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The main idea is to introduce a shift parameter in the linear denominator and consider \\[\n  \\Phi_{m,\\varepsilon}(\\lambda;\\alpha)\n  =\n  \\sum_{n=0}^{\\infty}\\varepsilon^n\n  \\frac{a_n(\\alpha)}{n+m+1+\\lambda},\n  \\qquad \\varepsilon\\in\\{1,-1\\}. \\] Expanding this expression in powers of \\(\\lambda\\) produces sums with denominator powers \\((n+m+1)^{-K}\\). We first discuss analytic interpolation in the denominator exponent and explain why positive integer exponents lead to t"},"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":"2607.10135","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-11T05:49:28Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"6bae0eccd3a7b618e538b46b1b6a3b9baf5e447a51854a1d30093e0e617890c8","abstract_canon_sha256":"d9ad644972da51d6cee03961e3b77b40195a71b8e6b6189589d93dac560da292"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T01:20:27.070540Z","signature_b64":"thgbHw6sIwmaGhawMDkkAGC5WMLdC09gQbaPUU7WeYSfTMLNNJIyeq4VTSjUNrZc2BSDljWULG/kCQU6ior6CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73adaac220c6776c1a27d8e21b30e525c306fb7cf495356992ea5aa4e4d16c9e","last_reissued_at":"2026-07-14T01:20:27.069719Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T01:20:27.069719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a Class of Hypergeometric Sums via Recurrences and Product Binomial-Harmonic Identities","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Narendra Bhandari","submitted_at":"2026-07-11T05:49:28Z","abstract_excerpt":"We study hypergeometric series with coefficients \\(\n  a_n(\\alpha)=\\frac{(\\alpha)_n(1-\\alpha)_n}{(n!)^2},\\) where \\(0< \\alpha < 1\\). The main idea is to introduce a shift parameter in the linear denominator and consider \\[\n  \\Phi_{m,\\varepsilon}(\\lambda;\\alpha)\n  =\n  \\sum_{n=0}^{\\infty}\\varepsilon^n\n  \\frac{a_n(\\alpha)}{n+m+1+\\lambda},\n  \\qquad \\varepsilon\\in\\{1,-1\\}. \\] Expanding this expression in powers of \\(\\lambda\\) produces sums with denominator powers \\((n+m+1)^{-K}\\). 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