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Ferrar and gives as by-products, transformation formulas of the form $F(z,\\alpha)=F(iz,\\beta)$, where $\\alpha\\beta=1$. As particular examples, we derive an extended version of the general theta transformation formula and generalizations of certain formulas of Ferrar and Hardy. A one-variable generalization of a well-known identity of Ramanujan is also given. We conclude wit"},"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":"1111.4799","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-11-21T09:46:25Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"3064ace45958e419e9bfd192d2a41ab2a6add60b2ed93337cc6c22bf5da5c164","abstract_canon_sha256":"aa9d8fafa378da5b1bf14af6496793048c172015dbf26c37437cedffc3ebfb64"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:07:55.052970Z","signature_b64":"jEwVFbfcg4/g9RFcr/JAipy+5t5+2SrFsAYi6SxIo9X4NlhSpw7XxaBe0ThuYJNyW9SpRXosWoVDc7NLA6S1BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"84ba50ac1b72c752aad2d8de5001397dbdb5cc8b5ed96cdcfb2ec5658eae56d1","last_reissued_at":"2026-05-18T04:07:55.052217Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:07:55.052217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Analogues of the general theta transformation formula","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Atul Dixit","submitted_at":"2011-11-21T09:46:25Z","abstract_excerpt":"A new class of integrals involving the confluent hypergeometric function ${}_1F_{1}(a;c;z)$ and the Riemann $\\Xi$-function is considered. It generalizes a class containing some integrals of S. Ramanujan, G.H. Hardy and W.L. Ferrar and gives as by-products, transformation formulas of the form $F(z,\\alpha)=F(iz,\\beta)$, where $\\alpha\\beta=1$. As particular examples, we derive an extended version of the general theta transformation formula and generalizations of certain formulas of Ferrar and Hardy. A one-variable generalization of a well-known identity of Ramanujan is also given. 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