Calculus of archimedean Rankin–Selberg integrals with recurrence relations

Let n n and alttext="n prime"> ′ encoding="application/x-tex">n’ be positive integers such that minus n prime element-of StartSet 0 comma 1 EndSet"> − ∈<!-- ∈ <mml:mo fence="false" stretchy="false">{ 0 , 1 stretchy="false">} encoding="application/x-tex">n-n’\in \{0,1\} . alttext="upper F"> F encoding="application/x-tex">F either alttext="double-struck upper R"> R encoding="application/x-tex">\mathbb {R} or C"> mathvariant="double-struck">C {C} K Subscript n"> K encoding="application/x-tex">K_n encoding="application/x-tex">K_{n’} maximal compact subgroups of alttext="normal G normal L left-parenthesis F right-parenthesis"> mathvariant="normal">G mathvariant="normal">L stretchy="false">( stretchy="false">) encoding="application/x-tex">\mathrm {GL}(n,F) {GL}(n’,F) , respectively. We give the explicit descriptions archimedean Rankin–Selberg integrals at minimal - -types for pairs principal series representations using their recurrence relations. Our results equals double-struck = encoding="application/x-tex">F=\mathbb can applied to arithmetic study critical values automorphic L"> L encoding="application/x-tex">L -functions.