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We investigate the complete asymptotic expansion for $x_{k}$ and prove that for any $n\\ge1$, as $k\\to \\infty$, \\begin{align*} x_k=-kq^{1-k}\\Big(1+\\sum_{i=1}^{n}C_i(q)k^{-1-i}+o(k^{-1-n})\\Big), \\end{align*} where $C_i(q)$ are some $q$ series which can be determined recursively. We show that each $C_{i}(q)\\in \\mathbb{Q}[A_0,A_1,A_2]$, where $A_{i}=\\sum_{m=1}^{\\infty}m^i\\sigma(m)q^m$ and "},"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":"1709.04357","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-09-13T14:40:16Z","cross_cats_sorted":["math-ph","math.CO","math.MP","math.NT"],"title_canon_sha256":"d765cf0fcd39f8e84e9c3721ae39a0f2c68c7c330af679177a99cfc8f2625955","abstract_canon_sha256":"97f50439fa251fea4835520c565473b9828ad9dd3f0abbb50c791c60063ab0bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:14.619881Z","signature_b64":"Y5cdmDUnwQisMSjl2/5AvBb+vQdf+07LFGGVCZUIychE4HVBKeRUcqj0F6Zg8elKJdQCMrhzDTYg9cD/zSOhCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"21c97ba4ca58d171e8c7061e61973297db665042654e19ff55b9a2b779788b28","last_reissued_at":"2026-05-18T00:35:14.619277Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:14.619277Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zeros of the deformed exponential function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CO","math.MP","math.NT"],"primary_cat":"math.CA","authors_text":"Cheng Zhang, Liuquan Wang","submitted_at":"2017-09-13T14:40:16Z","abstract_excerpt":"Let $f(x)=\\sum_{n=0}^{\\infty}\\frac{1}{n!}q^{n(n-1)/2}x^n$ ($0<q<1$) be the deformed exponential function. It is known that the zeros of $f(x)$ are real and form a negative decreasing sequence $(x_k)$ ($k\\ge 1$). We investigate the complete asymptotic expansion for $x_{k}$ and prove that for any $n\\ge1$, as $k\\to \\infty$, \\begin{align*} x_k=-kq^{1-k}\\Big(1+\\sum_{i=1}^{n}C_i(q)k^{-1-i}+o(k^{-1-n})\\Big), \\end{align*} where $C_i(q)$ are some $q$ series which can be determined recursively. We show that each $C_{i}(q)\\in \\mathbb{Q}[A_0,A_1,A_2]$, where $A_{i}=\\sum_{m=1}^{\\infty}m^i\\sigma(m)q^m$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.04357","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1709.04357","created_at":"2026-05-18T00:35:14.619392+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.04357v1","created_at":"2026-05-18T00:35:14.619392+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.04357","created_at":"2026-05-18T00:35:14.619392+00:00"},{"alias_kind":"pith_short_12","alias_value":"EHEXXJGKLDIX","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_16","alias_value":"EHEXXJGKLDIXD2GH","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_8","alias_value":"EHEXXJGK","created_at":"2026-05-18T12:31:12.930513+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/EHEXXJGKLDIXD2GHAYPGDFZSS7","json":"https://pith.science/pith/EHEXXJGKLDIXD2GHAYPGDFZSS7.json","graph_json":"https://pith.science/api/pith-number/EHEXXJGKLDIXD2GHAYPGDFZSS7/graph.json","events_json":"https://pith.science/api/pith-number/EHEXXJGKLDIXD2GHAYPGDFZSS7/events.json","paper":"https://pith.science/paper/EHEXXJGK"},"agent_actions":{"view_html":"https://pith.science/pith/EHEXXJGKLDIXD2GHAYPGDFZSS7","download_json":"https://pith.science/pith/EHEXXJGKLDIXD2GHAYPGDFZSS7.json","view_paper":"https://pith.science/paper/EHEXXJGK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.04357&json=true","fetch_graph":"https://pith.science/api/pith-number/EHEXXJGKLDIXD2GHAYPGDFZSS7/graph.json","fetch_events":"https://pith.science/api/pith-number/EHEXXJGKLDIXD2GHAYPGDFZSS7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EHEXXJGKLDIXD2GHAYPGDFZSS7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EHEXXJGKLDIXD2GHAYPGDFZSS7/action/storage_attestation","attest_author":"https://pith.science/pith/EHEXXJGKLDIXD2GHAYPGDFZSS7/action/author_attestation","sign_citation":"https://pith.science/pith/EHEXXJGKLDIXD2GHAYPGDFZSS7/action/citation_signature","submit_replication":"https://pith.science/pith/EHEXXJGKLDIXD2GHAYPGDFZSS7/action/replication_record"}},"created_at":"2026-05-18T00:35:14.619392+00:00","updated_at":"2026-05-18T00:35:14.619392+00:00"}