---
tag: 2B9G13MQK3
kind: definition
name: Shifted Legendre polynomials
msc: [33C45, 42C05]
defines: 9V3QA0KTW8
uses: [QFYGJVA996]
sources:
  - url: https://en.wikipedia.org/wiki/Legendre_polynomials
---

## Definition

Let $P_n$ denote the [[Legendre polynomial|QFYGJVA996]] 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 $P_n(1 - 2x) = (-1)^n \tilde{P}_n(x)$.