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(a) We solve $(E)$ when $\\sigma$ is an involutive automorphism, and we obtain some properties about solutions of $(E)$ when $\\sigma$ is an involutive anti-automorphism. (b) We obtain the Hyers Ulam stability of equation $(E)$. As an application, we prove the superstability of the functional equation $f(xy)+\\chi("},"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":"1505.06512","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-05-14T10:14:51Z","cross_cats_sorted":[],"title_canon_sha256":"d666e88213da37ae60316f3d58a1b7c0ce1d7d66b449e4d2b48f1a99bc75418c","abstract_canon_sha256":"1fa16ff720a14cb13c6a0600708c3cef3f7b352d548d8d2b65523ec514edf66f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:44.782710Z","signature_b64":"aWltm2c00UcU8xFR5xhOdfLXAFLepNl7d7IuuyICwb+86K7cvJb8xSF5Fp5gLPrMCpYWsM1DqR/Q8rNCkiv5Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"90bad5943b98135a0d6fcda4c6727257d882da5c7ed0a8102af92f83a30708a2","last_reissued_at":"2026-05-18T02:03:44.782155Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:44.782155Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Solutions and stability of variant of Wilson's functional equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Elqorachi Elhoucien, Redouani Ahmed","submitted_at":"2015-05-14T10:14:51Z","abstract_excerpt":"In this paper we will investigate the solutions and stability of the generalized variant of Wilson's functional equation $$ (E):\\;\\;\\;\\; f(xy)+\\chi(y)f(\\sigma(y)x)=2f(x)g(y),\\; x,y\\in G,$$ where $G$ is a group, $\\sigma$ is an involutive morphism of $G$ and $\\chi$ is a character of $G$. 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