{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:I7QOK6FQATBK2W5CYB6DSVKVIM","short_pith_number":"pith:I7QOK6FQ","canonical_record":{"source":{"id":"1303.3978","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-03-16T13:49:15Z","cross_cats_sorted":[],"title_canon_sha256":"d9bdfdbd4916270fd6664a6fc6b64a12ad71dfd73e15b22a100cbc7525670258","abstract_canon_sha256":"f801934a916827dce3ce6930429c9ae32a7ed28ff560054dab89064d9d435d7f"},"schema_version":"1.0"},"canonical_sha256":"47e0e578b004c2ad5ba2c07c3955554314f324e33c4c43e9deb5bd7c457b640a","source":{"kind":"arxiv","id":"1303.3978","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.3978","created_at":"2026-05-18T03:30:40Z"},{"alias_kind":"arxiv_version","alias_value":"1303.3978v1","created_at":"2026-05-18T03:30:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.3978","created_at":"2026-05-18T03:30:40Z"},{"alias_kind":"pith_short_12","alias_value":"I7QOK6FQATBK","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"I7QOK6FQATBK2W5C","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"I7QOK6FQ","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:I7QOK6FQATBK2W5CYB6DSVKVIM","target":"record","payload":{"canonical_record":{"source":{"id":"1303.3978","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-03-16T13:49:15Z","cross_cats_sorted":[],"title_canon_sha256":"d9bdfdbd4916270fd6664a6fc6b64a12ad71dfd73e15b22a100cbc7525670258","abstract_canon_sha256":"f801934a916827dce3ce6930429c9ae32a7ed28ff560054dab89064d9d435d7f"},"schema_version":"1.0"},"canonical_sha256":"47e0e578b004c2ad5ba2c07c3955554314f324e33c4c43e9deb5bd7c457b640a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:40.309526Z","signature_b64":"2CX/FGjQsqE0+NE7TXk7DDp3S2/vZOIv9jHI8NwFGfFEMg40zxSG85FG9biLuBafEmrlqSXyEI8LKsIMbVEiCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47e0e578b004c2ad5ba2c07c3955554314f324e33c4c43e9deb5bd7c457b640a","last_reissued_at":"2026-05-18T03:30:40.308652Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:40.308652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1303.3978","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":"2026-05-18T03:30:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a8Gnrro+aY+QKeoZ0oKxhaGfOgeITn9n5J2DaNR/4WsrzBBmQ/wXYaLVxEGjwhOr7rjpJ41uPySYb/DvSpjKCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T09:24:22.526785Z"},"content_sha256":"76684a7bd0d72c4c27102742abac20ff7b8a1e8f65b4423d2cce2fb13ee056bd","schema_version":"1.0","event_id":"sha256:76684a7bd0d72c4c27102742abac20ff7b8a1e8f65b4423d2cce2fb13ee056bd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:I7QOK6FQATBK2W5CYB6DSVKVIM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Erdelyi-Kober Fractional Integral Operators from a Statistical Perspective -I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"A.M. Mathai, H.J. Haubold","submitted_at":"2013-03-16T13:49:15Z","abstract_excerpt":"In this article we examine the densities of a product and a ratio of two real positive scalar random variables $x_1$ and $x_2$, which are statistically independently distributed, and we consider the density of the product $u_1=x_1x_2$ as well as the density of the ratio $u_2={{x_2}\\over{x_1}}$ and show that Kober operator of the second kind is available as the density of $u_1$ and Kober operator of the first kind is available as the density of $u_2$ when $x_1$ has a type-1 beta density and $x_2$ has an arbitrary density. We also give interpretations of Kober operators of the second and first k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3978","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":""},"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":"2026-05-18T03:30:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nH6xmIgeRFFhA8UBO3mYarO8jAFAwhdEdWyid/dV2HuE6G1u5t8eJYhhk2tKgnz4Xtxl0fjsLuoMJMncv2YZBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T09:24:22.527703Z"},"content_sha256":"03d08cc6681b92be0cab1c56c49e1a11e3a637a8b507dd8d138dbfe9517e88bd","schema_version":"1.0","event_id":"sha256:03d08cc6681b92be0cab1c56c49e1a11e3a637a8b507dd8d138dbfe9517e88bd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I7QOK6FQATBK2W5CYB6DSVKVIM/bundle.json","state_url":"https://pith.science/pith/I7QOK6FQATBK2W5CYB6DSVKVIM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I7QOK6FQATBK2W5CYB6DSVKVIM/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":"2026-05-31T09:24:22Z","links":{"resolver":"https://pith.science/pith/I7QOK6FQATBK2W5CYB6DSVKVIM","bundle":"https://pith.science/pith/I7QOK6FQATBK2W5CYB6DSVKVIM/bundle.json","state":"https://pith.science/pith/I7QOK6FQATBK2W5CYB6DSVKVIM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I7QOK6FQATBK2W5CYB6DSVKVIM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:I7QOK6FQATBK2W5CYB6DSVKVIM","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":"f801934a916827dce3ce6930429c9ae32a7ed28ff560054dab89064d9d435d7f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-03-16T13:49:15Z","title_canon_sha256":"d9bdfdbd4916270fd6664a6fc6b64a12ad71dfd73e15b22a100cbc7525670258"},"schema_version":"1.0","source":{"id":"1303.3978","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.3978","created_at":"2026-05-18T03:30:40Z"},{"alias_kind":"arxiv_version","alias_value":"1303.3978v1","created_at":"2026-05-18T03:30:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.3978","created_at":"2026-05-18T03:30:40Z"},{"alias_kind":"pith_short_12","alias_value":"I7QOK6FQATBK","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"I7QOK6FQATBK2W5C","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"I7QOK6FQ","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:03d08cc6681b92be0cab1c56c49e1a11e3a637a8b507dd8d138dbfe9517e88bd","target":"graph","created_at":"2026-05-18T03:30:40Z","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":"In this article we examine the densities of a product and a ratio of two real positive scalar random variables $x_1$ and $x_2$, which are statistically independently distributed, and we consider the density of the product $u_1=x_1x_2$ as well as the density of the ratio $u_2={{x_2}\\over{x_1}}$ and show that Kober operator of the second kind is available as the density of $u_1$ and Kober operator of the first kind is available as the density of $u_2$ when $x_1$ has a type-1 beta density and $x_2$ has an arbitrary density. We also give interpretations of Kober operators of the second and first k","authors_text":"A.M. Mathai, H.J. Haubold","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-03-16T13:49:15Z","title":"Erdelyi-Kober Fractional Integral Operators from a Statistical Perspective -I"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3978","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":"sha256:76684a7bd0d72c4c27102742abac20ff7b8a1e8f65b4423d2cce2fb13ee056bd","target":"record","created_at":"2026-05-18T03:30:40Z","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":"f801934a916827dce3ce6930429c9ae32a7ed28ff560054dab89064d9d435d7f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-03-16T13:49:15Z","title_canon_sha256":"d9bdfdbd4916270fd6664a6fc6b64a12ad71dfd73e15b22a100cbc7525670258"},"schema_version":"1.0","source":{"id":"1303.3978","kind":"arxiv","version":1}},"canonical_sha256":"47e0e578b004c2ad5ba2c07c3955554314f324e33c4c43e9deb5bd7c457b640a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"47e0e578b004c2ad5ba2c07c3955554314f324e33c4c43e9deb5bd7c457b640a","first_computed_at":"2026-05-18T03:30:40.308652Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:30:40.308652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2CX/FGjQsqE0+NE7TXk7DDp3S2/vZOIv9jHI8NwFGfFEMg40zxSG85FG9biLuBafEmrlqSXyEI8LKsIMbVEiCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:30:40.309526Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.3978","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:76684a7bd0d72c4c27102742abac20ff7b8a1e8f65b4423d2cce2fb13ee056bd","sha256:03d08cc6681b92be0cab1c56c49e1a11e3a637a8b507dd8d138dbfe9517e88bd"],"state_sha256":"ba664d3a974460d40154e27adbbd49dd46bd49b237ac6726b97b442301c5892b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MHIKKfBb3/TFKkCV3DjjB3j6/FmvaOqqMqGFigQyyhQRyyYsr/EXhzVlR6kuLefGUxlsCc8FZHo7ojbYWHbRCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T09:24:22.532620Z","bundle_sha256":"eb2da278fffbfd1b5a52a8ef2255f3d84b338d4072e99778becd396dd2107374"}}