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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.

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Does the $i$ here denote the square root of $-1$? Also, could you show a few examples for at least one pair of indices $n$ and $\ell$? It would help to know how you want the recursion to start -- otherwise, $a_j=0$ for all $j$ would seem to suffice. –  Barry Cipra Sep 7 '12 at 20:07
Why in particular is this recursion interesting? –  Per Alexandersson Sep 7 '12 at 21:36
Can anything be said from the differential equation for the generating function? (I'm guessing it's 4th order.) –  Pavel Safronov Sep 8 '12 at 22:57

1 Answer 1

up vote 0 down vote accepted

Racah are involved

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