Bilateral discrete and continuous orthogonality relations in the q⁻¹-symmetric Askey scheme

In the q⁻¹ q - 1 -symmetric Askey scheme, namely the q⁻¹ q - 1 -Askey–Wilson, continuous dual q⁻¹ q - 1 -Hahn, q⁻¹ q - 1 -Al-Salam–Chihara, continuous big q⁻¹ q - 1 -Hermite and continuous q⁻¹ q - 1 -Hermite polynomials, we compute bilateral discrete and continuous orthogonality relations. We also derive a q -beta integral which comes from the continuous orthogonality relation for the q⁻¹ q - 1 -Askey–Wilson polynomials. In the q→ 1⁻ q → 1 - limit, this q -beta integral corresponds to a beta integral of Ramanujan-type which we present and provide two proofs for.