Associated Legendre polynomials

These functions are denoted $P\_\backslash ell^\{m\}(x)$, where the superscript indicates the order and not a power of P. Their most straightforward definition is in terms

of derivatives of ordinary Legendre polynomials (m ≥ 0)

$$P\_\backslash ell^\{m\}(x)\; =\; (-1)^m\; (1-x^2)^\{m/2\}\; \backslash frac\{d^m\}\{dx^m\}\; \backslash left(\; P\_\backslash ell(x)\; \backslash right),$$

The {{math|(−1)^m}} factor in this formula is known as the Condon–Shortley phase. Some authors omit it. That the functions described by this equation satisfy the general Legendre differential equation with the indicated values of the parameters ℓ and m follows by differentiating m times the Legendre equation for {{math|P_ℓ}}:{{harvnb|Courant|Hilbert|1953|loc=V, §10}}.

$$\backslash left(1-x^2\backslash right)\; \backslash frac\{d^2\}\{dx^2\}P\_\backslash ell(x)\; -2x\backslash frac\{d\}\{dx\}P\_\backslash ell(x)+\; \backslash ell(\backslash ell+1)P\_\backslash ell(x)\; =\; 0.$$

Moreover, since by Rodrigues' formula,

$$P\_\backslash ell(x)\; =\; \backslash frac\{1\}\{2^\backslash ell\backslash ,\backslash ell!\}\; \backslash \; \backslash frac\{d^\backslash ell\}\{dx^\backslash ell\}\backslash left[(x^2-1)^\backslash ell\backslash right],$$

the P{{su|b=ℓ|p=m}} can be expressed in the form

$$P\_\backslash ell^\{m\}(x)\; =\; \backslash frac\{(-1)^m\}\{2^\backslash ell\; \backslash ell!\}\; (1-x^2)^\{m/2\}\backslash \; \backslash frac\{d^\{\backslash ell+m\}\}\{dx^\{\backslash ell+m\}\}(x^2-1)^\backslash ell.$$

This equation allows extension of the range of m to: {{math|−ℓ ≤ m ≤ ℓ}}. The definitions of {{math|P_ℓ^±m}}, resulting from this expression by substitution of {{math|±m}}, are proportional. Indeed, equate the coefficients of equal powers on the left and right hand side of

$$\backslash frac\{d^\{\backslash ell-m\}\}\{dx^\{\backslash ell-m\}\}\; (x^2-1)^\{\backslash ell\}\; =\; c\_\{lm\}\; (1-x^2)^m\; \backslash frac\{d^\{\backslash ell+m\}\}\{dx^\{\backslash ell+m\}\}(x^2-1)^\{\backslash ell\},$$

then it follows that the proportionality constant is

$$c\_\{lm\}\; =\; (-1)^m\; \backslash frac\{(\backslash ell-m)!\}\{(\backslash ell+m)!\}\; ,$$

so that

$$P^\{-m\}\_\backslash ell(x)\; =\; (-1)^m\; \backslash frac\{(\backslash ell-m)!\}\{(\backslash ell+m)!\}\; P^\{m\}\_\backslash ell(x).$$

The following alternative notations are also used in literature:{{Abramowitz_Stegun_ref|8|332}}

$$P\_\{\backslash ell\; m\}(x)\; =\; (-1)^m\; P\_\backslash ell^\{m\}(x)$$

Starting from the explicit form provided in the article of Legendre Polynomials

P_l(x)=2^l\sum_{k=0}^l x^k\binom{l}{k}\binom{(l+k-1)/2}{l}

one obtains with the standard rules for $m$-fold derivatives for powers

$$P\_l^m(x)=(-1)^\{m\}\; \backslash cdot\; 2^\{l\}\; \backslash cdot\; (1-x^2)^\{m/2\}\; \backslash cdot\; \backslash sum\_\{k=m\}^l\; \backslash frac\{k!\}\{(k-m)!\}\backslash cdot\; x^\{k-m\}\; \backslash cdot\; \backslash binom\{l\}\{k\}\; \backslash binom\{\backslash frac\{l+k-1\}\{2\}\}\{l\}$$

with simple monomials and the generalized form of the binomial coefficient. The sum effectively extends only over terms where $l-k$ is even, because for odd $l-k$ the binomial factor $\backslash binom\{(l+k-1)/2\}\{l\}$ is zero.

Summarizing results of Doha

{{Cite journal |last=Doha |first=E. H. |year=1991|title=The coefficients of differentiated expansions and derivatives of ultraspherical polynomials |journal=Computers & Mathematics with Applications |volume=21 |issue=2 |pages=115–122 |doi=10.1016/0898-1221(91)90089-M |issn=0898-1221}}

the expansion of derivatives into Legendre Polynomials defines coefficients $\backslash tau$

\frac{d^m}{dx^m}P_l(x)

= \sum_{t=0}^{\lfloor (l-m)/2\rfloor} \tau_{l,m,t} P_{l-m-2t}(x)

,

where

\tau_{l,m,t}

=

\epsilon_{l-t}

\frac{l-m-2t+1/2}{2l-2t+1}\frac{(2m)!}{2^mm!}

\binom{2l-2t+1}{2m}

\frac{m}{m+t}\binom{m+t}{t}

\frac{1}{\binom{l-t}{m}}

,

and where

\epsilon_q\equiv \begin{cases}

1, & q=0;\\

2, & q\ge 1

\end{cases}

is the Neumann factor.

The associated Legendre polynomials are not mutually orthogonal in general. For example, $P\_1^1$ is not orthogonal to $P\_2^2$. However, some subsets are orthogonal. Assuming 0 ≤ m ≤ ℓ, they satisfy the orthogonality condition for fixed m:

$$\backslash int\_\{-1\}^\{1\}\; P\_k\; ^\{m\}\; P\_\backslash ell\; ^\{m\}\; dx\; =\; \backslash frac\{2\; (\backslash ell+m)!\}\{(2\backslash ell+1)(\backslash ell-m)!\}\backslash \; \backslash delta\; \_\{k,\backslash ell\}$$

Where {{math|δ_k,ℓ}} is the Kronecker delta.

Also, they satisfy the orthogonality condition for fixed {{mvar|ℓ}}:

$$\backslash int\_\{-1\}^\{1\}\; \backslash frac\{P\_\backslash ell\; ^\{m\}\; P\_\backslash ell\; ^\{n\}\}\{1-x^2\}dx\; =\; \backslash begin\{cases\}$$

0 & \text{if } m\neq n \\

\frac{(\ell+m)!}{m(\ell-m)!} & \text{if } m=n\neq0 \\

\infty & \text{if } m=n=0

\end{cases}

The differential equation is clearly invariant under a change in sign of m.

The functions for negative m were shown above to be proportional to those of positive m:

$$P\_\backslash ell\; ^\{-m\}\; =\; (-1)^m\; \backslash frac\{(\backslash ell-m)!\}\{(\backslash ell+m)!\}\; P\_\backslash ell\; ^\{m\}$$

(This followed from the Rodrigues' formula definition. This definition also makes the various recurrence formulas work for positive or negative {{mvar|m}}.)

$$\backslash text\{If\}\backslash quad\; |m|\; >\; \backslash ell\backslash ,\backslash quad\backslash text\{then\}\backslash quad\; P\_\backslash ell^\{m\}\; =\; 0.\backslash ,$$

The differential equation is also invariant under a change from {{mvar|ℓ}} to {{math|−ℓ − 1}}, and the functions for negative {{mvar|ℓ}} are defined by

$$P\_\{-\backslash ell\}\; ^\{m\}\; =\; P\_\{\backslash ell-1\}\; ^\{m\},\backslash \; (\backslash ell=1,\backslash ,2,\backslash ,\; \backslash dots).$$

From their definition, one can verify that the Associated Legendre functions are either even or odd according to

$$P\_\backslash ell\; ^\{m\}\; (-x)\; =\; (-1)^\{\backslash ell\; -\; m\}\; P\_\backslash ell\; ^\{m\}(x)$$

File:Mplwp legendreP04a0.svg

File:Mplwp legendreP15a1.svg

File:Mplwp legendreP26a2.svg

The first few associated Legendre functions, including those for negative values of m, are:

$$P\_\{0\}^\{0\}(x)=1$$

$$\backslash begin\{align\}$$

P_{1}^{-1}(x)&=-\tfrac{1}{2}P_{1}^{1}(x) \\

P_{1}^{0}(x)&=x \\

P_{1}^{1}(x)&=-(1-x^2)^{1/2}

\end{align}

$$\backslash begin\{align\}$$

P_{2}^{-2}(x)&=\tfrac{1}{24}P_{2}^{2}(x) \\

P_{2}^{-1}(x)&=-\tfrac{1}{6}P_{2}^{1}(x) \\

P_{2}^{0}(x)&=\tfrac{1}{2}(3x^{2}-1) \\

P_{2}^{1}(x)&=-3x(1-x^2)^{1/2} \\

P_{2}^{2}(x)&=3(1-x^2)

\end{align}

$$\backslash begin\{align\}$$

P_{3}^{-3}(x)&=-\tfrac{1}{720}P_{3}^{3}(x) \\

P_{3}^{-2}(x)&=\tfrac{1}{120}P_{3}^{2}(x) \\

P_{3}^{-1}(x)&=-\tfrac{1}{12}P_{3}^{1}(x) \\

P_{3}^{0}(x)&=\tfrac{1}{2}(5x^3-3x) \\

P_{3}^{1}(x)&=\tfrac{3}{2}(1-5x^{2})(1-x^2)^{1/2} \\

P_{3}^{2}(x)&=15x(1-x^2) \\

P_{3}^{3}(x)&=-15(1-x^2)^{3/2}

\end{align}

$$\backslash begin\{align\}$$

P_{4}^{-4}(x)&=\tfrac{1}{40320}P_{4}^{4}(x) \\

P_{4}^{-3}(x)&=-\tfrac{1}{5040}P_{4}^{3}(x) \\

P_{4}^{-2}(x)&=\tfrac{1}{360}P_{4}^{2}(x) \\

P_{4}^{-1}(x)&=-\tfrac{1}{20}P_{4}^{1}(x) \\

P_{4}^{0}(x)&=\tfrac{1}{8}(35x^{4}-30x^{2}+3) \\

P_{4}^{1}(x)&=-\tfrac{5}{2}(7x^3-3x)(1-x^2)^{1/2} \\

P_{4}^{2}(x)&=\tfrac{15}{2}(7x^2-1)(1-x^2) \\

P_{4}^{3}(x)&= - 105x(1-x^2)^{3/2} \\

P_{4}^{4}(x)&=105(1-x^2)^{2}

\end{align}

These functions have a number of recurrence properties:

$$(\backslash ell-m+1)P\_\{\backslash ell+1\}^\{m\}(x)\; =\; (2\backslash ell+1)xP\_\{\backslash ell\}^\{m\}(x)\; -\; (\backslash ell+m)P\_\{\backslash ell-1\}^\{m\}(x)$$

$$2mxP\_\{\backslash ell\}^\{m\}(x)=-\backslash sqrt\{1-x^2\}\backslash left[P\_\{\backslash ell\}^\{m+1\}(x)+(\backslash ell+m)(\backslash ell-m+1)P\_\{\backslash ell\}^\{m-1\}(x)\backslash right]$$

$$\backslash frac\{1\}\{\backslash sqrt\{1-x^2\}\}P\_\backslash ell^m(x)\; =\; \backslash frac\{-1\}\{2m\}\; \backslash left[\; P\_\{\backslash ell-1\}^\{m+1\}(x)\; +\; (\backslash ell+m-1)(\backslash ell+m)P\_\{\backslash ell-1\}^\{m-1\}(x)\; \backslash right]$$

$$\backslash frac\{1\}\{\backslash sqrt\{1-x^2\}\}P\_\backslash ell^m(x)\; =\; \backslash frac\{-1\}\{2m\}\; \backslash left[\; P\_\{\backslash ell+1\}^\{m+1\}(x)\; +\; (\backslash ell-m+1)(\backslash ell-m+2)P\_\{\backslash ell+1\}^\{m-1\}(x)\; \backslash right]$$

$$\backslash sqrt\{1-x^2\}P\_\backslash ell^m(x)\; =\; \backslash frac1\{2\backslash ell+1\}\; \backslash left[\; (\backslash ell-m+1)(\backslash ell-m+2)\; P\_\{\backslash ell+1\}^\{m-1\}(x)\; -\; (\backslash ell+m-1)(\backslash ell+m)\; P\_\{\backslash ell-1\}^\{m-1\}(x)\; \backslash right]$$

$$\backslash sqrt\{1-x^2\}P\_\backslash ell^m(x)\; =\; \backslash frac\{-1\}\{2\backslash ell+1\}\; \backslash left[\; P\_\{\backslash ell+1\}^\{m+1\}(x)\; -\; P\_\{\backslash ell-1\}^\{m+1\}(x)\; \backslash right]$$

$$\backslash sqrt\{1-x^2\}P\_\backslash ell^\{m+1\}(x)\; =\; (\backslash ell-m)xP\_\{\backslash ell\}^\{m\}(x)\; -\; (\backslash ell+m)P\_\{\backslash ell-1\}^\{m\}(x)$$

$$\backslash sqrt\{1-x^2\}P\_\backslash ell^\{m+1\}(x)\; =\; (\backslash ell-m+1)P\_\{\backslash ell+1\}^m(x)\; -\; (\backslash ell+m+1)xP\_\backslash ell^m(x)$$

$$\backslash sqrt\{1-x^2\}\backslash frac\{d\}\{dx\}\{P\_\backslash ell^m\}(x)\; =\; \backslash frac12\; \backslash left[\; (\backslash ell+m)(\backslash ell-m+1)P\_\backslash ell^\{m-1\}(x)\; -\; P\_\backslash ell^\{m+1\}(x)\; \backslash right]$$

$$(1-x^2)\backslash frac\{d\}\{dx\}\{P\_\backslash ell^m\}(x)\; =\; \backslash frac1\{2\backslash ell+1\}\; \backslash left[\; (\backslash ell+1)(\backslash ell+m)P\_\{\backslash ell-1\}^m(x)\; -\; \backslash ell(\backslash ell-m+1)P\_\{\backslash ell+1\}^m(x)\; \backslash right]$$

$$(x^2-1)\backslash frac\{d\}\{dx\}\{P\_\{\backslash ell\}^\{m\}\}(x)\; =\; \{\backslash ell\}xP\_\{\backslash ell\}^\{m\}(x)\; -\; (\backslash ell+m)P\_\{\backslash ell-1\}^\{m\}(x)$$

$$(x^2-1)\backslash frac\{d\}\{dx\}\{P\_\{\backslash ell\}^\{m\}\}(x)\; =\; -(\backslash ell+1)xP\_\{\backslash ell\}^\{m\}(x)\; +\; (\backslash ell-m+1)P\_\{\backslash ell+1\}^\{m\}(x)$$

$$(x^2-1)\backslash frac\{d\}\{dx\}\{P\_\{\backslash ell\}^\{m\}\}(x)\; =\; \backslash sqrt\{1-x^2\}P\_\{\backslash ell\}^\{m+1\}(x)\; +\; mxP\_\{\backslash ell\}^\{m\}(x)$$

$$(x^2-1)\backslash frac\{d\}\{dx\}\{P\_\{\backslash ell\}^\{m\}\}(x)\; =\; -(\backslash ell+m)(\backslash ell-m+1)\backslash sqrt\{1-x^2\}P\_\{\backslash ell\}^\{m-1\}(x)\; -\; mxP\_\{\backslash ell\}^\{m\}(x)$$

$$(\backslash ell-m-1)(\backslash ell-m)P\_\{\backslash ell\}^\{m\}(x)\; =\; -P\_\{\backslash ell\}^\{m+2\}(x)\; +\; P\_\{\backslash ell-2\}^\{m+2\}(x)\; +\; (\backslash ell+m)(\backslash ell+m-1)P\_\{\backslash ell-2\}^\{m\}(x)$$

Helpful identities (initial values for the first recursion):

$$P\_\{\backslash ell\; +1\}^\{\backslash ell\; +1\}(x)\; =\; -\; (2\backslash ell+1)\; \backslash sqrt\{1-x^2\}\; P\_\{\backslash ell\}^\{\backslash ell\}(x)$$

$$P\_\{\backslash ell\}^\{\backslash ell\}(x)\; =\; (-1)^\backslash ell\; (2\backslash ell-1)!!\; (1-\; x^2)^\{(\backslash ell/2)\}$$

$$P\_\{\backslash ell\; +1\}^\{\backslash ell\}(x)\; =\; x\; (2\backslash ell+1)\; P\_\{\backslash ell\}^\{\backslash ell\}(x)$$

with {{math|!!}} the double factorial.

The integral over the product of three associated Legendre polynomials (with orders matching as shown below) is a necessary ingredient when developing products of Legendre polynomials into a series linear in the Legendre polynomials. For instance, this turns out to be necessary when doing atomic calculations of the Hartree–Fock variety where matrix elements of the Coulomb operator are needed. For this we have Gaunt's formula From John C. Slater Quantum Theory of Atomic Structure, McGraw-Hill (New York, 1960), Volume I, page 309, which cites the original work of J. A. Gaunt, Philosophical Transactions of the Royal Society of London, A228:151 (1929){{cite journal|first1=Yu-Lin|last1=Xu|title=Fast evaluation of the Gaunt coefficients|journal=Math. Comp.|year=1996|volume=65|number=216|pages=1601-1612|doi=10.1090/S0025-5718-96-00774-0}}

$$\backslash begin\{align\}$$

\frac{1}{2} \int_{-1}^1 P_l^u(x) P_m^v(x) P_n^w(x) dx

={}&{}(-1)^{s-m-w}\frac{(m+v)!(n+w)!(2s-2n)!s!}{(m-v)!(s-l)!(s-m)!(s-n)!(2s+1)!} \\

&{}\times \ \sum_{t=p}^q (-1)^t \frac{(l+u+t)!(m+n-u-t)!}{t!(l-u-t)!(m-n+u+t)!(n-w-t)!}

\end{align}

This formula is to be used under the following assumptions:

1. the degrees are non-negative integers $l,m,n\backslash ge0$
2. all three orders are non-negative integers $u,v,w\backslash ge\; 0$
3. $u$ is the largest of the three orders
4. the orders sum up $u=v+w$
5. the degrees obey $m\backslash ge\; n$

Other quantities appearing in the formula are defined as

$$2s\; =\; l+m+n$$

$$p\; =\; \backslash max(0,\backslash ,n-m-u)$$

$$q\; =\; \backslash min(m+n-u,\backslash ,l-u,\backslash ,n-w)$$

The integral is zero unless

1. the sum of degrees is even so that $s$ is an integer
2. the triangular condition is satisfied $m+n\backslash ge\; l\; \backslash ge\; m-n$

Dong and Lemus (2002)Dong S.H., Lemus R., (2002), [http://www.sciencedirect.com/science/article/pii/S0893965902800040 "The overlap integral of three associated Legendre polynomials"], Appl. Math. Lett. 15, 541-546. generalized the derivation of this formula to integrals over a product of an arbitrary number of associated Legendre polynomials.

{{main|Legendre function}}

These functions may actually be defined for general complex parameters and argument:{{cite journal|first1=H. A.|last1=Mavromatis|first2=R. S. |last2=Alassar|title=A generalized formula for the integral of three Associated Legendre Polynomials|year=1999|journal=Appl. Math. Lett.|volume=12|number=3|pages=101-105|doi=10.1016/S0893-9659(98)00180-3}}

$$P\_\{\backslash lambda\}^\{\backslash mu\}(z)\; =\; \backslash frac\{1\}\{\backslash Gamma(1-\backslash mu)\}\; \backslash left[\backslash frac\{1+z\}\{1-z\}\backslash right]^\{\backslash mu/2\}\; \backslash ,\_2F\_1\; (-\backslash lambda,\; \backslash lambda+1;\; 1-\backslash mu;\; \backslash frac\{1-z\}\{2\})$$

where $\backslash Gamma$ is the gamma function and $\_2F\_1$ is the hypergeometric function

$$\backslash ,\_2F\_1\; (\backslash alpha,\; \backslash beta;\; \backslash gamma;\; z)\; =\; \backslash frac\{\backslash Gamma(\backslash gamma)\}\{\backslash Gamma(\backslash alpha)\backslash Gamma(\backslash beta)\}\; \backslash sum\_\{n=0\}^\backslash infty\backslash frac\{\backslash Gamma(n+\backslash alpha)\backslash Gamma(n+\backslash beta)\}\{\backslash Gamma(n+\backslash gamma)\backslash \; n!\}z^n,$$

They are called the Legendre functions when defined in this more general way. They satisfy the same differential equation as before:

$$(1-z^2)\backslash ,y\text{'}\text{'}\; -2zy\text{'}\; +\; \backslash left(\backslash lambda[\backslash lambda+1]\; -\; \backslash frac\{\backslash mu^2\}\{1-z^2\}\backslash right)\backslash ,y\; =\; 0.\backslash ,$$

Since this is a second order differential equation, it has a second solution, $Q\_\backslash lambda^\{\backslash mu\}(z)$, defined as:

$$Q\_\{\backslash lambda\}^\{\backslash mu\}(z)\; =\; \backslash frac\{\backslash sqrt\{\backslash pi\}\backslash \; \backslash Gamma(\backslash lambda+\backslash mu+1)\}\{2^\{\backslash lambda+1\}\backslash Gamma(\backslash lambda+3/2)\}\backslash frac\{1\}\{z^\{\backslash lambda+\backslash mu+1\}\}(1-z^2)^\{\backslash mu/2\}\; \backslash ,\_2F\_1\; \backslash left(\backslash frac\{\backslash lambda+\backslash mu+1\}\{2\},\; \backslash frac\{\backslash lambda+\backslash mu+2\}\{2\};\; \backslash lambda+\backslash frac\{3\}\{2\};\; \backslash frac\{1\}\{z^2\}\backslash right)$$

$P\_\backslash lambda^\{\backslash mu\}(z)$ and $Q\_\backslash lambda^\{\backslash mu\}(z)$ both obey the various recurrence formulas given previously.

These functions are most useful when the argument is reparameterized in terms of angles, letting $x\; =\; \backslash cos\backslash theta$:

$$P\_\backslash ell^\{m\}(\backslash cos\backslash theta)\; =\; (-1)^m\; (\backslash sin\; \backslash theta)^m\backslash \; \backslash frac\{d^m\}\{d(\backslash cos\backslash theta)^m\}\backslash left(P\_\backslash ell(\backslash cos\backslash theta)\backslash right)$$

Using the relation $(1\; -\; x^2)^\{1\; /\; 2\}\; =\; \backslash sin\backslash theta$, the list given above yields the first few polynomials, parameterized this way, as:

$$\backslash begin\{align\}$$

P_0^0(\cos\theta) & = 1 \\[8pt]

P_1^0(\cos\theta) & = \cos\theta \\[8pt]

P_1^1(\cos\theta) & = -\sin\theta \\[8pt]

P_2^0(\cos\theta) & = \tfrac{1}{2} (3\cos^2\theta-1) \\[8pt]

P_2^1(\cos\theta) & = -3\cos\theta\sin\theta \\[8pt]

P_2^2(\cos\theta) & = 3\sin^2\theta \\[8pt]

P_3^0(\cos\theta) & = \tfrac{1}{2} (5\cos^3\theta-3\cos\theta) \\[8pt]

P_3^1(\cos\theta) & = -\tfrac{3}{2} (5\cos^2\theta-1)\sin\theta \\[8pt]

P_3^2(\cos\theta) & = 15\cos\theta\sin^2\theta \\[8pt]

P_3^3(\cos\theta) & = -15\sin^3\theta \\[8pt]

P_4^0(\cos\theta) & = \tfrac{1}{8} (35\cos^4\theta-30\cos^2\theta+3) \\[8pt]

P_4^1(\cos\theta) & = - \tfrac{5}{2} (7\cos^3\theta-3\cos\theta)\sin\theta \\[8pt]

P_4^2(\cos\theta) & = \tfrac{15}{2} (7\cos^2\theta-1)\sin^2\theta \\[8pt]

P_4^3(\cos\theta) & = -105\cos\theta\sin^3\theta \\[8pt]

P_4^4(\cos\theta) & = 105\sin^4\theta

\end{align}

The orthogonality relations given above become in this formulation:

for fixed m, $P\_\backslash ell^m(\backslash cos\backslash theta)$ are orthogonal, parameterized by θ over $[0,\; \backslash pi]$, with weight $\backslash sin\; \backslash theta$:

$$\backslash int\_0^\backslash pi\; P\_k^\{m\}(\backslash cos\backslash theta)\; P\_\backslash ell^\{m\}(\backslash cos\backslash theta)\backslash ,\backslash sin\backslash theta\backslash ,d\backslash theta\; =\; \backslash frac\{2\; (\backslash ell+m)!\}\{(2\backslash ell+1)(\backslash ell-m)!\}\backslash \; \backslash delta\; \_\{k,\backslash ell\}$$

Also, for fixed ℓ:

In terms of θ, $P\_\backslash ell^\{m\}(\backslash cos\; \backslash theta)$ are solutions of

$$\backslash frac\{d^\{2\}y\}\{d\backslash theta^2\}\; +\; \backslash cot\; \backslash theta\; \backslash frac\{dy\}\{d\backslash theta\}\; +\; \backslash left[\backslash lambda\; -\; \backslash frac\{m^2\}\{\backslash sin^2\backslash theta\}\backslash right]\backslash ,y\; =\; 0\backslash ,$$

More precisely, given an integer m$\backslash ge$0, the above equation has

nonsingular solutions only when $\backslash lambda\; =\; \backslash ell(\backslash ell+1)\backslash ,$ for ℓ

an integer ≥ m, and those solutions are proportional to

$P\_\backslash ell^\{m\}(\backslash cos\; \backslash theta)$.

{{main|Spherical harmonics}}

In many occasions in physics, associated Legendre polynomials in terms of angles occur where spherical symmetry is involved. The colatitude angle in spherical coordinates is

the angle $\backslash theta$ used above. The longitude angle, $\backslash phi$, appears in a multiplying factor. Together, they make a set of functions called spherical harmonics. These functions express the symmetry of the two-sphere under the action of the Lie group SO(3).{{cn|date=July 2022}}

What makes these functions useful is that they are central to the solution of the equation

$\backslash nabla^2\backslash psi\; +\; \backslash lambda\backslash psi\; =\; 0$ on the surface of a sphere. In spherical coordinates θ (colatitude) and φ (longitude), the Laplacian is

$$\backslash nabla^2\backslash psi\; =\; \backslash frac\{\backslash partial^2\backslash psi\}\{\backslash partial\backslash theta^2\}\; +\; \backslash cot\; \backslash theta\; \backslash frac\{\backslash partial\; \backslash psi\}\{\backslash partial\; \backslash theta\}\; +\; \backslash csc^2\; \backslash theta\backslash frac\{\backslash partial^2\backslash psi\}\{\backslash partial\backslash phi^2\}.$$

When the partial differential equation

$$\backslash frac\{\backslash partial^2\backslash psi\}\{\backslash partial\backslash theta^2\}\; +\; \backslash cot\; \backslash theta\; \backslash frac\{\backslash partial\; \backslash psi\}\{\backslash partial\; \backslash theta\}\; +\; \backslash csc^2\; \backslash theta\backslash frac\{\backslash partial^2\backslash psi\}\{\backslash partial\backslash phi^2\}\; +\; \backslash lambda\; \backslash psi\; =\; 0$$

is solved by the method of separation of variables, one gets a φ-dependent part $\backslash sin(m\backslash phi)$ or $\backslash cos(m\backslash phi)$ for integer m≥0, and an equation for the θ-dependent part

$$\backslash frac\{d^\{2\}y\}\{d\backslash theta^2\}\; +\; \backslash cot\; \backslash theta\; \backslash frac\{dy\}\{d\backslash theta\}\; +\; \backslash left[\backslash lambda\; -\; \backslash frac\{m^2\}\{\backslash sin^2\backslash theta\}\backslash right]\backslash ,y\; =\; 0\backslash ,$$

for which the solutions are $P\_\backslash ell^\{m\}(\backslash cos\; \backslash theta)$ with $\backslash ell\{\backslash ge\}m$

and $\backslash lambda\; =\; \backslash ell(\backslash ell+1)$.

Therefore, the equation

$$\backslash nabla^2\backslash psi\; +\; \backslash lambda\backslash psi\; =\; 0$$

has nonsingular separated solutions only when $\backslash lambda\; =\; \backslash ell(\backslash ell+1)$,

and those solutions are proportional to

$$P\_\backslash ell^\{m\}(\backslash cos\; \backslash theta)\backslash \; \backslash cos\; (m\backslash phi)\backslash \; \backslash \; \backslash \; \backslash \; 0\; \backslash le\; m\; \backslash le\; \backslash ell$$

and

For each choice of ℓ, there are {{nowrap|2ℓ + 1}} functions

for the various values of m and choices of sine and cosine.

They are all orthogonal in both ℓ and m when integrated over the

surface of the sphere.

The solutions are usually written in terms of complex exponentials:

$$Y\_\{\backslash ell,\; m\}(\backslash theta,\; \backslash phi)\; =\; \backslash sqrt\{\backslash frac\{(2\backslash ell+1)(\backslash ell-m)!\}\{4\backslash pi(\backslash ell+m)!\}\}\backslash \; P\_\backslash ell^\{m\}(\backslash cos\; \backslash theta)\backslash \; e^\{im\backslash phi\}\backslash qquad\; -\backslash ell\; \backslash le\; m\; \backslash le\; \backslash ell.$$

The functions $Y\_\{\backslash ell,\; m\}(\backslash theta,\; \backslash phi)$ are the spherical harmonics, and the quantity in the square root is a normalizing factor.

Recalling the relation between the associated Legendre functions of positive and negative m, it is easily shown that the spherical harmonics satisfy the identityThis identity can also be shown by relating the spherical harmonics to Wigner D-matrices and use of the time-reversal property of the latter. The relation between associated Legendre functions of ±m can then be proved from the complex conjugation identity of the spherical harmonics.

$$Y\_\{\backslash ell,\; m\}^*(\backslash theta,\; \backslash phi)\; =\; (-1)^m\; Y\_\{\backslash ell,\; -m\}(\backslash theta,\; \backslash phi).$$

The spherical harmonic functions form a complete orthonormal set of functions in the sense of Fourier series. Workers in the fields of geodesy, geomagnetism and spectral analysis use a different phase and normalization factor than given here (see spherical harmonics).

When a 3-dimensional spherically symmetric partial differential equation is solved by the method of separation of variables in spherical coordinates, the part that remains after removal of the radial part is typically

of the form

$$\backslash nabla^2\backslash psi(\backslash theta,\; \backslash phi)\; +\; \backslash lambda\backslash psi(\backslash theta,\; \backslash phi)\; =\; 0,$$

and hence the solutions are spherical harmonics.

The Legendre polynomials are closely related to hypergeometric series. In the form of spherical harmonics, they express the symmetry of the two-sphere under the action of the Lie group SO(3). There are many other Lie groups besides SO(3), and analogous generalizations of the Legendre polynomials exist to express the symmetries of semi-simple Lie groups and Riemannian symmetric spaces. Crudely speaking, one may define a Laplacian on symmetric spaces; the eigenfunctions of the Laplacian can be thought of as generalizations of the spherical harmonics to other settings.

By solving the Laplace equation in higher dimensions (with a potential that does not fall of $\backslash sim\; 1/r$) Legendre Polynonials in higher than 3D can be defined.{{cite journal|first1=L. M. B. C.|last1=Campos|first2=F. S. R. P.|last2=Cunha|title=On hyperspherical Legendre polynomials and higher dimensional multipole expansions|journal=J. Inequal. Spec. Func.|year=2012|volume=3|number=3|url=http://ilirias.com/jiasf/repository/docs/JIASF3-3-1.pdf}}

• Angular momentum
• Gaussian quadrature
• Legendre polynomials
• Spherical harmonics
• Whipple's transformation of Legendre functions
• Laguerre polynomials
• Hermite polynomials

{{Reflist}}

• {{citation|last1=Arfken|first1=G.B.|last2=Weber|first2=H.J.|title=Mathematical methods for physicists|year=2001|publisher=Academic Press|isbn=978-0-12-059825-0}}; Section 12.5. (Uses a different sign convention.)
• {{citation|last=Belousov|first=S. L.|year=1962|title=Tables of normalized associated Legendre polynomials|series=Mathematical tables|volume=18|publisher=Pergamon Press}}.
• {{citation|first1=E. U.|last1=Condon|first2=G. H.|last2=Shortley|title=The Theory of Atomic Spectra|year=1970|location=Cambridge, England|publisher=Cambridge University Press|oclc=5388084}}; Chapter 3.
• {{citation|first1=Richard|last1=Courant|authorlink1=Richard Courant|first2=David|last2=Hilbert|authorlink2=David Hilbert|year=1953|title=Methods of Mathematical Physics, Volume 1|publisher=Interscience Publischer, Inc|location=New York}}.
• {{dlmf|first=T. M. |last=Dunster|id=14|title=Legendre and Related Functions}}
• {{citation|first=A.R.|last=Edmonds|title=Angular Momentum in Quantum Mechanics|year=1957|publisher=Princeton University Press|isbn=978-0-691-07912-7|url-access=registration|url=https://archive.org/details/angularmomentumi0000edmo}}; Chapter 2.
• {{cite journal|first=J. A. |last=Gaunt|year=1929|title=IV. The triples of helium|journal=Phil. Trans. Royal Soc. A|volume =228|number=659-669|pages=151|doi=10.1098/rsta.1929.0004|doi-access=free}}
• {{citation|first=F. B.|last=Hildebrand|authorlink=Francis B. Hildebrand|title=Advanced Calculus for Applications|year=1976|publisher=Prentice Hall|isbn=978-0-13-011189-0}}.
• {{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
• {{cite journal|last1=Schach|first1=S. R.|doi=10.1137/0507007|title=New Identities for Legendre Associated Functions of Integral Order and Degree|journal=SIAM J. Math. Anal.|year=1976|volume=7|number=1|pages=59–69}}

• [http://mathworld.wolfram.com/AssociatedLegendrePolynomial.html Associated Legendre polynomials in MathWorld]
• [http://mathworld.wolfram.com/LegendrePolynomial.html Legendre polynomials in MathWorld]
• [https://dlmf.nist.gov/14 Legendre and Related Functions in DLMF]

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Category:Atomic physics

Category:Orthogonal polynomials