ChebyshevT

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed n

For fixed z

Values at infinities

General characteristics

Domain and analyticity

The function (Subscript[T, n])(z)is defined over ℕ⊗ℂ. For fixed n, the function (Subscript[T, n])(z) is a polynomial in z of degree n.

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

The function (Subscript[T, n])(z) is polynomial and has pole of order n at z =(OverTilde[∞]).

Branch points

With respect to z

The function (Subscript[T, n])(z) does not have branch points.

Branch cuts

With respect to z

The function (Subscript[T, n])(z) does not have branch cuts.

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Expansions at z==0

Expansions at z==1

Expansions at z==-1

Expansions at z==∞

Expansions at n==∞

Other series representations

Integral representations

Integral representations of negative integer order

Rodrigues-type formula.

Limit representations

Generating functions

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Multiple arguments

Brychkov Yu.A. (2007)

Products, sums, and powers of the direct function

Products of the direct function

Addition formulas

Related transformations

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations between contiguous functions

Complex characteristics

Real part

Imaginary part

Argument

Conjugate value

Differentiation

Low-order differentiation

With respect to z

Symbolic differentiation

With respect to z

Fractional integro-differentiation

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Definite integration

Involving the direct function

Summation

Infinite summation

Operations

Orthogonality, completeness, and Fourier expansions

The set of functions (Subscript[T, n])(x), n=0,1,…, forms a complete, orthogonal (with weight (2/π)(1/(Sqrt[1-(x^2)]))) system on the interval (-1,1).

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric2F1

Through hypergeometric functions of two variables

Through Meijer G

Classical cases for the direct function itself

Classical cases involving algebraic functions

Classical cases involving unit step UnitStep

Generalized cases involving algebraic functions

Generalized cases involving unit step UnitStep

Through other functions

Involving some hypergeometric-type functions

Involving spheroidal functions

Representations through equivalent functions

With related functions

With elementary functions

Theorems

Expansions in generalized Fourier series

f(x)==(∑_(k=0)^∞)(Subscript[c, k])(Subscript[ψ, k])(x)/;   (Subscript[c, k ])==(∫_(-1)^1)f(t)(Subscript[ψ, k])(t)dt, (Subscript[ψ, k])(x)==(Sqrt[(2/π)])(((1-(x^2)))^(-(1/4))) (Subscript[T, k])(x), k∈ℕ.

Minimizing property of Chebyshev polynomials

Chebyshev polynomials (Subscript[T, n])(x) have the smallest absolute values among all polynomials of degree n with leading coefficient 1:

(max_(-1≤x≤1))|(∑_(k=0)^n)(Subscript[a, i])(x^i)|≥(max_(-1≤x≤1))|(((Subscript[T, n])(x))/([(x^n)]((Subscript[T, n])(x))))|.

Another minimizing property of Chebyshev polynomials

(∫_(-1)^1)(((P(x))^2)/(Sqrt[1-(x^2)]))dx≥(∫_(-1)^1)((((Subscript[T, n])(x))^2)/(Sqrt[1-(x^2)]))dx==(π/2)/;         P(x)==(2^(n-1))(x^n)+(∑_(k=0)^(n-1))(Subscript[c, i])(x^i)∧(Subscript[c, i])∈ℝ

Distribution of the zeros of Chebyshev polynomials of high order

In the limit n->∞ , the zeros of (Subscript[T, n])(x) fulfill the arcsin distribution; in other words, the relative number of zeros (Subscript[m, n])(a, b)/n in the interval (a,b) is (arcsin(b)-arcsin(a))/π.

History

– P. L. Chebyshev (1854,1855,1859)

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