{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:AWJB77MP4FGFPDHT6J6TUWFA6T","short_pith_number":"pith:AWJB77MP","canonical_record":{"source":{"id":"2606.05329","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CA","submitted_at":"2026-06-03T18:15:47Z","cross_cats_sorted":[],"title_canon_sha256":"965dc92594a0ace41406562f1892c320d04f25e7f115fa30b923a26ef45a2aae","abstract_canon_sha256":"1b640658dd27467059c6d9c9d41f26f1cc2187de4dc9c0d921a6536a5d70f484"},"schema_version":"1.0"},"canonical_sha256":"05921ffd8fe14c578cf3f27d3a58a0f4d0c6f9ed4f851d0cc1bc331d4d19c269","source":{"kind":"arxiv","id":"2606.05329","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.05329","created_at":"2026-06-05T00:13:54Z"},{"alias_kind":"arxiv_version","alias_value":"2606.05329v1","created_at":"2026-06-05T00:13:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.05329","created_at":"2026-06-05T00:13:54Z"},{"alias_kind":"pith_short_12","alias_value":"AWJB77MP4FGF","created_at":"2026-06-05T00:13:54Z"},{"alias_kind":"pith_short_16","alias_value":"AWJB77MP4FGFPDHT","created_at":"2026-06-05T00:13:54Z"},{"alias_kind":"pith_short_8","alias_value":"AWJB77MP","created_at":"2026-06-05T00:13:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:AWJB77MP4FGFPDHT6J6TUWFA6T","target":"record","payload":{"canonical_record":{"source":{"id":"2606.05329","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CA","submitted_at":"2026-06-03T18:15:47Z","cross_cats_sorted":[],"title_canon_sha256":"965dc92594a0ace41406562f1892c320d04f25e7f115fa30b923a26ef45a2aae","abstract_canon_sha256":"1b640658dd27467059c6d9c9d41f26f1cc2187de4dc9c0d921a6536a5d70f484"},"schema_version":"1.0"},"canonical_sha256":"05921ffd8fe14c578cf3f27d3a58a0f4d0c6f9ed4f851d0cc1bc331d4d19c269","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-05T00:13:54.134173Z","signature_b64":"qwwyzmWwrWU0JzKT7k6X+Qa3xIebvksHgnkraCTUv0JmOY8QMgenqQol6lqRYwq6RR/q+VF+p7Spa6GxtrB/Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"05921ffd8fe14c578cf3f27d3a58a0f4d0c6f9ed4f851d0cc1bc331d4d19c269","last_reissued_at":"2026-06-05T00:13:54.133556Z","signature_status":"signed_v1","first_computed_at":"2026-06-05T00:13:54.133556Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.05329","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-05T00:13:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"95/LUFaabjqbpZuMCxPs0SSJ02EFDB0nXGi1TBGUDmo2PyXxF4GAkjqH2DIRPVA9cPk83WfEEGXeG8R/UFYKDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T10:40:22.532435Z"},"content_sha256":"686e623ca6eee6ba1b320f29fe7292d606a8fedee80775204b182b7b9d3871f7","schema_version":"1.0","event_id":"sha256:686e623ca6eee6ba1b320f29fe7292d606a8fedee80775204b182b7b9d3871f7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:AWJB77MP4FGFPDHT6J6TUWFA6T","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Macdonald-type function and its relation with index transforms and orthogonal polynomials","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Semyon Yakubovich","submitted_at":"2026-06-03T18:15:47Z","abstract_excerpt":"We continue to investigate properties of the function $M_\\nu(z)$ which is associated with the Macdonald function $K_\\nu(z)$ in terms of the corresponding Fourier integral. In particular, recurrence relations for this function and its derivatives are obtained, involving properties of the associated Laguerre polynomials. Multiple orthogonal polynomials related to the scaled Macdonald-type weights $ \\hat{\\rho}_{\\nu}(x)= 2 x^{\\nu/2} M_\\nu\\left(2\\sqrt x\\right), x >0$ are investigated."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05329","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.05329/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-05T00:13:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bFspIuPEeKtZssRVZKRPTDmz2JP6Wbty89S5r7AEdw3P0XkCM+q2+AthZ/Gx9uUDZp5+GqZ4Xs4x5A3jqD0MCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T10:40:22.532826Z"},"content_sha256":"355bc5516f571fe86c0b2898f326369ba08fd84bb6c891019d489ee98e1aa54e","schema_version":"1.0","event_id":"sha256:355bc5516f571fe86c0b2898f326369ba08fd84bb6c891019d489ee98e1aa54e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AWJB77MP4FGFPDHT6J6TUWFA6T/bundle.json","state_url":"https://pith.science/pith/AWJB77MP4FGFPDHT6J6TUWFA6T/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AWJB77MP4FGFPDHT6J6TUWFA6T/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-26T10:40:22Z","links":{"resolver":"https://pith.science/pith/AWJB77MP4FGFPDHT6J6TUWFA6T","bundle":"https://pith.science/pith/AWJB77MP4FGFPDHT6J6TUWFA6T/bundle.json","state":"https://pith.science/pith/AWJB77MP4FGFPDHT6J6TUWFA6T/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AWJB77MP4FGFPDHT6J6TUWFA6T/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:AWJB77MP4FGFPDHT6J6TUWFA6T","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1b640658dd27467059c6d9c9d41f26f1cc2187de4dc9c0d921a6536a5d70f484","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CA","submitted_at":"2026-06-03T18:15:47Z","title_canon_sha256":"965dc92594a0ace41406562f1892c320d04f25e7f115fa30b923a26ef45a2aae"},"schema_version":"1.0","source":{"id":"2606.05329","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.05329","created_at":"2026-06-05T00:13:54Z"},{"alias_kind":"arxiv_version","alias_value":"2606.05329v1","created_at":"2026-06-05T00:13:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.05329","created_at":"2026-06-05T00:13:54Z"},{"alias_kind":"pith_short_12","alias_value":"AWJB77MP4FGF","created_at":"2026-06-05T00:13:54Z"},{"alias_kind":"pith_short_16","alias_value":"AWJB77MP4FGFPDHT","created_at":"2026-06-05T00:13:54Z"},{"alias_kind":"pith_short_8","alias_value":"AWJB77MP","created_at":"2026-06-05T00:13:54Z"}],"graph_snapshots":[{"event_id":"sha256:355bc5516f571fe86c0b2898f326369ba08fd84bb6c891019d489ee98e1aa54e","target":"graph","created_at":"2026-06-05T00:13:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.05329/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We continue to investigate properties of the function $M_\\nu(z)$ which is associated with the Macdonald function $K_\\nu(z)$ in terms of the corresponding Fourier integral. In particular, recurrence relations for this function and its derivatives are obtained, involving properties of the associated Laguerre polynomials. Multiple orthogonal polynomials related to the scaled Macdonald-type weights $ \\hat{\\rho}_{\\nu}(x)= 2 x^{\\nu/2} M_\\nu\\left(2\\sqrt x\\right), x >0$ are investigated.","authors_text":"Semyon Yakubovich","cross_cats":[],"headline":"","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CA","submitted_at":"2026-06-03T18:15:47Z","title":"On the Macdonald-type function and its relation with index transforms and orthogonal polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05329","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:686e623ca6eee6ba1b320f29fe7292d606a8fedee80775204b182b7b9d3871f7","target":"record","created_at":"2026-06-05T00:13:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1b640658dd27467059c6d9c9d41f26f1cc2187de4dc9c0d921a6536a5d70f484","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CA","submitted_at":"2026-06-03T18:15:47Z","title_canon_sha256":"965dc92594a0ace41406562f1892c320d04f25e7f115fa30b923a26ef45a2aae"},"schema_version":"1.0","source":{"id":"2606.05329","kind":"arxiv","version":1}},"canonical_sha256":"05921ffd8fe14c578cf3f27d3a58a0f4d0c6f9ed4f851d0cc1bc331d4d19c269","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"05921ffd8fe14c578cf3f27d3a58a0f4d0c6f9ed4f851d0cc1bc331d4d19c269","first_computed_at":"2026-06-05T00:13:54.133556Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-05T00:13:54.133556Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qwwyzmWwrWU0JzKT7k6X+Qa3xIebvksHgnkraCTUv0JmOY8QMgenqQol6lqRYwq6RR/q+VF+p7Spa6GxtrB/Aw==","signature_status":"signed_v1","signed_at":"2026-06-05T00:13:54.134173Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.05329","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:686e623ca6eee6ba1b320f29fe7292d606a8fedee80775204b182b7b9d3871f7","sha256:355bc5516f571fe86c0b2898f326369ba08fd84bb6c891019d489ee98e1aa54e"],"state_sha256":"e90b48ef37d5f6fd7434e075d26f2c648dff96039c5df86e98f634233437909f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hx+TdS6H7l/FEl73xc/KJLcV462fRZ6YzlgcK6FDgrah1YFUrAtYOcvl6irrkcKZQ6TFpBL5bwXSoMZ2a4S1CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T10:40:22.534790Z","bundle_sha256":"e5061d3e766d5c186fd756e5c08073b552d056dcd04e3b271197d34c3abdd92e"}}