#Generating Function of the m-point Strodt polynomials
x,t=var('x,t')
def mptStGF(m,x,t):
Q=0
for j in range(m):
Q=Q+exp(j*t/(m-1))
return m*exp(x*t)/Q
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#Pulls the nth m-point Strodt Polynomial from its generating function
def P(m,n,x):
f=taylor(mptStGF(m,x,t),t,0,n)
out=f.coefficient(t^n)*factorial(n)
if n==0:
out=1
return out
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#List of first few 2-pt Strodt polynomials
for n in range(11):
P(2,n,x)
1 x - 1/2 x^2 - x x^3 - 3/2*x^2 + 1/4 x^4 - 2*x^3 + x x^5 - 5/2*x^4 + 5/2*x^2 - 1/2 x^6 - 3*x^5 + 5*x^3 - 3*x x^7 - 7/2*x^6 + 35/4*x^4 - 21/2*x^2 + 17/8 x^8 - 4*x^7 + 14*x^5 - 28*x^3 + 17*x x^9 - 9/2*x^8 + 21*x^6 - 63*x^4 + 153/2*x^2 - 31/2 x^10 - 5*x^9 + 30*x^7 - 126*x^5 + 255*x^3 - 155*x 1 x - 1/2 x^2 - x x^3 - 3/2*x^2 + 1/4 x^4 - 2*x^3 + x x^5 - 5/2*x^4 + 5/2*x^2 - 1/2 x^6 - 3*x^5 + 5*x^3 - 3*x x^7 - 7/2*x^6 + 35/4*x^4 - 21/2*x^2 + 17/8 x^8 - 4*x^7 + 14*x^5 - 28*x^3 + 17*x x^9 - 9/2*x^8 + 21*x^6 - 63*x^4 + 153/2*x^2 - 31/2 x^10 - 5*x^9 + 30*x^7 - 126*x^5 + 255*x^3 - 155*x |
#The next plots show asymptotic (in n) to trig functions
plot(P(2,20,x),xmin=-5,xmax=5,ymin=-10^9,ymax=10^9)
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plot(P(2,40,x),xmin=-5,xmax=5,ymin=-2*10^28,ymax=2*10^28)
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plot(P(3,20,x),xmin=-5,xmax=5,ymin=-10^7,ymax=10^7)
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#PlotScaledRoots returns a plot of the roots of the (scaled) m-pt Strodt poly
#Note: If the polynomials were not squarefree, this program would loop forever
#Note: There is no particular reason for the color and size of the points
from sage.rings.polynomial.complex_roots import complex_roots
def PlotScaledRootsP(m,n):
L=len(complex_roots(P(m,n,n*x),skip_squarefree=True))
l=[]
for j in range(L):
l.append((real(complex_roots(P(m,n,n*x),skip_squarefree=True)[j][0]).center(), imaginary(complex_roots(P(m,n,n*x),skip_squarefree=True)[j][0]).center()))
return point(l,rgbcolor=hue(1),size=30)
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#In the next few lines, we look at roots of the 2 and 3-point Strodt polynomials
#Note: I'm not really sure why this takes so long to evaluate
PlotScaledRootsP(2,20)
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PlotScaledRootsP(2,25)
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PlotScaledRootsP(3,25)
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