Sunday, September 25, 2022

Proving Chebyshev Polynomial Closed Formulas for n = 0, n = 1, and n = 2

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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