2.2 Racah

Racah Hahn.

  
If we take and let in the definition (1.2.1) of the Racah polynomials, we obtain the Hahn polynomials defined by (1.5.1). Hence

The Hahn polynomials can also be obtained from the Racah polynomials by taking in the definition (1.2.1) and letting :

Another way to do this is to take and in the definition (1.2.1) of the Racah polynomials and then take the limit . In that case we obtain the Hahn polynomials given by (1.5.1) in the following way :

Racah Dual Hahn.

  
If we take and let in (1.2.1), then we obtain the dual Hahn polynomials from the Racah polynomials. So we have

And if we take and let in (1.2.1), then we also obtain the dual Hahn polynomials :

Finally, if we take and in the definition (1.2.1) of the Racah polynomials and take the limit we find the dual Hahn polynomials given by (1.6.1) in the following way :