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 nth-order type I Chebyshev polynomial is

```     T0(x) = 1
T1(x) = x
Tn(x) = 2⋅x⋅Tn-1(x) - Tn-2(x)
```

The first few polynomials are:

```
ORDER     POLYNOMIAL
-----     -----------------------
0       T0(x) = 1
1       T1(x) = x
2       T2(x) = 2⋅x2 - 1
3       T3(x) = 4⋅x3 - 3⋅x
4       T4(x) = 8⋅x4 - 8⋅x2 + 1
5       T5(x) = 16⋅x5 - 20⋅x3 + 5⋅x
6       T6(x) = 32⋅x6 - 48⋅x4 + 18⋅x2 - 1
7       T7(x) = 64⋅x7 - 112⋅x5 + 52⋅x3 - 7⋅x
8       T8(x) = 128⋅x8 - 256⋅x6 + 152⋅x4 - 32⋅x2 + 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 nth-order type II Chebyshev polynomial is

```     U0(x) = 1
U1(x) = 2x
Un(x) = 2⋅x⋅Un-1(x) - Un-2(x)
```

The first few polynomials are:

```
ORDER     POLYNOMIAL
-----     -----------------------
0       U0(x) = 1
1       U1(x) = 2x
2       U2(x) = 4⋅x2 - 1
3       U3(x) = 8⋅x3 - 4⋅x
4       U4(x) = 16⋅x4 - 12⋅x2 + 1
5       U5(x) = 32⋅x5 - 32⋅x3 + 6⋅x
6       U6(x) = 64⋅x6 - 80⋅x4 + 24⋅x2 - 1
7       U7(x) = 128⋅x7 - 192⋅x5 + 80⋅x3 - 8⋅x
8       U8(x) = 256⋅x8 - 448⋅x6 + 240⋅x4 - 40⋅x2 + 1
```

REFERENCES

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