New integral representations for a hypergeometric function (Q2754804)

The authors obtain an integral representation of the function NEWLINE\[NEWLINE \varphi_\lambda^{(\alpha ,\beta)}(r)=F\left( \frac{\alpha +\beta +1+i\lambda }{2},\frac{\alpha +\beta +1-i\lambda }{2};\alpha +1; -\sinh^2 r\right),\;r>0, NEWLINE\]NEWLINE where \(F\) is the Gauss hypergeometric function, \(\lambda \in \mathbb C\), \(\beta >-1/2\), \(\alpha -\beta\in \mathbb Z_+\). An asymptotic expansion of \(\varphi_\lambda^{(n,1)}(r)\) (\(n\in \mathbb N\)) for \(\lambda \to \infty\) is also found. These functions can be interpreted as generalized spherical functions on some non-compact symmetric spaces.