Archimedean zeta integrals on SO₂ₙ × GLₙ and Sp₂ₙ × GLₙ

We evaluate the archimedean Rankin-Selberg zeta integrals associated with spherical principal series representations of SO₂ₙ( R) × GLₙ( R) and Sp₂ₙ( R) × GLₙ( R), where SO₂ₙ is split and n≥2. We obtain explicit formulas expressing these integrals as quotients of products of gamma functions. Our approach uses Mellin-Barnes representations of spherical Whittaker functions, together with new integral identities for their Mellin transforms and transformations of Barnes integrals. In the even orthogonal case, we establish a recurrence in the rank and evaluate the integral by induction. The symplectic case then follows from an explicit identity relating the two families of zeta integrals.