The Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted . They are used as an approximation to a least squares fit, and are a special case of the Gegenbauer polynomial with . They are also intimately connected with trigonometric multipleangle formulas. The Chebyshev polynomials of the first kind are denoted , and are implemented in the Wolfram Language as ChebyshevT[n, x]. They are normalized such that . The first few polynomials are illustrated above for and , 2, ..., 5.
The Chebyshev polynomial of the first kind can be defined by the contour integral
(1)

where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).
The first few Chebyshev polynomials of the first kind are
(2)
 
(3)
 
(4)
 
(5)
 
(6)
 
(7)
 
(8)

When ordered from smallest to largest powers, the triangle of nonzero coefficients is 1; 1; , 2; , 4; 1, , 8; 5, , 16, ... (OEIS A008310).
A beautiful plot can be obtained by plotting radially, increasing the radius for each value of , and filling in the areas between the curves (Trott 1999, pp. 10 and 84).
The Chebyshev polynomials of the first kind are defined through the identity
(9)

or
(10)

The Chebyshev polynomials of the first kind can be obtained from the generating functions
(11)
 
(12)

and
(13)
 
(14)

for and (Beeler et al. 1972, Item 15). (A closely related generating function is the basis for the definition of Chebyshev polynomial of the second kind.)
A direct representation in terms of powers of square roots is given by
(15)

The polynomials can also be defined in terms of the sums
(16)
 
(17)

where is a binomial coefficient and is the floor function, or the product
(18)

(Zwillinger 1995, p. 696).
also satisfy the curious determinant equation
(19)

(Nash 1986).
The Chebyshev polynomials of the first kind are a special case of the Jacobi polynomials with ,
(20)
 
(21)

where is a hypergeometric function (Koekoek and Swarttouw 1998).
Zeros occur when
(22)

for , 2, ..., . Extrema occur for
(23)

where . At maximum, , and at minimum, .
The Chebyshev polynomials are orthogonal polynomials with respect to the weighting function
(24)

where is the Kronecker delta. Chebyshev polynomials of the first kind satisfy the additional discrete identity
(25)

where for , ..., are the zeros of .
They also satisfy the recurrence relations
(26)
 
(27)

for , as well as
(28)
 
(29)

(Watkins and Zeitlin 1993; Rivlin 1990, p. 5).
They have a complex integral representation
(30)

and a Rodrigues representation
(31)

Using a fast Fibonacci transform with multiplication law
(32)

gives
(33)

Using GramSchmidt orthonormalization in the range (,1) with weighting function gives
(34)
 
(35)
 
(36)
 
(37)
 
(38)
 
(39)
 
(40)

etc. Normalizing such that gives the Chebyshev polynomials of the first kind.
The Chebyshev polynomial of the first kind is related to the Bessel function of the first kind and modified Bessel function of the first kind by the relations
(41)

(42)

Letting allows the Chebyshev polynomials of the first kind to be written as
(43)
 
(44)

The second linearly dependent solution to the transformed differential equation
(45)

is then given by
(46)
 
(47)

which can also be written
(48)

where is a Chebyshev polynomial of the second kind. Note that is therefore not a polynomial.
The triangle of resultants is given by , , , , , ... (OEIS A054375).
The polynomials
(49)

of degree , the first few of which are
(50)
 
(51)
 
(52)
 
(53)
 
(54)

are the polynomials of degree which stay closest to in the interval . The maximum deviation is at the points where
(55)

for , 1, ..., (Beeler et al. 1972).