Chebyshev Polynomial of the Second Kind/Examples/U7

Example of Chebyshev Polynomial of the Second Kind

The $7$th Chebyshev polynomial of the second kind is:

$\map {U_7} x = 128 x^7 - 192 x^5 + 80 x^3 - 8 x$

Proof

\(\ds \map {U_n} x\)

\(=\)

\(\ds 2 x \, \map {U_{n - 1} } x - \map {U_{n - 2} } x\)

Recurrence Formula for Chebyshev Polynomials of the Second Kind

\(\ds \leadsto \ \ \)

\(\ds \map {U_7} x\)

\(=\)

\(\ds 2 x \, \map {U_6} x - \map {U_5} x\)

setting $n = 7$

\(\ds \)

\(=\)

\(\ds 2 x \paren {64 x^6 - 80 x^4 + 24 x^2 - 1} - \paren {32 x^5 - 32 x^3 + 6 x}\)

$6$th Chebyshev Polynomial of the Second Kind and $5$th Chebyshev Polynomial of the Second Kind

\(\ds \)

\(=\)

\(\ds 128 x^7 - 192 x^5 + 80 x^3 - 8 x\)

simplifying

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 30$: Chebyshev Polynomials: Special Chebyshev Polynomials of the Second Kind: $30.30$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 31$: Chebyshev Polynomials: Special Chebyshev Polynomials of the Second Kind: $31.30.$