Degenerations of the elliptic analogue of Euler–Gauss hypergeometric function

The elliptic analogue of Euler–Gauss hypergeometric function depends on two complex moduli $p$ and $q$ and seven parameters $t_j$. We consider its various degeneration limits. For fixed parameters, in the simplest $p=0$ limit one gets Rahman's (unit circle) integral representation for a combination of two non-terminating ${}_{10}\varphi_9$ $q$-hypergeometric series. In the limit, when all $p$, $q$ and $t_j$ simulataneously tend to $1$, one gets the hyperbolic (Mellin–Barnes type) integral expressed in terms of Faddeev's quantum dilogarithm (the hyperbolic gamma function). Both these limiting integrals are degenerated for $q=1$ to special ${}_9F_8$ hypergeometric functions. However, the hyperbolic integrals have more intricate limits leading to infinite bilateral sums of the plain hypergeometric integrals, which will be a main emphasis of the talk. These are the Mellin–Barnes representations of complex hypergeometric functions related to principal series representations of the group $\mathrm{SL}(2,\mathbb{C})$. Thus, all known forms of classical hypergeometric functions are unified at the elliptic level to one unique object. The talk is partially based on recent joint works with Gor Sarkissian.