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11.2 Properties of Legendre polynomials

11.2.1 Generating function

Let F(x,t) be a function of the two variables x and t that can be expressed as a Taylor’s series in t, {\mathop{\mathop{∑ }}\nolimits }_{n}{c}_{n}(x){t}^{n}. The function F is then called a generating function of the functions {c}_{n}.

Example 11.1: 

Show that F(x,t) = {1\over 1−xt} is a generating function of the polynomials {x}^{n}.

Solution: 

Look at

{1\over 1 − xt} ={ \mathop{∑ }}_{n=0}^{∞}{x}^{n}{t}^{n}\kern 2.77695pt \kern 2.77695pt (|xt| < 1).
(11.16)

Example 11.2: 

Show that F(x,t) =\mathop{ exp}\nolimits \left ({tx−t\over 2t} \right ) is the generating function for the Bessel functions,

F(x,t) =\mathop{ exp}\nolimits ({tx − t\over 2t} ) ={ \mathop{∑ }}_{n=0}^{∞}{J}_{ n}(x){t}^{n}.
(11.17)

Example 11.3: 

(The case of most interest here)

F(x,t) = {1\over \sqrt{1 − 2xt + {t}^{2}}} ={ \mathop{∑ }}_{n=0}^{∞}{P}_{ n}(x){t}^{n}.
(11.18)

11.2.2 Rodrigues’ Formula

{P}_{n}(x) = {1\over { 2}^{n}n!} {{d}^{n}\over d{x}^{n}}{({x}^{2} − 1)}^{n}.
(11.19)

11.2.3 A table of properties

  1. {P}_{n}(x) is even or odd if n is even or odd.
  2. {P}_{n}(1) = 1.
  3. {P}_{n}(−1) = {(−1)}^{n}.
  4. (2n + 1){P}_{n}(x) = {P'}_{n+1}(x) − {P'}_{n−1}(x).
  5. (2n + 1)x{P}_{n}(x) = (n + 1){P}_{n+1}(x) + n{P}_{n−1}(x).
  6. {\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }_{−1}^{x}{P}_{n}(x')dx' = {1\over 2n+1}\left [{P}_{n+1}(x) − {P}_{n−1}(x)\right ].

Let us prove some of these relations, first Rodrigues’ formula. We start from the simple formula

({x}^{2} − 1) {d\over dx}{({x}^{2} − 1)}^{n} − 2nx{({x}^{2} − 1)}^{n} = 0,
(11.20)

which is easily proven by explicit differentiation. This is then differentiated n + 1 times,

\begin{eqnarray} {{d}^{n+1}\over d{x}^{n+1}}\left [({x}^{2} − 1) {d\over dx}{({x}^{2} − 1)}^{n} − 2nx{({x}^{2} − 1)}^{n}\right ]&& %& \\ & =& n(n + 1) {{d}^{n}\over d{x}^{n}}{({x}^{2} − 1)}^{n} + 2(n + 1)x {{d}^{n+1}\over d{x}^{n+1}}{({x}^{2} − 1)}^{n} + ({x}^{2} − 1) {{d}^{n+2}\over d{x}^{n+2}}{({x}^{2} − 1)}^{n}%& \\ & & −2n(n + 1) {{d}^{n}\over d{x}^{n}}{({x}^{2} − 1)}^{n} − 2nx {{d}^{n+1}\over d{x}^{n+1}}{({x}^{2} − 1)}^{n} %& \\ & =& −n(n + 1) {{d}^{n}\over d{x}^{n}}{({x}^{2} − 1)}^{n} + 2x {{d}^{n+1}\over d{x}^{n+1}}{({x}^{2} − 1)}^{n} + ({x}^{2} − 1) {{d}^{n+2}\over d{x}^{n+2}}{({x}^{2} − 1)}^{n} %& \\ & =& −\left [ {d\over dx}(1 − {x}^{2}) {d\over dx}\left \{ {{d}^{n}\over d{x}^{n}}{({x}^{2} − 1)}^{n}\right \} + n(n + 1)\left \{ {{d}^{n}\over d{x}^{n}}{({x}^{2} − 1)}^{n}\right \}\right ] = 0. %&(11.21) \\ \end{eqnarray}

We have thus proven that {{d}^{n}\over d{x}^{n}}{({x}^{2} − 1)}^{n} satisfies Legendre’s equation. The normalisation follows from the evaluation of the highest coefficient,

{{d}^{n}\over d{x}^{n}}{x}^{2n} = {2n!\over n!} {x}^{n},
(11.22)

and thus we need to multiply the derivative with {1\over { 2}^{n}n!} to get the properly normalised {P}_{n}.

Let’s use the generating function to prove some of the other properties: 2.:

F(1,t) = {1\over 1 − t} ={ \mathop{∑ }}_{n}{t}^{n}
(11.23)

has all coefficients one, so {P}_{n}(1) = 1. Similarly for 3.:

F(−1,t) = {1\over 1 + t} ={ \mathop{∑ }}_{n}{(−1)}^{n}{t}^{n}.
(11.24)

Property 5. can be found by differentiating the generating function with respect to t:

\begin{eqnarray} {d\over dt} {1\over \sqrt{1 − 2tx + {t}^{2}}}& =& {d\over dt}{\mathop{∑ }}_{n=0}^{∞}{t}^{n}{P}_{ n}(x) %& \\ {x − t\over { (1 − 2tx + {t}^{2})}^{3∕2}}& =& {\mathop{∑ }}_{n=0}n{t}^{n−1}{P}_{ n}(x) %& \\ {x − t\over 1 − 2xt + {t}^{2}}{ \mathop{∑ }}_{n=0}^{∞}{t}^{n}{P}_{ n}(x)& =& {\mathop{∑ }}_{n=0}n{t}^{n−1}{P}_{ n}(x) %& \\ {\mathop{∑ }}_{n=0}^{∞}{t}^{n}x{P}_{ n}(x) −{\mathop{∑ }}_{n=0}^{∞}{t}^{n+1}{P}_{ n}(x)& =& {\mathop{∑ }}_{n=0}^{∞}n{t}^{n−1}{P}_{ n}(x) − 2{\mathop{∑ }}_{n=0}^{∞}n{t}^{n}x{P}_{ n}(x) +{ \mathop{∑ }}_{n=0}^{∞}n{t}^{n+1}{P}_{ n}(x)%& \\ {\mathop{∑ }}_{n=0}^{∞}{t}^{n}(2n + 1)x{P}_{ n}(x)& =& {\mathop{∑ }}_{n=0}^{∞}(n + 1){t}^{n}{P}_{ n+1}(x) +{ \mathop{∑ }}_{n=0}^{∞}n{t}^{n}{P}_{ n−1}(x) %&(11.25)\\ \end{eqnarray}

Equating terms with identical powers of t we find

(2n + 1)x{P}_{n}(x) = (n + 1){P}_{n+1}(x) + n{P}_{n−1}(x).
(11.26)

Proofs for the other properties can be found using similar methods.

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