Feynman integral in ${\mathbb{R}}^1\oplus {\mathbb{R}}^m$ and complex expansion of ${}_2{F}_1$

Closed form expressions are proposed for the Feynman integral $$I_{D,m}(p,q)=\int \frac{d^{m}y}{(2\pi {)}^m}\int \frac{d^{D}x}{(2\pi {)}^D}\frac{1}{(x-p\mathrm{/}2{)}^2+(y-q\mathrm{/}2{)}^4}\frac{1}{(x+p\mathrm{/}2{)}^2+(y+q\mathrm{/}2{)}^4}$$ over $d=D+m$ dimensional space with $(x,y),,(p,q)\in \mathbb R^D \oplus \mathbb R^m$,inthespecialcase$D=1$\mathrm{.}Weshowthat$I_{1,m}(p,q)$ can be expressed in different forms involving real and imaginary parts of the complex variable Gauss hypergeometric function ${}_2{F}_1$, as well as generalized hypergeometric ${}_2{F}_2$ and ${}_3{F}_2$, Horn $H_4$ and Appell $F_2$ functions. Several interesting relations are derived between the real and imaginary parts of ${}_2{F}_1$ and the function $H_4$.