Dual addition formulas associated with dual product formulas

We observe that the linearization coefficients for Gegenbauer polynomials are the orthogonality weights for Racah polynomials with special parameters. Then it turns out that the linearization sum with such a Racah polynomial as extra factor inserted, can also be evaluated. The corresponding Fourier-Racah expansion is an addition-type formula which is dual to the well-known addition formula for Gegenbauer polynomials. The limit to the case of Hermite polynomials of this dual addition formula is also considered. Similar results as for Gegenbauer polynomials, although only formal, are given by taking the Ruijsenaars- Hallnás dual product formula for Gegenbauer functions as a starting point and by working with Wilson polynomials.