Chebyshev Polynomials - Definition

Definition

The Chebyshev polynomials of the first kind are defined by the recurrence relation

\begin{align} T_0(x) & = 1 \\ T_1(x) & = x \\ T_{n+1}(x) & = 2xT_n(x) - T_{n-1}(x). \end{align}

The conventional generating function for Tn is

The exponential generating function is

The generating function relevant for 2-dimensional potential theory and multipole expansion is

The Chebyshev polynomials of the second kind are defined by the recurrence relation

\begin{align} U_0(x) & = 1 \\ U_1(x) & = 2x \\ U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x). \end{align}

One example of a generating function for Un is

Read more about this topic:  Chebyshev Polynomials

Famous quotes containing the word definition:

    “Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
    —Nadine Gordimer (b. 1923)

    “Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
    —The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

    “One definition of man is “an intelligence served by organs.””
    —Ralph Waldo Emerson (1803–1882)