Legendre functions in number theory

Question

I have heard that Legendre functions are important in number theory. Can any one tell me how?

The Legendre function of the first kind $P_s$ is defined by \begin{eqnarray*}P_s(x) =& \frac{1}{2\pi}\int_{-\pi}^{\pi}\left(x+\sqrt{x^2-1}\cos\theta\right)^s d\theta
\newline =&\frac{1}{\pi}\int_0^1\left(x+\sqrt{x^2-1}(2t-1)\right)^s\frac{dt}{\sqrt{t(1-t)}},& s\in\mathbb{C}, x\ge 1. \end{eqnarray*}

Answer 1

Perhaps this has something for you: http://wis.kuleuven.be/analyse/walter/coimbra.pdf

This uses the Legendre polynomials in proving irrationality of certain numbers.

Answer 2

If you accept enumerative combinatorics as part of number theory, here is a nice application of the Legendre polynomials.

Count the number of ways to insert paired, grammatical and non-repeating parentheses enclosing at least two letters in a word of length $n > 1$. Define $a_1 = 1$ and ignore outmost paratheses. For $n = 2$, there is only one way $(ab) = ab$. For $n = 3$, we have $abc, (ab)c, a(bc)$. For $n = 4$, we have $abcd$, $(ab)cd$, $a(bc)d$, $ab(cd)$, $(ab)(cd)$, $a(bcd)$, $a(b(cd))$, $a((bc)d)$, $(abc)d$, $(a(bc))d$, and $((ab)c)d$. And, so on.

The sequence continues {$1$, $1$, $3$, $11$, $45$, $197$, $903$, $4279 \dots$ } (A001003) and the terms are called Super Catalan numbers or Little Schroeder numbers. In general, \begin{eqnarray} a_n = \tfrac{1}{n} \sum_{k =0}^{n-2} \binom{2n-k-2}{n-1} \binom{n-2}{k} = \tfrac{1}{4n}(3P_{n-1}(3) - P_{n-2}(3)) + \tfrac{1}{2} \epsilon(n), \end{eqnarray} where $\epsilon$ is the unit arithmetic function, which is $1$ when the argument is $1$ and $0$ otherwise.

In fact, there are at least $10$ different combinatorial problems enumerated by this sequence, involving restricted lattice paths, integer compositions and even matrix row-sums. The Online Encyclopedia of Integer Sequences has the present state of knowledge about these fascinating numbers; see there reference section of http://oeis.org/A001003 for relevant papers and books.