Proving Chebyshev Polynomial Closed Formulas for n = 0, n = 1, and n = 2
Chebyshev Polynomials of the First Kind
Recurrence Definition:
T_0(x) = 1
T_1(x) = x
T_n+1(x) = 2 * x * T_n(x) - T_n-1(x)
Closed Definition:
T_n(x) = 1/2 * [ (x - √(x^2 - 1))^n + (x + √(x^2 - 1))^n ]
Let: w = √(x^2 - 1)
T_n(x) = 1/2 * [ (x - w)^n + (x + w)^n ]
n = 0
T_0(x)
= 1/2 * [ (x - w)^0 + (x + w)^0 ]
= 1/2 * [ 1 + 1 ]
= 1
n = 1
T_1(x)
= 1/2 * [ (x - w)^1 + (x + w)^1 ]
= 1/2 * [ x - w + x + w ]
= 1/2 * [ 2 * x]
= x
n = 2
T_2(x)
= 1/2 * [ (x - w)^2 + (x + w)^2 ]
= 1/2 * [ x^2 - 2*w + w^2 + x^2 + 2*w^2 + w^2 ]
= 1/2 * [ 2 * x^2 + 2 * w^2 ]
= x^2 + x^2 - 1
= 2 * x^2 - 1
Chebyshev Polynomials of the Second Kind
Recurrence Definition:
U_0(x) = 1
U_1(x) = 2 * x
U_n+1(x) = 2 * x * U_n(x) - U_n-1(x)
Closed Definition:
U_n(x) = [ (x + √(x^2 - 1))^(n + 1) - (x - √(x^2 - 1))^(n + 1) ] ÷ [ 2 * √(x^2 - 1) ]
Let: w = √(x^2 - 1)
U_n(x) = [ (x + w)^(n + 1) - (x - w)^(n + 1) ] ÷ [ 2 * w ]
n = 0
U_0(x)
= [ (x + w)^(1) - (x - w)^(1) ] ÷ [ 2 * w ]
= [ x + w - x + w ] ÷ (2 * w)
= (2 * w) ÷ (2 * w)
= 1
n = 1
U_1(x)
= [ (x + w)^(2) - (x - w)^(2) ] ÷ [ 2 * w ]
= [ (x^2 + 2 * x * w + w^2) - (x^2 - 2 * x * w + w^2) ] ÷ (2 * w)
= [ 4 * x * w ] ÷ (2 * w)
= 2 * x
n = 2
U_2(x)
= [ (x + w)^(3) - (x - w)^(3) ] ÷ [ 2 * w ]
= [ x^3 + 3*x^2*w + 3*x*w^2 + w^3 - (x^3 - 3*x^2*w + 3*x*w^2 - w^3)] ÷ [ 2*w ]
= [ x^3 + 3*x^2*w + 3*x*w^2 + w^3 - x^3 + 3*x^2*w - 3*x*w^2 + w^3] ÷ [ 2*w ]
= [ 6*x^2*w + 2*w^3 ] ÷ [ 2*w ]
= [ 6*x*√(x^2 - 1) + 2*(x^2 - 1)^(3/2) ] ÷ [ 2*√(x^2 - 1) ]
= [ 6*x*√(x^2 - 1) + 2*(x^2 - 1)*√(x^2- 1) ] ÷ [ 2*√(x^2 - 1) ]
= [ 6*x*√(x^2 - 1) + 2*(x^2 - 1)*√(x^2- 1) ] ÷ [ 2*√(x^2 - 1) ]
= [ 6*x*√(x^2 - 1) + (2*x^2 - 2)*√(x^2- 1) ] ÷ [ 2*√(x^2 - 1) ]
= [ (8*x - 2)*√(x^2 - 1) ] ÷ [ 2*√(x^2 - 1) ]
= 4*x^2 - 1
Good that the closed formulas hold up, at least for n = 0, 1, 2. The closed formulas would be good if you don't want to use recurrence relations.
Sources:
"Chebyshev polynomials" Wikipedia. https://en.wikipedia.org/wiki/Chebyshev_polynomials Last Updated July 20, 2022. Last Accessed June 21, 2022
Oldman, Keith, Jan Myland, & Jerome Spainer An Atlas of Functions: with Equator, the Atlas Function Calculator 2nd Edition Springer: New York, NY. 2009 ISBN 978-0-387-48806-6
Eddie
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