Editing Legendre polynomials (section)

== Variants with transformed argument ==

=== Shifted Legendre polynomials ===

The '''shifted Legendre polynomials''' are defined as
<math display="block">\widetilde{P}_n(x) = P_n(2x-1) \,.</math>
Here the "shifting" function {{math|''x'' ↦ 2''x'' − 1}} is an [[affine transformation]] that [[bijection|bijectively maps]] the interval {{closed-closed|0, 1}} to the interval {{closed-closed|−1, 1}}, implying that the polynomials {{math|''P̃<sub>n</sub>''(''x'')}} are orthogonal on {{closed-closed|0, 1}}:
<math display="block">\int_0^1 \widetilde{P}_m(x) \widetilde{P}_n(x)\,dx = \frac{1}{2n + 1} \delta_{mn} \,.</math>

An explicit expression for the shifted Legendre polynomials is given by
<math display="block">\widetilde{P}_n(x) = (-1)^n \sum_{k=0}^n \binom{n}{k} \binom{n+k}{k} (-x)^k \,.</math>

The analogue of [[Rodrigues' formula]] for the shifted Legendre polynomials is
<math display="block">\widetilde{P}_n(x) = \frac{1}{n!} \frac{d^n}{dx^n} \left(x^2 -x \right)^n \,.</math>

The first few shifted Legendre polynomials are:
{| class="wikitable" style="text-align: right;"
! <math>n</math> !! <math>\widetilde{P}_n(x)</math>
|-
| 0 || <math>1</math>
|-
| 1 || <math>2x-1</math>
|-
| 2 || <math>6x^2-6x+1</math>
|-
| 3 || <math>20x^3-30x^2+12x-1</math>
|-
| 4 || <math>70x^4-140x^3+90x^2-20x+1</math>
|-
| 5 || <math>252x^5 -630x^4 +560x^3 - 210 x^2 + 30 x - 1</math>
|-
|}

=== Legendre rational functions ===
{{main|Legendre rational functions}}

The [[Legendre rational functions]] are a sequence of [[orthogonal functions]] on [0, ∞). They are obtained by composing the [[Cayley transform]] with Legendre polynomials.

A rational Legendre function of degree ''n'' is defined as:
<math display="block">R_n(x) = \frac{\sqrt{2}}{x+1}\,P_n\left(\frac{x-1}{x+1}\right)\,.</math>

They are [[eigenfunction]]s of the singular [[Sturm–Liouville problem]]:
<math display="block">\left(x+1\right) \frac{d}{dx} \left(x \frac{d}{dx} \left[\left(x+1\right) v(x)\right]\right) + \lambda v(x) = 0</math>
with eigenvalues <math display="block">\lambda_n=n(n+1)\,.</math>