A connection formula between the Ramanujan function and the $q$-Airy function

We show a connection formula between two different $q$-Airy functions. One is called the Ramanujan function which appears in Ramanujan's "Lost notebook". Another one is called the $q$-Airy function that obtained in the study of the second $q$-Painlevé equation. We use the $q$-Borel transformation and the $q$-Laplace transformation following C. Zhang to obtain the connection formula.

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