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The conjectural newform theory for generic representations of $p$-adic ${\rmSO}_{2n+1}$ was formulated by P.-Y. Tsai in her thesis in which Tsai alsoverified the conjecture when the representations are supercuspidal. The mainpurpose of this work is to compute the Rankin-Selberg integrals for ${\rmSO}_{2n+1}\times{\rm GL}_r$ with $1\le r\le n$ attached to newforms and alsooldforms under the validity of the conjecture.

Rankin-Selberg integrals for ${\rm SO}_{2n+1}\times{\rm GL}_r$ attached to Newforms and Oldforms