AI 中文总结
本文推广拉马努金-$q$-艾里连接公式至单参数变形，通过$q$-波雷尔反演和离散求和，给出整函数无穷远处的发散渐近展开及连接系数，并证明余项估计。
AI 中文摘要
拉马努金-$q$-艾里连接公式将原点处的收敛级数与拉马努金整函数在无穷远处的行为联系起来。我们将此连接推广到一个单参数变形，该变形将拉马努金（二阶）$q$-差分算子嵌入到三阶方程族中。通过作为$q$-波雷尔反演的围道积分，我们给出了无穷远处的行为，其形式为发散局部展开式，连接系数唯一确定。离散$q$-波雷尔-拉普拉斯求和产生一个收敛的双边幂级数表示，其对求和路径的依赖性反映了$q$-斯托克斯现象。我们证明了余项估计，确立了无穷远处形式上一般发散的展开式作为该整函数的渐近描述。
英文摘要
The Ramanujan$-$$q$-Airy connection formula relates the convergent series at the origin to the behaviour at infinity of the Ramanujan entire function. We extend this connection to a one-parameter deformation, which embeds the Ramanujan (second-order) $q$-difference operator in a family of third-order equations. By contour integral as $q$-Borel inversion, we give behaviour at infinity in terms of divergent local expansions with connection coefficients uniquely determined. Discrete $q$-Borel$-$Laplace summation yields a convergent, bilateral power series representation, whose dependence on summation path reflects the $q$-Stokes phenomenon. We prove remainder estimates establishing the formal, generally divergent, expansion at infinity as an asymptotic description of the entire function.
