{"paper":{"title":"Probabilistic implications of symmetries of q-Hermite and Al-Salam-Chihara polynomials","license":"","headline":"","cross_cats":["math.CV"],"primary_cat":"math.PR","authors_text":"Pawe{\\l} J. Szab{\\l}owski","submitted_at":"2007-08-03T18:50:33Z","abstract_excerpt":"We prove the existence of stationary random fields with linear regressions for $q>1$ and thus close an open question posed by W. Bryc et al.. We prove this result by describing a discrete 1 dimensional conditional distribution and then checking Chapman-Kolmogorov equation. Support of this distribution consist of zeros of certain Al-Salam-Chihara polynomials. To find them we refer to and expose known result concerning addition of $q-$ exponential function. This leads to generalization of a well known formula $(x+y)^{n}% =\\sum_{i=0}^{n}\\binom{n}{k}i^{k}H_{n-k}(x) H_{k}(-iy) ,$ where $H_{k}(x) $ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0708.0563","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"}