# Archimedean zeta integrals on $GL_n \times GL_m$ and $SO_{2n+1} \times GL_m$

@article{Ishii2011ArchimedeanZI, title={Archimedean zeta integrals on \$GL\_n \times GL\_m\$ and \$SO\_\{2n+1\} \times GL\_m\$}, author={Taku Ishii and Eric Stade}, journal={arXiv: Number Theory}, year={2011} }

In this paper, we evaluate archimedean zeta integrals for automorphic $L$-functions on $GL_n \times GL_{n-1+\ell}$ and on $ SO_{2n+1} \times GL_{n+\ell}$, for $\ell=-1$, $0$, and $1$. In each of these cases, the zeta integrals in question may be expressed as Mellin transforms of products of class one Whittaker functions. Here, we obtain explicit expressions for these Mellin transforms in terms of Gamma functions and Barnes integrals.

## One Citation

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We demonstrate how a known Whittaker function integral identity arises from the t = 0 and q! 1 limit of the Macdonald polynomial eigenrelation satisfied by Noumi's q-integral operator.

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