Chebyshev polynomials of the first kind are used in the design of Chebyshev Type I lowpass filters that are specified by maximum ripple within the passband.

The recurrence relationship for the n^{th}-order type I Chebyshev polynomial is

T_{0}(x) = 1
T_{1}(x) = x
T_{n}(x) = 2⋅x⋅T_{n-1}(x) - T_{n-2}(x)

The first few polynomials are:

ORDER POLYNOMIAL
----- -----------------------
0 T_{0}(x) = 1
1 T_{1}(x) = x
2 T_{2}(x) = 2⋅x^{2} - 1
3 T_{3}(x) = 4⋅x^{3} - 3⋅x
4 T_{4}(x) = 8⋅x^{4} - 8⋅x^{2} + 1
5 T_{5}(x) = 16⋅x^{5} - 20⋅x^{3} + 5⋅x
6 T_{6}(x) = 32⋅x^{6} - 48⋅x^{4} + 18⋅x^{2} - 1
7 T_{7}(x) = 64⋅x^{7} - 112⋅x^{5} + 52⋅x^{3} - 7⋅x
8 T_{8}(x) = 128⋅x^{8} - 256⋅x^{6} + 152⋅x^{4} - 32⋅x^{2} + 1

Chebyshev polynomials of the second kind are used in the design of Chebyshev Type II lowpass filters that are specified by maximum ripple above the cutoff frequency.

The recurrence relationship for the n^{th}-order type II Chebyshev polynomial is

U_{0}(x) = 1
U_{1}(x) = 2x
U_{n}(x) = 2⋅x⋅U_{n-1}(x) - U_{n-2}(x)

The first few polynomials are:

ORDER POLYNOMIAL
----- -----------------------
0 U_{0}(x) = 1
1 U_{1}(x) = 2x
2 U_{2}(x) = 4⋅x^{2} - 1
3 U_{3}(x) = 8⋅x^{3} - 4⋅x
4 U_{4}(x) = 16⋅x^{4} - 12⋅x^{2} + 1
5 U_{5}(x) = 32⋅x^{5} - 32⋅x^{3} + 6⋅x
6 U_{6}(x) = 64⋅x^{6} - 80⋅x^{4} + 24⋅x^{2} - 1
7 U_{7}(x) = 128⋅x^{7} - 192⋅x^{5} + 80⋅x^{3} - 8⋅x
8 U_{8}(x) = 256⋅x^{8} - 448⋅x^{6} + 240⋅x^{4} - 40⋅x^{2} + 1

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REFERENCES
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- George Arfken,
**Mathematical Methods for Physicists**, 2nd Ed., Academic Press, (c)1970, p.626