A Relativistic Conical Function and its Whittaker Limits

In previous work we introduced and studied a function $R(a_{+},a_{-},c;v,\hat{v})$ that is a generalization of the hypergeometric function ${}_2F_1$ and the Askey–Wilson polynomials. When the coupling vector $c\in\mathbb C^4$ is specialized to $(b,0,0,0)$, $b\in\mathbb C$, we obtain a function $\mathcal R (a_{+},a_{-},b;v,2\hat{v})$ that generalizes the conical function specialization of ${}_2F_1$ and the $q$-Gegenbauer polynomials. The function $\mathcal R$ is the joint eigenfunction of four analytic difference operators associated with the relativistic Calogero–Moser system of $A_1$ type, whereas the function $R$ corresponds to $BC_1$, and is the joint eigenfunction of four hyperbolic Askey–Wilson type difference operators. We show that the $\mathcal R$-function admits five novel integral representations that involve only four hyperbolic gamma functions and plane waves. Taking their nonrelativistic limit, we arrive at four representations of the conical function. We also show that a limit procedure leads to two commuting relativistic Toda Hamiltonians and two commuting dual Toda Hamiltonians, and that a similarity transform of the function $\mathcal R$ converges to a joint eigenfunction of the latter four difference operators.