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We give the explicit bivariate distribution $\\Pr \\{N^\\nu(s)=k, N^\\nu(t)=r \\}$, for $t \\ge s$, $r \\ge k$, the mean $\\mathbb{E}\\, \\mathcal{N}^{\\alpha,\\nu}(t)$ and the variance $\\mathbb{V}\\text{ar}\\, \\mathcal{N}^{\\alpha,\\nu}(t)$. We study the process $\\mathcal{N}^{\\alpha,1}(t)$ for which we are able to produce explicit re"},"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":"1303.6687","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-26T22:35:53Z","cross_cats_sorted":[],"title_canon_sha256":"15a3d4d96c3a9ad0d8b5115f93e5fc24ff54998e53d751ac446c8b61dbd44cb3","abstract_canon_sha256":"c926212122d206b69b1163771973a5ec52b568a70e1b2840afc560d50f9f1371"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:06.786660Z","signature_b64":"JGDla0kTTe0+fVBk0KqmZoUzh5D60Urf/yJoMPyXKXEDLlzM8r05NNhkxOzt1bgdq7iV15g4nJs9URSgJar1CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9fe45f241452efb6a242524496a42efe6dae68101adffaa93a2599873c5e267b","last_reissued_at":"2026-05-18T02:57:06.786250Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:06.786250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Integral of Fractional Poisson Processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Enzo Orsingher, Federico Polito","submitted_at":"2013-03-26T22:35:53Z","abstract_excerpt":"In this paper we consider the Riemann--Liouville fractional integral $\\mathcal{N}^{\\alpha,\\nu}(t)= \\frac{1}{\\Gamma(\\alpha)} \\int_0^t (t-s)^{\\alpha-1}N^\\nu(s) \\, \\mathrm ds $, where $N^\\nu(t)$, $t \\ge 0$, is a fractional Poisson process of order $\\nu \\in (0,1]$, and $\\alpha > 0$. We give the explicit bivariate distribution $\\Pr \\{N^\\nu(s)=k, N^\\nu(t)=r \\}$, for $t \\ge s$, $r \\ge k$, the mean $\\mathbb{E}\\, \\mathcal{N}^{\\alpha,\\nu}(t)$ and the variance $\\mathbb{V}\\text{ar}\\, \\mathcal{N}^{\\alpha,\\nu}(t)$. 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