Wolfram Math (http://mathworld.wolfram.com/InverseCosecant.html) had a breakdown of Inverse Cosecant (arcsc), a series that looks almost identical to the inverse Sine (arcsin). It (the series) seemed straight forward enough, but I'm trying to understand the generating Sigma/Sum below for that series:
$$csc^{-1}=-\sum_{n=1}^{\infty}\frac{i^{n+1}P_{n-1}(0)}{n}x^{-n}$$
I'm tripped up here 3 times in a row:
1st: the "i^n+1" What is the "i"? Is it imaginary "i"? And if so, what do I do when the sums include positive and negative i on n=2,4,6, etc.? I assume if this is imaginary "i" there's a rule I don't know that makes this work in a series.
2nd & 3rd: P_n-1 and (0) here, according to wolfram:
"P_n(x) is a Legendre Polynomial and (x)_n is a Pochhammer symbol."
At this point, I'm not sure where the Legendre Polynomial ends and where the Pochhammer symbol begins. My textbooks on Elementary Differential Equations and Advanced Engineering Mathematics cover Legendre Polynomials but I couldn't find any reference explaining Pochhammer symbols. Arcsine is similar, and again, lack of understanding Pochhammer symbols is a road block to my understanding:
$$sin^{-1}=\sum_{n=0}^{\infty}\frac{(\frac{1}{2})_n}{(2n+1)n!}x^{2n+1}$$
I believe that part that looks like a fraction (1/2) is pochhammer notation, and not a real fraction. Wolfram again says "where (x)_n is a pochhammer symbol"
EDIT (for inverse Cosecant):
this formula below (for each term) makes more sense. I only need to know what the P and (0) stand for and how they convert into the following:
$$=\frac{(\frac{1}{2})_n}{(n-1)!(2n-1)}x^{1-2n}$$
With it though, the explanation of fractions multiplying [for interpreting the (1/2)_n] works.
Best Answer
Note that this is an expansion at $x=\infty!\;$ $P_{n-1}(0)$ is the value at $x=0$ not a mixture of Legendre polynomial and Pochhammer symbol. $(\frac{1}{2})_{n-1}$ is a Pochhammer symbol and means $(\frac{1}{2})_{n-1} = \frac{1}{2} \times \frac{3}{2}\times \frac{5}{2} \cdots$ or generally for $a\in \mathbb{R}:$ $$(a)_n=a(a+1)(a+2)\cdots(a+n-1),$$ by convention $(a)_0=1.$ For odd $n$ the term $i^{n+1}$ in $i^{n+1}P_{n-1}(0)$ is real, and for even $n=2m$ the Legendre polynomial $P_{2m-1}(x)$ is zero at $x=0$