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Degrees and leading coefficients of the Chebyshev polynomials

Statement

For $n \geq 1$ , $T_{n}$ has degree $n$ and leading coefficient $2^{n - 1}$ , while $U_{n}$ has degree $n$ and leading coefficient $2^{n}$ . The conventions and prerequisite facts used below are recorded in Chebyshev polynomials of the first and second kinds by their three-term recurrences, The principle of mathematical induction, Integer powers $a^{m}$ .

Facts & Assumptions

Given: A natural $n$ .

[L1]

Chebyshev polynomials of the first and second kinds by their three-term recurrences defines $T_{0} = 1$ , $T_{1} = x$ , $T_{n + 1} = 2 x T_{n} - T_{n - 1}$ and $U_{0} = 1$ , $U_{1} = 2 x$ , $U_{n + 1} = 2 x U_{n} - U_{n - 1}$ .

Proof

technique · induction

1.1L1basealgebra

The initial values in [L1] give $T_{0} = U_{0} = 1$ , both of degree $0$ and leading coefficient $1$ , and $T_{1} = X$ , $U_{1} = 2 X$ , of degree $1$ and leading coefficients $1$ and $2$ . Applying the recurrences once gives $T_{2} = 2 X^{2} - 1$ and $U_{2} = 4 X^{2} - 1$ , of degree $2$ and leading coefficients $2$ and $4$ . Thus the asserted formulas hold at the consecutive indices $1, 2$ ; the index- $0$ values used in obtaining index $2$ have been established separately, without applying the formula $2^{n - 1}$ at $n = 0$ .

1.2ih

Fix $n \geq 1$ and assume the degree and leading-coefficient assertions at the consecutive indices $n, n + 1$ .

2.1L1step 1.2algebra

Use $T_{n + 2} = 2 X T_{n + 1} - T_{n}$ and $U_{n + 2} = 2 X U_{n + 1} - U_{n}$ from [L1]. The first term in each recurrence has degree $n + 2$ , whereas the subtracted term has degree $n$ . No leading-term cancellation is possible. The next leading coefficients are $2 \cdot 2^{n} = 2^{n + 1}$ for $T_{n + 2}$ and $2 \cdot 2^{n + 1} = 2^{n + 2}$ for $U_{n + 2}$ . This proves the two formulas at index $n + 2$ .

3.1discharge-induction∎

This proves the stated degree and leading-coefficient formulas at every index.

Depends on

Used by

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

NIST Digital Library of Mathematical Functions, Chapter 18 (standard reference, not scraped)