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$T_{0}$ through $T_{4}$ and $U_{0}$ through $U_{3}$ from the Chebyshev recurrences

Example

The recurrences give $T_{0} = 1$ , $T_{1} = x$ , $T_{2} = 2 x^{2} - 1$ , $T_{3} = 4 x^{3} - 3 x$ , $T_{4} = 8 x^{4} - 8 x^{2} + 1$ , and $U_{0} = 1$ , $U_{1} = 2 x$ , $U_{2} = 4 x^{2} - 1$ , $U_{3} = 8 x^{3} - 4 x$ . The conventions and prerequisite facts used below are recorded in Chebyshev polynomials of the first and second kinds by their three-term recurrences, Degrees and leading coefficients of the Chebyshev polynomials.

Facts & Assumptions

Given: The defining recurrences.

Verification

1.1

Apply the $T_{n}$ recurrence successively and collect like powers through $T_{4}$ .

algebra

2.1

Apply the $U_{n}$ recurrence successively and collect like powers through $U_{3}$ .

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)

T0T0​ through T4T4​ and U0U0​ through U3U3​ from the Chebyshev recurrences