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We derive an initial value problem for a first order differential equation whose solution $\\alpha(x)$ characterizes the limit behavior of $a_{\\nu,k}$ in the following sense: $$ \\lim_{k \\to \\infty} \\frac{a_{kx,k}}{k} = \\alpha(x), \\quad x \\geq 0. $$ Moreover, we show that $$ a_{\\nu,k} < \\frac{\\pi k}{R-1} + \\frac{\\pi \\nu}{2R}. $$ We use $\\alpha(x)$ to obtain an explicit expression of the Pleijel constant for planar annuli and compute some o"},"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":"1803.09972","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-03-27T09:10:43Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"6659ec9f419f58d5ae2c16e8f445e4621a461f18e5bc580e9cb439b56c255bf9","abstract_canon_sha256":"999532a30a73d58bbb56ab6bbf98ddf986e1f76656973f86cf5c5a48fa40cdfd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:59.485480Z","signature_b64":"nU0PrCKfyu01gmK8Mi+EDCwHkuVborYQh1nV/D/gpyLmNs8QJcJ8VdRoaj2l6liv/agn3x79x7cYogbMwVk8Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ad8af84986f2b8ff8b69e48b35fe959b6823e9abfb8e6fef2521e17f9dcecd4","last_reissued_at":"2026-05-17T23:57:59.484922Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:59.484922Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic relation for zeros of cross-product of Bessel functions and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.CA","authors_text":"Vladimir Bobkov","submitted_at":"2018-03-27T09:10:43Z","abstract_excerpt":"Let $a_{\\nu,k}$ be the $k$-th positive zero of the cross-product of Bessel functions $J_\\nu(R z) Y_\\nu(z) - J_\\nu(z) Y_\\nu(R z)$, where $\\nu\\geq 0$ and $R>1$. We derive an initial value problem for a first order differential equation whose solution $\\alpha(x)$ characterizes the limit behavior of $a_{\\nu,k}$ in the following sense: $$ \\lim_{k \\to \\infty} \\frac{a_{kx,k}}{k} = \\alpha(x), \\quad x \\geq 0. $$ Moreover, we show that $$ a_{\\nu,k} < \\frac{\\pi k}{R-1} + \\frac{\\pi \\nu}{2R}. $$ We use $\\alpha(x)$ to obtain an explicit expression of the Pleijel constant for planar annuli and compute some o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.09972","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1803.09972","created_at":"2026-05-17T23:57:59.484999+00:00"},{"alias_kind":"arxiv_version","alias_value":"1803.09972v2","created_at":"2026-05-17T23:57:59.484999+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.09972","created_at":"2026-05-17T23:57:59.484999+00:00"},{"alias_kind":"pith_short_12","alias_value":"FLMK7BEYN4VY","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_16","alias_value":"FLMK7BEYN4VY76FW","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_8","alias_value":"FLMK7BEY","created_at":"2026-05-18T12:32:22.470017+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FLMK7BEYN4VY76FWTZELGX7JLG","json":"https://pith.science/pith/FLMK7BEYN4VY76FWTZELGX7JLG.json","graph_json":"https://pith.science/api/pith-number/FLMK7BEYN4VY76FWTZELGX7JLG/graph.json","events_json":"https://pith.science/api/pith-number/FLMK7BEYN4VY76FWTZELGX7JLG/events.json","paper":"https://pith.science/paper/FLMK7BEY"},"agent_actions":{"view_html":"https://pith.science/pith/FLMK7BEYN4VY76FWTZELGX7JLG","download_json":"https://pith.science/pith/FLMK7BEYN4VY76FWTZELGX7JLG.json","view_paper":"https://pith.science/paper/FLMK7BEY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1803.09972&json=true","fetch_graph":"https://pith.science/api/pith-number/FLMK7BEYN4VY76FWTZELGX7JLG/graph.json","fetch_events":"https://pith.science/api/pith-number/FLMK7BEYN4VY76FWTZELGX7JLG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FLMK7BEYN4VY76FWTZELGX7JLG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FLMK7BEYN4VY76FWTZELGX7JLG/action/storage_attestation","attest_author":"https://pith.science/pith/FLMK7BEYN4VY76FWTZELGX7JLG/action/author_attestation","sign_citation":"https://pith.science/pith/FLMK7BEYN4VY76FWTZELGX7JLG/action/citation_signature","submit_replication":"https://pith.science/pith/FLMK7BEYN4VY76FWTZELGX7JLG/action/replication_record"}},"created_at":"2026-05-17T23:57:59.484999+00:00","updated_at":"2026-05-17T23:57:59.484999+00:00"}