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Stetk{\\ae}r obtained the solutions of the functional equation $f(xy)-f(\\sigma(y)x)=g(x)h(y)$ where $\\sigma$ is an involutive automorphism and $f,g,h$ are complex-valued functions, in the setting of a group $G$ and a monoid $M$.\n  Our main goal is to determine the complex-valued solutions of the following more general version of this equation, viz $f(xy)-\\mu(y)f(\\sigma(y)x)=g(x)h(y)$ where $\\mu: G\\longrightarrow \\mathbb{C}$ is a multiplicative function such that $\\mu(x\\sigma(x))=1$ for all $x\\in G$. As an application we find the complex-valued solutions $(f,g,h)$ 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":"1603.02065","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-02-22T11:32:02Z","cross_cats_sorted":[],"title_canon_sha256":"d17bc1d88fae6273e03e9486a0d438529cf1162c53aa05ef1a2251491e9b0adc","abstract_canon_sha256":"82a2326de0b88b3dfa8a5b2c328d4e65acdad6b79beeb59f3022a123f7666801"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:31.717676Z","signature_b64":"RBnh4Pvx+lGLwXCtFGXILmijb8PzuhRPwqyLkYcBNnWuDLAxK918DScDm1D4NshnTKq6R+DL55rLvsFmg4dfCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91cde45d40eac8e538d0645183bc85c0689b712fe85714787d42bd61bef6eefa","last_reissued_at":"2026-05-18T01:19:31.717209Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:31.717209Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A class of functional equations on monoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Bouikhalene Belaid, Elqorachi Elhoucien","submitted_at":"2016-02-22T11:32:02Z","abstract_excerpt":"In \\cite{05} B. Ebanks and H. Stetk{\\ae}r obtained the solutions of the functional equation $f(xy)-f(\\sigma(y)x)=g(x)h(y)$ where $\\sigma$ is an involutive automorphism and $f,g,h$ are complex-valued functions, in the setting of a group $G$ and a monoid $M$.\n  Our main goal is to determine the complex-valued solutions of the following more general version of this equation, viz $f(xy)-\\mu(y)f(\\sigma(y)x)=g(x)h(y)$ where $\\mu: G\\longrightarrow \\mathbb{C}$ is a multiplicative function such that $\\mu(x\\sigma(x))=1$ for all $x\\in G$. 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