On gamma factors of generic representations of ${\mathrm{U}}_{2n+1}\times {\mathrm{R}\mathrm{e}\mathrm{s}}_{E\mathrm{/}F}{\mathrm{G}\mathrm{L}}_r$

In this article we prove that the gamma factors attached to generic representations of ${\mathrm{U}}_{2n+1}\times {\mathrm{R}\mathrm{e}\mathrm{s}}_{E\mathrm{/}F}{\mathrm{G}\mathrm{L}}_r$ over a local field $F$ defined by various approaches coincide when $F$ is non-archimedean and under additional assumptions on $n,r$ and $E$ when $F$ is archimedean.