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The Askey-Wilson algebra is realized in terms of the elements of the quantumalgebras $U_q(\mathfrak{su}(2))$ or $U_q(\mathfrak{su}(1,1))$. A newrealization of the Racah algebra in terms of the Lie algebras$\mathfrak{su}(2)$ or $\mathfrak{su}(1,1)$ is given also. Details for differentspecializations are provided. The advantage of these new realizations is thatone generator of the Askey-Wilson (or Racah) algebra becomes diagonal in theusual representation of the quantum algebras whereas the second one istridiagonal. This allows to recover easily the recurrence relations of theassociated orthogonal polynomials of the Askey scheme. These realizationsinvolve rational functions of the Cartan generator of the quantum algebras,they are linear with respect to the other generators and depend on the Casimirelement of the quantum algebras.

New realizations of algebras of the Askey-Wilson type in terms of Lie and quantum algebras