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We use two different $q$-Borel-Laplace resummation methods to obtain our connection formulae. We also introduce the $q$-Borel-Laplace transformation of level $r-1$, which are higher order extension of these transformations. These methods are useful to obtain an asymptotic formula of a divergent series ${}_r\\varphi_0(0,0,\\dots ,0;-;q,x)$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1404.2541","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-04-09T16:54:20Z","cross_cats_sorted":[],"title_canon_sha256":"19b4f4b76805246e4f9202e97c179b17e18a7c0ac469c2af7ed8452c6d208f14","abstract_canon_sha256":"54a05e250656b4fae6c3c869fc8f465aaec29e1eb92da65725df93d4265c7d55"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:31.270544Z","signature_b64":"t/EBbrtAEUkTfyX0Ih60CFWwy98EGFLXjvTYtLrylQgfGrBB1lqHtl/+hqevh78FGk05Mqjof3vctI4m18ilBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec2f4582ab38a7273073693b26cc22ab89032cbbbb61da54d0f9da12c45d6b43","last_reissued_at":"2026-05-18T02:54:31.269881Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:31.269881Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Stokes phenomenon for the Ramanujan's $q$-difference equation and its higher order extension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Takeshi Morita","submitted_at":"2014-04-09T16:54:20Z","abstract_excerpt":"We show connection formulae of local solutions of the Ramanujan equation between the origin and the infinity. 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