Editing Legendre polynomials (section)

== Additional properties ==
Legendre polynomials have definite parity. That is, they are [[Even and odd functions|even or odd]],<ref>{{harvnb|Arfken|Weber|2005|loc=p.753}}</ref> according to
<math display="block">P_n(-x) = (-1)^n P_n(x) \,.</math>

Another useful property is
<math display="block">\int_{-1}^1 P_n(x)\,dx = 0 \text{ for } n\ge1,</math>
which follows from considering the orthogonality relation with <math>P_0(x) = 1</math>. It is convenient when a Legendre series <math display="inline">\sum_i a_i P_i</math> is used to approximate a function or experimental data: the ''average'' of the series over the interval {{closed-closed|−1, 1}} is simply given by the leading expansion coefficient <math>a_0</math>.

The underivative is<ref>
{{cite journal|first1=Orion|last1=Ciftja|title=Integrals of Legendre Polynomials over half range and their relation to the electrostatic potential in hemispherical geometry|year=2022|journal=Results in Physics|volume=40|page=105838|doi=10.1016/j.rinp.2022.105838|bibcode=2022ResPh..4005838C |doi-access=free}}
</ref>

<math>
\int P_n(x)dx=\frac{1}{2n+1}[P_{n+1}(x)-P_{n-1}(x)],\quad n\ge 1.
</math>

Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials' definitions are "standardized" (sometimes called "normalization", but the actual norm is not 1) by being scaled so that
<math display="block">P_n(1) = 1 \,.</math>

The derivative at the end point is given by
<math display="block">P_n'(1) = \frac{n(n+1)}{2} \,. </math>

The product expansion is <ref>{{cite journal|first1=L.|last1=Carlitz|title=Some integrals containing products of legendre polynomials|year=1961|journal=Archiv Mathem.|volume=12|pages=334–340|doi=10.1007/BF01650571}}</ref>

<math>
P_m(x)P_n(x)=\sum_{r=0}^{\min(m,n)}\frac{A_rA_{m-r}A_{n-r}}{A_{m+n-r}}\frac{2m+2n-4r+1}{2m+2n-2r+1}P_{m+n-2r}(x)
</math>

where <math>A_r\equiv (2r-1)!!/r!</math>.

The [[Askey–Gasper inequality]] for Legendre polynomials reads
<math display="block">\sum_{j=0}^n P_j(x) \ge 0 \quad \text{for }\quad x\ge -1 \,.</math>

The Legendre polynomials of a [[scalar product]] of [[unit vectors]] can be expanded with [[spherical harmonics]] using
<math display="block">P_\ell \left(r \cdot r'\right) = \frac{4\pi}{2\ell + 1} \sum_{m=-\ell}^\ell Y_{\ell m}(\theta,\varphi) Y_{\ell m}^*(\theta',\varphi')\,,</math>
where the unit vectors {{math|''r''}} and {{math|''r''′}} have [[spherical coordinates]] {{math|(''θ'', ''φ'')}} and {{math|(''θ''′, ''φ''′)}}, respectively.

The product of two Legendre polynomials <ref>{{cite journal|author = Leonard C. Maximon|title = A generating function for the product of two Legendre polynomials|journal = Norske Videnskabers Selskab Forhandlinger | volume = 29 | year = 1957 | pages = 82–86 | url=https://www.researchgate.net/publication/269015726}}</ref> 
<math display="block">
 \sum_{p=0}^\infty t^{p}P_p(\cos\theta_1)P_p(\cos\theta_2)=\frac2\pi\frac{\mathbf K\left( 2\sqrt{\frac{t\sin\theta_1\sin\theta_2}{t^2-2t\cos\left( \theta_1+\theta_2 \right)+1}} \right)}{\sqrt{t^2-2t\cos\left( \theta_1+\theta_2 \right)+1}}\,,</math>
where <math>K(\cdot)</math> is the [[complete elliptic integral of the first kind]].

The formulas of Dirichlet-Mehler:<ref>{{Cite journal |date=1 July 1837 |title=Sur les séries dont le terme général dépend de deux angles, et qui servent à exprimer des fonctions arbitraires entre des limites donnée. |url=https://www.degruyter.com/document/doi/10.1515/crll.1837.17.35/html |journal=Journal für die reine und angewandte Mathematik (Crelles Journal) |volume=1837 |issue=17 |pages=35–56 |doi=10.1515/crll.1837.17.35 |issn=0075-4102}}</ref><ref>{{Cite journal |last=Mehler |first=F. G. |date=June 1881 |title=Ueber eine mit den Kugel- und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Elektricitätsvertheilung |url=http://link.springer.com/10.1007/BF01445847 |journal=Mathematische Annalen |language=de |volume=18 |issue=2 |pages=161–194 |doi=10.1007/BF01445847 |issn=0025-5831}}</ref><ref name=":0" />{{Pg|page=86|location=Eq. 4.8.6, Eq. 4.8.7}}<ref>{{Cite web |title=DLMF: §18.10 Integral Representations ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials |url=https://dlmf.nist.gov/18.10 |access-date=18 March 2025 |website=dlmf.nist.gov}}</ref><math display="block">P_n(\cos \theta)
= \frac{2}{\pi} \int_0^\theta \frac{\cos \left(n+\frac{1}{2}\right) \phi}{(2 \cos \phi - 2 \cos \theta)^{\frac{1}{2}}} d \phi 
= \frac{2}{\pi} \int_\theta^\pi \frac{\sin \left(n+\frac{1}{2}\right) \phi}{(2 \cos \theta-2 \cos \phi)^{\frac{1}{2}}} d \phi</math>which has generalizations for associated Legendre polynomials.<ref>{{Cite journal |date=1896-12-31 |title=II. On a type of spherical harmonics of unrestricted degree, order, and argument |url=https://royalsocietypublishing.org/doi/10.1098/rspl.1895.0075 |journal=Proceedings of the Royal Society of London |language=en |volume=59 |issue=353–358 |pages=189–196 |doi=10.1098/rspl.1895.0075 |issn=0370-1662}}</ref><ref>{{Citation |last=Gasper |first=George |title=Formulas of the dirichlet-mehler type |date=1975 |work=Fractional Calculus and Its Applications |series=Lecture Notes in Mathematics |volume=457 |pages=207–215 |editor-last=Ross |editor-first=Bertram |url=http://link.springer.com/10.1007/BFb0067105 |access-date=2025-03-18 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |doi=10.1007/bfb0067105 |isbn=978-3-540-07161-7}}</ref>

The Fourier-Legendre series:<ref>Lord Rayleigh, Theory of sound, Volume II, p. 273</ref><math display="block">e^{i t x}=\sum_{n=0}^{\infty}(2 n+1) i^n \sqrt{\frac{\pi}{2 t}} J_{n+\frac{1}{2}}(t) P_n(x)</math>where <math>J</math> is the [[Bessel function of the first kind]].

===Recurrence relations===
As discussed above, the Legendre polynomials obey the three-term recurrence relation known as Bonnet's recursion formula given by
<math display="block"> (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x)</math>
and
<math display="block"> \frac{x^2-1}{n} \frac{d}{dx} P_n(x) = xP_n(x) - P_{n-1}(x) </math>
or, with the alternative expression, which also holds at the endpoints
<math display="block"> \frac{d}{dx} P_{n+1}(x) = (n+1)P_n(x) + x \frac{d}{dx}P_{n}(x) \,.</math>

Useful for the integration of Legendre polynomials is
<math display="block">(2n+1) P_n(x) = \frac{d}{dx} \bigl( P_{n+1}(x) - P_{n-1}(x) \bigr) \,.</math>

From the above one can see also that
<math display="block">\frac{d}{dx} P_{n+1}(x) = (2n+1) P_n(x) + \bigl(2(n-2)+1\bigr) P_{n-2}(x) + \bigl(2(n-4)+1\bigr) P_{n-4}(x) + \cdots</math>
or equivalently
<math display="block">\frac{d}{dx} P_{n+1}(x) = \frac{2 P_n(x)}{\left\| P_n \right\|^2} + \frac{2 P_{n-2}(x)}{\left\| P_{n-2} \right\|^2} + \cdots</math>
where {{math|{{norm|''P<sub>n</sub>''}}}} is the norm over the interval {{math|−1 ≤ ''x'' ≤ 1}}
<math display="block">\| P_n \| = \sqrt{\int_{-1}^1 \bigl(P_n(x)\bigr)^2 \,dx} = \sqrt{\frac{2}{2 n + 1}} \,.</math>More generally, all orders of derivatives are expressible as a sum of Legendre polynomials:<ref>{{Cite journal |last=Doha |first=E. H. |date=1991-01-01 |title=The coefficients of differentiated expansions and derivatives of ultraspherical polynomials |url=https://dx.doi.org/10.1016/0898-1221%2891%2990089-M |journal=Computers & Mathematics with Applications |volume=21 |issue=2 |pages=115–122 |doi=10.1016/0898-1221(91)90089-M |issn=0898-1221}}</ref><math display="block">\begin{aligned}
&\begin{aligned}
& \frac{d^q}{dx^q} P_{q+2 j}(x)=\frac{2^{q-1}}{(q-1)!} \sum_{i=0}^j(4 i+1) \frac{(q+j-i-1)!\Gamma\left(q+j+i+\frac{1}{2}\right)}{(j-i)!\Gamma(j+i+3 / 2)} P_{2 i}(x) \\
& \quad=\frac{1}{2^{q-2}(q-1)!} \sum_{i=0}^j(4 i+1) \frac{(q+j-i-1)!(2 q+2 j+2 i-1)!}{(j-i)!(2 j+2 i+2)!} \frac{(j+i+1)!}{(q+j+i-1)!} P_{2 i}(x)
\end{aligned}\\
&\begin{aligned}
& \frac{d^q}{dx^q} P_{q+2 j+1}(x)=\frac{2^{q-1}}{(q-1)!} \sum_{i=0}^j(4 i+3) \frac{(q+j-i-1)!\Gamma(q+j+i+3 / 2)}{(j-i)!\Gamma(j+i+5 / 2)} P_{2 i+1}(x) \\
& \quad=\frac{1}{2^{q-2}(q-1)!} \sum_{i=0}^j(4 i+3) \frac{(q+j-i-1)!(2 q+2 j+2 i+1)!}{(j-i)!(2 j+2 i+4)!} \frac{(j+i+2)!}{(q+j+i)!} P_{2 i+1}(x)
\end{aligned}
\end{aligned}</math>

===Asymptotics===

Asymptotically, for <math>\ell \to \infty</math>, the Legendre polynomials can be written as <ref name=":0">{{Cite book |last=Szegő |first=Gábor |title=Orthogonal polynomials |date=1975 |publisher=American Mathematical Society |isbn=0821810235 |edition=4th |location=Providence |oclc=1683237}}</ref>{{Pg|location=Theorem 8.21.2|page=194}}
<math display="block">\begin{align}
P_\ell (\cos \theta) &= \sqrt{\frac{\theta}{\sin\left(\theta\right)}}
\left\{J_0{\left[\left(\ell+\tfrac{1}{2}\right)\theta\right]} 
- \frac{\left(\frac{1}{\theta}-\cot\theta\right)}{8(\ell+\frac{1}{2})}
J_1{\left[\left(\ell+\tfrac{1}{2}\right)\theta\right]} 
\right\} + 
\mathcal{O}\left(\ell^{-2}\right) \\[1ex]
&= \sqrt{\frac{2}{\pi \ell\sin\left(\theta\right)}}\cos\left[\left(\ell + \tfrac{1}{2} \right)\theta - \tfrac{\pi}{4}\right] + \mathcal{O}\left(\ell^{-3/2}\right), \quad \theta \in (0,\pi),
\end{align}</math>
and for arguments of magnitude greater than 1<ref>{{Cite web|url=https://dlmf.nist.gov/14.15.E13|title = DLMF: 14.15 Uniform Asymptotic Approximations}}</ref>
<math display="block">\begin{align}
P_\ell \left(\cosh\xi\right) &= \sqrt{\frac{\xi}{\sinh\xi}} I_0\left(\left(\ell+\frac{1}{2}\right)\xi\right)\left(1+\mathcal{O}\left(\ell^{-1}\right)\right)\,,\\
P_\ell \left(\frac{1}{\sqrt{1-e^2}}\right) &= \frac{1}{\sqrt{2\pi\ell e}} \frac{(1+e)^\frac{\ell+1}{2}}{(1-e)^\frac{\ell}{2}} + \mathcal{O}\left(\ell^{-1}\right)
\end{align}</math>
where {{math|''J''<sub>0</sub>}}, {{math|''J''<sub>1</sub>}}, and {{math|''I''<sub>0</sub>}} are [[Bessel functions]].

=== Zeros ===
All <math> n</math> zeros of <math>P_n(x)</math> are real, distinct from each other, and lie in the interval <math>(-1,1)</math>. Furthermore, if we regard them as dividing the interval <math>[-1,1]</math> into <math> n+1 </math> subintervals, each subinterval will contain exactly one zero of <math>P_{n+1}</math>. This is known as the interlacing property. Because of the parity property it is evident that if <math>x_k</math> is a zero of <math>P_n(x)</math>, so is <math>-x_k</math>. These zeros play an important role in [[numerical integration]] based on [[Gaussian quadrature]]. The specific quadrature based on the <math>P_n</math>'s is known as [[Gauss-Legendre quadrature]].

The zeros of <math>P_n(\cos \theta)</math> are distributed nearly uniformly over the range of <math>\theta \in (0, \pi)</math>, in the sense that there is one zero <math>\theta \in \left(\frac{\pi(k + 1/2)}{n + 1/2}, \frac{\pi(k + 1)}{n + 1/2}\right)</math> per <math>k = 0, 1, \dots, n-1</math>.<ref>{{Cite journal |last=Askey |first=Richard |date=November 1969 |title=Mehler's Integral for P_n (cos θ) |url=https://www.tandfonline.com/doi/abs/10.1080/00029890.1969.12000407 |journal=The American Mathematical Monthly |language=en |volume=76 |issue=9 |pages=1046–1049 |doi=10.1080/00029890.1969.12000407 |issn=0002-9890}}</ref> This can be proved by looking at the first formula of Dirichlet-Mehler.<ref>{{Cite journal |last=Bruns |first=H. |date=1881 |title=Zur Theorie der Kugelfunctionen. |url=https://www.degruyter.com/document/doi/10.1515/crll.1881.90.322/html |journal=CRLL |language=en |volume=1881 |issue=90 |pages=322–328 |doi=10.1515/crll.1881.90.322 |issn=1435-5345}}</ref>

From this property and the facts that <math> P_n(\pm 1) \ne 0 </math>, it follows that <math> P_n(x) </math> has <math> n-1 </math> local minima and maxima in <math> (-1,1) </math>. Equivalently, <math> dP_n(x)/dx </math> has <math> n -1 </math> zeros in <math> (-1,1) </math>.

===Pointwise evaluations===
The parity and normalization implicate the values at the boundaries <math> x=\pm 1 </math> to be
<math display="block">
  P_n(1) = 1
  \,, \quad
  P_n(-1) =  (-1)^n
</math>
At the origin <math> x=0 </math> one can show that the values are given by
<math display="block">
  P_{2n}(0) = \frac{(-1)^{n}}{4^n} \binom{2n}{n} = \frac{(-1)^{n}}{2^{2n}} \frac{(2n)!}{\left(n!\right)^2}
= (-1)^n\frac{(2n-1)!!}{(2n)!!}
</math><math display="block">
  P_{2n+1}(0) = 0
</math>