Shifted Legendre polynomials

Definition

Let $P_n$ denote the Legendre polynomial of degree $n \ge 0$ . The shifted Legendre polynomial of degree $n$ is the polynomial $\tilde{P}_n(x) = P_n(2x - 1).$ Equivalently, $\tilde{P}_n(x) = (-1)^n \sum_{k=0}^{n} \binom{n}{k} \binom{n+k}{k} (-x)^k = \frac{1}{n!} \frac{d^n}{dx^n} \left( x^2 - x \right)^n.$

Remarks

The first ones are $\tilde{P}_0 = 1$ , $\tilde{P}_1 = 2x - 1$ and $\tilde{P}_2 = 6x^2 - 6x + 1$ . Check for $\tilde{P}_2$ : with $P_2(t) = (3t^2 - 1)/2$ and $t = 2x - 1$ , one gets $(12x^2 - 12x + 2)/2 = 6x^2 - 6x + 1$ . They are orthogonal on $[0, 1]$ : $\int_0^1 \tilde{P}_m(x) \tilde{P}_n(x) \, dx = \frac{\delta_{mn}}{2n + 1}$ . For instance $\int_0^1 (2x - 1) \, dx = 0$ . Non-example: the unshifted polynomials are not orthogonal on $[0, 1]$ , since $\int_0^1 P_0(x) P_1(x) \, dx = \int_0^1 x \, dx = \frac12 \ne 0$ . Some authors, and some software libraries, use the reversed convention $P_n(1 - 2x) = (-1)^n \tilde{P}_n(x)$ .

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Legendre polynomials concept
Shifted Legendre polynomials definition

Tag	`2B9G13MQK3`
URI	`m2://2B9G13MQK3`
Kind	definition
Writing	W0 stub
MSC	33C45, 42C05
Defines	Shifted Legendre polynomials
Uses	Legendre polynomials

Shifted Legendre polynomials

Definition

Remarks

Reading path

Further reading

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