Let $P\_{n}(x)$ the $n-th$ [Legendre polynomial](https://en.wikipedia.org/wiki/Legendre_polynomials). It is well-knonw that $$\int\_{-1}^1 P\_n(x) P\_m(x) P\_h(x) \, dx=2\left(\begin{array}{ccc}
n & m & h\\
0 & 0 & 0
\end{array}\right)^{2}\tag{1}$$ where $\left(\begin{array}{ccc}
n & m & h\\
0 & 0 & 0
\end{array}\right)$ is the [Wigner 3-j symbol](https://en.wikipedia.org/wiki/3-j_symbol). I know that there exist some possible choices for the generalization of the Wigner 3-j symbol to complex numbers but my question is: is it possible to extend othe Wigner 3-j symbol to complex numbers such that type (1) equation holds? In other words, is it possible to find a generalization of the 3-j symbols such that $$\int\_{-1}^1 P\_a(x)P\_b(x)P\_c(x) \, dx = f(a,b,c) \left(\begin{array}{ccc}
a & b & c\\
0 & 0 & 0
\end{array}\right)^{2}$$
for a suitable function $f$ depending on $a,b,c\in\mathbb{C}$ and where $P\_{a}(x)$ is the $C^{2}(-1,1)$ solution of the Legendre differential equation with parameter $a\in\mathbb{C}$, that is,


$$P\_{a}(x)=\,\_{2}F\_{1}\left(a+1,-a;1;\frac{1-x}{2}\right)?$$



Generalized Wigner 3-j symbol and Legendre functions
**Summary:** this is a topic of active research [1,2,3,4]; the answer to the general case is not known; answers exist for $a\in\mathbb{C}$, $b,c,\in\mathbb{N}$, and for $a,b,c\in\mathbb{C}$ with $b=c$.




---


**Case 1:** Consider first the case that one of the three parameters $a,b,c$ is complex, while the other two are still integers. For the Legendre function I use the representation
$$P\_\mu(z)=\, \_2F\_1\left(-\mu,\mu+1;1;\frac{1-z}{2}\right),\;\;|1-z|<2,$$
and for the 3j symbol squared
$$\left(\begin{array}{ccc}
a & b & c\\
0 & 0 & 0
\end{array}\right)^{2}=\frac{\binom{a+b-c}{\frac{1}{2} (a+b-c)} \binom{a-b+c}{\frac{1}{2} (a-b+c)} \binom{-a+b+c}{\frac{1}{2} (-a+b+c)}}{(a+b+c+1) \binom{a+b+c}{\frac{1}{2} (a+b+c)}}.$$
Then
$$\int\_{-1}^1 P\_{a}(x)P\_{b}(x)P\_c(x)\,dx=2\cos ^2\left(\tfrac{1}{2} \pi (a+b+c)\right)\left(\begin{array}{ccc}
a & b & c\\
0 & 0 & 0
\end{array}\right)^{2},$$
$$\text{for}\;\;a\in\mathbb{C},b\in\mathbb{N},c\in\mathbb{N}.$$
(I checked this with Mathematica, which can evaluate the integral in closed form.)


So in this case of one complex variable the function $f(a,b,c)$ in the OP equals $2\cos ^2\left(\tfrac{1}{2} \pi (a+b+c)\right)$. Note that if all three variables are integers, the $\cos^2$ factor can be set to unity, since the 3j symbol equals zero for $a+b+c$ odd.


**Case 2:** If all three parameters are complex, a closed form expression in terms of generalized 3j symbols (or equivalently, generalized Clebsch-Gordan coefficients) exists for the integral
$$T\_{\mu,\nu}=\int\_{-1}^1 P\_{\mu}(x)P\_{\nu}(x)P\_\nu(-x)\,dx,\;\;\mu,\nu\in\mathbb{C}.$$
Note the sign change of one argument of the Legendre function. For integer $\nu=n$ one would simply have $P\_n(-x)=(-1)^nP\_n(x)$, but for noninteger $\nu$ the sign change is essential.


The result, derived by Yajun Zhou [1] (see also [2,3,4]) is given below, it is lengthy. A simple case is $\mu=\nu$, when we have
$$\int\_{-1}^1 P\_{\nu}(x)P\_{\nu}(x)P\_\nu(-x)\,dx=\frac{1}{3} \bigl(2+3 \cos \pi \nu+\cos 2 \pi \nu\bigr)\left(\begin{array}{ccc}
\nu & \nu & \nu\\
0 & 0 & 0
\end{array}\right)^{2},\;\;\nu\in\mathbb{C}.$$
For integer $\nu$ the integral vanishes for $\nu$ odd, and for $\nu$ even the usual formula is recovered.


Here is the general result:
\begin{align}
T\_{\mu,\nu}={}&\frac{2}{\pi^2}\frac{\sin(\mu\pi)\sin(\nu\pi)}{\mu(\mu+1)}\left[ \, \frac{1}{\nu}\,{\_4F\_3}\left(\left.\begin{array}{c}
1,\frac{1-\mu}{2},\frac{\mu+2 }{2},-\nu \\[4pt]
\frac{2-\mu }{2},\frac{\mu+3 }{2},1-\nu \\
\end{array}\right| 1\right) -\frac{1}{\nu+1} \,{ \_4F\_3}\left(\left.\begin{array}{c}
1,\frac{1-\mu}{2},\frac{\mu+2 }{2},\nu+1\ \\[4pt]
\frac{2-\mu }{2},\frac{\mu+3 }{2},\nu+2 \\
\end{array}\right| 1\right) \right].
\end{align}
The combination of two hypergeometric functions ensures the symmetry $T\_{\mu,\nu}=T\_{-\mu-1,\nu}=T\_{\mu,-\nu-1}$, required by $P\_{\mu}(z)=P\_{-\mu-1}(z)$.


---


1. Yajun Zhou, [Legendre
Functions, Spherical Rotations, and Multiple Elliptic Integrals](https://arxiv.org/abs/1301.1735)
2. Marco Cantarini, [A
note on Clebsch-Gordan integral, Fourier-Legendre expansions and
closed form for hypergeometric series](https://arxiv.org/abs/2107.11969)
3. John Campbell, [New Clebsch-Gordan-type integrals involving threefold products of complete elliptic integrals](https://arxiv.org/abs/2302.05819)
4. Yajun Zhou, [On Some Integrals Over the Product of Three Legendre Functions](https://arxiv.org/abs/1304.1606)


