For given $n,\ell\in\mathbb N\_0$, I am interested in studying the following recursion relation for some $\mu\in\mathbb R$:
$$\sqrt{-1} \tfrac{j(\ell-j+1)(n-j+1)(n+j+1)}{2(2j-1)(2j+1)} a\_{j-1} - \tfrac {j(j+1)}2 a\_j
 - \sqrt{-1} \tfrac{(j+1)(\ell+j+2)}2 a\_{j+1} = \mu a\_j,$$
for $j\ge0$, assuming $a\_{-1}=0$.


The trivial solution is $a\_j=0,\ \forall j\in\mathbb N\_0$. The value of $a\_0$ determines the entire sequence, no matter what is $\mu$.


Since this three term recurrence relation is given by matrix valued spherical functions of the symmetric pair $(\mathrm{SO }(4) ,\mathrm{SO }(3) )$ one expects to be solvable by a set of orthogonal polynomials on a finite discrete set, for some $\mu$.


Does anybody recognize this expression or know any family of polynomials solving this?


From already thank you very much.



Three term recurrence relation.
Racah are involved


