work of Robert Jentzsch, 1917

3 Partial Sums of Geometric Series p n (z) = n k=0 z k = 1 + z + z z n = z n+1 1 z 1 p n (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

4 Taylor Polynomials for f (z) p n (z) = n k=0 f (k) (0) k! z k No simple formula to get the zeros of p n (z) Assume that the Taylor series k=0 f (k) (0) k! z k converges for z < R Question: What happens to the zeros of p n (z) as n?

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

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

7 Figure: ζ(x) Degree 100

8 Figure: x k /sin(k) Degree 400

9 Figure: x k /sin(k) Degree 7,000

10 Taylor Polynomials for Exponential and Cosine Work of Szegö

11 Figure: (70x) k /k! Degree 70

12 Figure: Szegö Curve ze 1 z =1

13 Figure: (70x) k /k! Degree 70 with limiting curve

14 Figure: Exponential with zero attractor - degree 1000

15 Figure: Scaled Cosine Polynomial with limiting curve Degree 70

16 Figure: Cosine with zero attractor - degree 1000

17 Taylor Polynomials for Linear Combinations of Exponentials Work of Bleher and Mallison, e (8+2 i)z (9 12 i) e (4+7 i)z ( 7+4 i)z + (2 + i) e + ( i) e (6 7 i)z (6 4 i)z + (8 5 i) e + (3 9 i) e (4+4 i)z ( 2 4 i)z +2ie

18 Figure: Linear Combination of exp s - degree 250

19 Figure: Linear Combination of exp s - degree 1000

20 Bernoulli Polynomials Work of Boyer and Goh, 2007

21 Bernoulli Polynomials Generating Function te xt e t 1 = n=0 B n (x)t n Formula for Sums of Powers of Integers m k=1 k n = B n+1(m) B n+1 (1) m +1

22 Table of Bernoulli Polynomials B 1 (x) = x 1/2 B 2 (x) = x 2 x +1/6 B 3 (x) = x 3 3/2 x 2 +1/2x B 4 (x) = x 4 2 x 3 + x 2 1/30 B 5 (x) = x 5 5/2 x 4 +5/3x 3 1/6 x B 6 (x) = x 6 3 x 5 +5/2x 4 1/2 x 2 +1/42 B 7 (x) = x 7 7/2 x 6 +7/2x 5 7/6 x 3 +1/6x B 8 (x) = x 8 4 x /3 x 6 7/3 x 4 +2/3x 2 1/30

23 Figure: Bernoulli Polynomial Degree 70

24 Figure: Bernoulli Polynomial Degree 70 with limiting curve

25 Figure: Bernoulli Polynomial Degree 500

26 Figure: Bernoulli Polynomial Degree 1000

27 Figure: Bernoulli Polynomial Degree 500 with limiting curve

28 Figure: Bernoulli Polynomial Degree 1,000 with limiting curve

29 Appell Polynomials Work of Boyer and Goh, 2010

30 Appell Polynomials Basic Example: Example: e xt g(t) = n k=0 n=0 P n (x)t n, P n (x) = xn n! P n (x) =P n 1(x) with g(t) = 1 x n, with generating function g(t) = 1 t n! Another Method: Given any sequence {a n }, the polynomial family below is Appell: P n (x) = n k=0 a n k x k k!

31 Figure: Appell Polynomial Degree 100 g(t) = (t 1)(t 2 + 2)

32 Figure: Two Szegö Curves

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

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

35 Figure: Appell Polynomial Degree 400 g(t) = (t 1/(1.2e i3π/16 ))(t 1/(1.3e i7π/16 ))(t 1/1.5)

36 Figure: Appell Polynomial g(t) = (t 1/(1.2e i3π/16 ))(t 1/(1.3e i7π/16 ))(t 1/1.5)

37 Figure: Appell Polynomials

38 Figure: Appell Polynomials

39 Polynomials Satisfying a Linear Recurrence k p n+1 (z) = q j (z)p n j (z) j=1 Example p n+1 (z) = (z +1 i) p n +(z + 1) (z i) p n 1 + ( z ) p n 2 Example p n+1 (z) = [(z +1 i) + (z + 1) (z i)]p n 1 + ( z ) p n 2

40 Fibonacci Type Polynomials Fibonacci numbers: F n+2 = F n+1 + F n have polynomial version: F n+1 (x) =xf n (x)+f n 1 (x), Their zeros are all purely imaginary. F 1 (x) = 1, F 2 (x) =x. More general versions Tribonacci : T n+3 (x) = x 2 T n+2 (x)+xt n+1 + T n (x), T 0 (x) = 0, T 1 (x) = 1, T 2 (x) =x 2

41 The following two examples have the following recurrences. Both have the same initial conditions: p 0 (z) =z 6 z 4 +i, p 1 (z) =z i +2, p 2 (z) = (2+i) 2 (z 2 8) Example One p n+1 (z) = [(z i) + (z + 1)] (z i) p n 1 (z) + (z ) p n 2 (z) Example Two p n+1 (z) = (z+1+i) p n (x)+(z+1) (z i) p n 1 (z)+(z 3 +10) p n 2 (z)

42 Figure: Tribonacci Polynomial - Degree 238

43 Example One Figure: Generalized Fibonacci - Degree 76

44 Figure: Generalized Fibonacci - Degree 506

45 Figure: Generalized Fiboncaci - Degree 1006

46 Example Two Figure: Another Example: Generalized Fibonaaci - Degree 506

47 Jacobi Polynomials K Driver and P. Duren, 1999 P (α,β) n (z) = 1 n! Γ(α + n + 1) Γ(α + β + n + 1) n m=0 ( ) n Γ(α + β + n + m + 1) m Γ(α + m + 1) α = kn +1, β = n 1, with k =2 ( z 1 2 ) m Lemniscate: z 1 k z +1 = ( ) 2 k+1 k k k +1

48 Figure: Jacobi Polynomial - Degree 50

49 Figure: Jacobi Polynomial - Degree 500

50 Figure: Jacobi Polynomial - Degree 700

51 Mandelbrot Polynomials with Fractal Zeros p n+1 (x) =xp n (x) 2 +1, p 0 (x) = 1

52 Figure: Mandelbot Polynomial - Degree = 1023

53 Figure: Mandelbot Polynomial - Degree = 2, 048

54 Figure: Mandelbot Polynomial - Degree = 4, 095

55 Polynomials Associated with Painlevé Equations Peter A Clarkson and Elizabeth L Mansfield, 2003

56 Painlevé Differential Equations and VorobevYablonskii Polynomials Suppose that Q n (z) satisfies the recursion relation Q n+1 Q n 1 = zq 2 n 4[Q nq n (Q n )2 ], Q 0 (z) = 1, Q 1 (z) =z. Then the rational function satisfies P II w(z; n) = d dz ln Q n 1(z) Q n (z) w =2w 3 + zw + α, α = n Z +. Further, w(z; 0) = 0 and w(z; n) = w(z; n). The VorobevYablonskii polynomials are monic with degree n(n + 1)/2.

57 Figure: Painleve Polynomial - Degree 325

58 Figure: Painleve Polynomial - Degree 1, 275

59 Richard Stanley Examples from Combinatorics, 2001

60 Chromatic Polynomial of a Graph A complete bipartite graph G has its vertices broken into two disjoint subsets A and B so that every vertex in A is connected by an edge with every vertex in B. A coloring of a graph with r colors is an assignment that uses all the possible colors so that if vertices v and w are connected by an edge they must have different colors. The chromatic polynomial p G (x) of a graph G is determined by its values on the positive integers: p G (n) = # all colorings of G using n colors

61 Figure: Chromatic Polynomial for a Complete Bipartite Graph Degree 1,000

62 q-catalan Numbers Catalan numbers: C n = 1 n+1( 2n n ) with recurrence C n+1 = n C i C n 1 i=0 Counts properly parathenized expressions or nondecreasing binary paths

63 q-catalan Polynomials C n (q) C n+1 (q) = deg(c n ) = n C i (q)c n i (q)q (i+1)(n i), C 0 (q) = 1, i=0 ( ) n, C n (1) = C n 2 Geometric-Combinatorial Meaning C n (q) = P:path q area(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.

64 Figure: q-catalan Polynomial - Degree 190

65 Figure: q-catalan Polynomial - Degree 11,175

66 Partition Polynomials Work of Boyer and Goh, 2007

67 Polynomial Partition Polynomials Partition Numbers 4 = 4, 4 = 3+1, 4 = 2+2, 4 = 2+1+1, 4 = p 1 (4) = 1, p 2 (4) = 2, p 3 (4) = 1, p 4 (4) = 1 Hardy-Ramanujan Asymptotics Partition Polynomials p(n) 1 4n 3 eπ 2n/3. F n (x) = n p k (n)x k k=1

68 Figure: Partition Polynomial - degree 200

69 Figure: Digits of Partition Polynomial - degree 500

70 Figure: Digits of Partition Polynomial - degree 80,000

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

72 Figure: Partition Polynomial Attractor

73 Figure: Partition Polynomial Attractor in Upper Half Plane

74 Figure: Partition Polynomial Attractor in Second Quadrant

75 Figure: Triple Point for Partition Polynomial - degree 400

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

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

78 Figure: Region I for Partition Polynomial

79 Figure: Region II for Partition Polynomial

80 Figure: Region III for Partition Polynomial

are the q-versions of n, n! and . The falling factorial is (x) k = x(x 1)(x 2)... (x k + 1).

are the q-versions of n, n! and . The falling factorial is (x) k = x(x 1)(x 2)... (x k + 1). Lecture A jacques@ucsd.edu Notation: N, R, Z, F, C naturals, reals, integers, a field, complex numbers. p(n), S n,, b(n), s n, partition numbers, Stirling of the second ind, Bell numbers, Stirling of the