{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:HJXGWNPJVYUTN2JTIJSIKWD5JZ","short_pith_number":"pith:HJXGWNPJ","canonical_record":{"source":{"id":"1112.6031","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-12-27T23:13:42Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"452c7c559249573c1ce44aaf875380aced0578ad8412ac3a1164f3bc5861d330","abstract_canon_sha256":"c8da01c85191a8ee890f00ee84b1dd344e7fe79b612f7f0e672df823b6b57eb8"},"schema_version":"1.0"},"canonical_sha256":"3a6e6b35e9ae2936e933426485587d4e705a2d7b71268057d4463ad69f97679d","source":{"kind":"arxiv","id":"1112.6031","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.6031","created_at":"2026-05-18T02:23:35Z"},{"alias_kind":"arxiv_version","alias_value":"1112.6031v2","created_at":"2026-05-18T02:23:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.6031","created_at":"2026-05-18T02:23:35Z"},{"alias_kind":"pith_short_12","alias_value":"HJXGWNPJVYUT","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"HJXGWNPJVYUTN2JT","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"HJXGWNPJ","created_at":"2026-05-18T12:26:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:HJXGWNPJVYUTN2JTIJSIKWD5JZ","target":"record","payload":{"canonical_record":{"source":{"id":"1112.6031","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-12-27T23:13:42Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"452c7c559249573c1ce44aaf875380aced0578ad8412ac3a1164f3bc5861d330","abstract_canon_sha256":"c8da01c85191a8ee890f00ee84b1dd344e7fe79b612f7f0e672df823b6b57eb8"},"schema_version":"1.0"},"canonical_sha256":"3a6e6b35e9ae2936e933426485587d4e705a2d7b71268057d4463ad69f97679d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:23:35.905154Z","signature_b64":"apwbr526Xe3UYZdlM67L6Q/cYHbK2i7QZ/5t+4r+CYeuBJvGZ9w1i84roNKwtdRJXniC5i2l6XD7LE0H7wjqCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a6e6b35e9ae2936e933426485587d4e705a2d7b71268057d4463ad69f97679d","last_reissued_at":"2026-05-18T02:23:35.904416Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:23:35.904416Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.6031","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:23:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jojvOfMNgypw/Mfu3inNDyjbijiVcAWN6wxXWjZ+TvHAGPUKrlk+Qk5CNiKBQw00LOGrSZqYmO4+BCDsgAmRBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T13:02:01.215676Z"},"content_sha256":"a1eb27ef3fef3f3d58d68f1cc9b2d841b76b1fa577782dbe21ccd6f502cf00dd","schema_version":"1.0","event_id":"sha256:a1eb27ef3fef3f3d58d68f1cc9b2d841b76b1fa577782dbe21ccd6f502cf00dd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:HJXGWNPJVYUTN2JTIJSIKWD5JZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mellin Transforms of the Generalized Fractional Integrals and Derivatives","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Udita N. Katugampola","submitted_at":"2011-12-27T23:13:42Z","abstract_excerpt":"We obtain the Mellin transforms of the generalized fractional integrals and derivatives that generalize the Riemann-Liouville and the Hadamard fractional integrals and derivatives. We also obtain interesting results, which combine generalized $\\delta_{r,m}$ operators with generalized Stirling numbers and Lah numbers. For example, we show that $\\delta_{1,1}$ corresponds to the Stirling numbers of the $2^{nd}$ kind and $\\delta_{2,1}$ corresponds to the unsigned Lah numbers. Further, we show that the two operators $\\delta_{r,m}$ and $\\delta_{m,r}$, $r,m\\in\\mathbb{N}$, generate the same sequence g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.6031","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T02:23:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y0VVTH428/XLsh9cs9/b/cLiDBcG8UO1Ut0HXM9CEbY6+PL0NFtpVvC3tmIrNb3Ulmx3SacJu+bdfUIzoBuQCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T13:02:01.216374Z"},"content_sha256":"ea9e2a91ca81bd1a4896aa6393daea38104fa7c00f20095ff070033768334936","schema_version":"1.0","event_id":"sha256:ea9e2a91ca81bd1a4896aa6393daea38104fa7c00f20095ff070033768334936"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HJXGWNPJVYUTN2JTIJSIKWD5JZ/bundle.json","state_url":"https://pith.science/pith/HJXGWNPJVYUTN2JTIJSIKWD5JZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HJXGWNPJVYUTN2JTIJSIKWD5JZ/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":"2026-05-27T13:02:01Z","links":{"resolver":"https://pith.science/pith/HJXGWNPJVYUTN2JTIJSIKWD5JZ","bundle":"https://pith.science/pith/HJXGWNPJVYUTN2JTIJSIKWD5JZ/bundle.json","state":"https://pith.science/pith/HJXGWNPJVYUTN2JTIJSIKWD5JZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HJXGWNPJVYUTN2JTIJSIKWD5JZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:HJXGWNPJVYUTN2JTIJSIKWD5JZ","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":"c8da01c85191a8ee890f00ee84b1dd344e7fe79b612f7f0e672df823b6b57eb8","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-12-27T23:13:42Z","title_canon_sha256":"452c7c559249573c1ce44aaf875380aced0578ad8412ac3a1164f3bc5861d330"},"schema_version":"1.0","source":{"id":"1112.6031","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.6031","created_at":"2026-05-18T02:23:35Z"},{"alias_kind":"arxiv_version","alias_value":"1112.6031v2","created_at":"2026-05-18T02:23:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.6031","created_at":"2026-05-18T02:23:35Z"},{"alias_kind":"pith_short_12","alias_value":"HJXGWNPJVYUT","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"HJXGWNPJVYUTN2JT","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"HJXGWNPJ","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:ea9e2a91ca81bd1a4896aa6393daea38104fa7c00f20095ff070033768334936","target":"graph","created_at":"2026-05-18T02:23:35Z","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"},"paper":{"abstract_excerpt":"We obtain the Mellin transforms of the generalized fractional integrals and derivatives that generalize the Riemann-Liouville and the Hadamard fractional integrals and derivatives. We also obtain interesting results, which combine generalized $\\delta_{r,m}$ operators with generalized Stirling numbers and Lah numbers. For example, we show that $\\delta_{1,1}$ corresponds to the Stirling numbers of the $2^{nd}$ kind and $\\delta_{2,1}$ corresponds to the unsigned Lah numbers. Further, we show that the two operators $\\delta_{r,m}$ and $\\delta_{m,r}$, $r,m\\in\\mathbb{N}$, generate the same sequence g","authors_text":"Udita N. Katugampola","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-12-27T23:13:42Z","title":"Mellin Transforms of the Generalized Fractional Integrals and Derivatives"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.6031","kind":"arxiv","version":2},"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:a1eb27ef3fef3f3d58d68f1cc9b2d841b76b1fa577782dbe21ccd6f502cf00dd","target":"record","created_at":"2026-05-18T02:23:35Z","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":"c8da01c85191a8ee890f00ee84b1dd344e7fe79b612f7f0e672df823b6b57eb8","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-12-27T23:13:42Z","title_canon_sha256":"452c7c559249573c1ce44aaf875380aced0578ad8412ac3a1164f3bc5861d330"},"schema_version":"1.0","source":{"id":"1112.6031","kind":"arxiv","version":2}},"canonical_sha256":"3a6e6b35e9ae2936e933426485587d4e705a2d7b71268057d4463ad69f97679d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3a6e6b35e9ae2936e933426485587d4e705a2d7b71268057d4463ad69f97679d","first_computed_at":"2026-05-18T02:23:35.904416Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:23:35.904416Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"apwbr526Xe3UYZdlM67L6Q/cYHbK2i7QZ/5t+4r+CYeuBJvGZ9w1i84roNKwtdRJXniC5i2l6XD7LE0H7wjqCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:23:35.905154Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.6031","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a1eb27ef3fef3f3d58d68f1cc9b2d841b76b1fa577782dbe21ccd6f502cf00dd","sha256:ea9e2a91ca81bd1a4896aa6393daea38104fa7c00f20095ff070033768334936"],"state_sha256":"24748c8bdfae7c68e66c677a155c43c7b53c26a8f80070faa3e5c479afb5a094"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LIIt44+0DNw6GUQHVy0PVMBXlMcuncwFByXbHaLaAjcsvU1WHoezmA1G9RivIhmOu9y1fJ6dH4NK4/eirbkKCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T13:02:01.220231Z","bundle_sha256":"004a59d0311804e03948c23d60378770be936d8924b13e619d2965e78da83660"}}