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ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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T0T_0 through T4T_4 and U0U_0 through U3U_3 from the Chebyshev recurrences

Example

The recurrences give T0=1T_0=1, T1=xT_1=x, T2=2x2−1T_2=2x^2-1, T3=4x3−3xT_3=4x^3-3x, T4=8x4−8x2+1T_4=8x^4-8x^2+1, and U0=1U_0=1, U1=2xU_1=2x, U2=4x2−1U_2=4x^2-1, U3=8x3−4xU_3=8x^3-4x. 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 TnT_n recurrence successively and collect like powers through T4T_4.

algebra
2.1

Apply the UnU_n recurrence successively and collect like powers through U3U_3.

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 24 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources