Asymptotic expansion for the hypergeometric-integral kernel in the small-argument regime

Derive an asymptotic expansion for the integral kernel I(m,x) when x is much smaller than T^{-2+\epsilon}, a regime in which the argument of the associated confluent hypergeometric functions remains bounded by T^{\epsilon}.

Background

The paper analyzes the off-diagonal term $\Sigma_2(r;h)$ arising in the twisted second moment of Maass-form symmetric-square $L$-functions. The analysis depends on the behavior of the integral transform $I(m,x)$, whose parameter $x$ is divided into distinct ranges.

For $x$ near or above the transition scale, the authors use asymptotic formulas or direct estimates for the relevant hypergeometric functions. However, in the smaller range $x\ll T^{-2+\epsilon}$, the argument of the confluent hypergeometric functions does not become sufficiently large for the asymptotic expansion used elsewhere. The authors therefore replace the unavailable expansion with a bound obtained by expressing the relevant confluent hypergeometric functions in terms of modified Bessel functions. Establishing an asymptotic expansion in this regime would provide more precise information about $I(m,x)$ beyond the estimates used in the paper.