Generating Function for Chebyshev Polynomials of the Second Kind

Theorem

Let $\map {U_n} x$ denote the $n$th Chebyshev polynomial of the second kind.

Then the generating function for $U_n$ is:

$\ds \dfrac 1 {1 - 2 t x + t^2} = \sum_{n \mathop = 0}^\infty \map {U_n} x t^n$

Proof

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Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 30$: Chebyshev Polynomials: Generating Function for $\map {U_n} x$: $30.31$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 31$: Chebyshev Polynomials: Generating Function for $\map {U_n} x$: $31.31.$