{
  "id": "1003.2998",
  "version": "v1",
  "published": "2010-03-15T19:20:19.000Z",
  "updated": "2010-03-15T19:20:19.000Z",
  "title": "Meixner class of non-commutative generalized stochastic processes with freely independent values II. The generating function",
  "authors": [
    "M. Bozejko",
    "E. Lytvynov"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $T$ be an underlying space with a non-atomic measure $\\sigma$ on it. In [{\\it Comm.\\ Math.\\ Phys.}\\ {\\bf 292} (2009), 99--129] the Meixner class of non-commutative generalized stochastic processes with freely independent values, $\\omega=(\\omega(t))_{t\\in T}$, was characterized through the continuity of the corresponding orthogonal polynomials. In this paper, we derive a generating function for these orthogonal polynomials. The first question we have to answer is: What should serve as a generating function for a system of polynomials of infinitely many non-commuting variables? We construct a class of operator-valued functions $Z=(Z(t))_{t\\in T}$ such that $Z(t)$ commutes with $\\omega(s)$ for any $s,t\\in T$. Then a generating function can be understood as $G(Z,\\omega)=\\sum_{n=0}^\\infty \\int_{T^n}P^{(n)}(\\omega(t_1),...,\\omega(t_n))Z(t_1)...Z(t_n)\\sigma(dt_1)...\\sigma(dt_n)$, where $P^{(n)}(\\omega(t_1),...,\\omega(t_n))$ is (the kernel of the) $n$-th orthogonal polynomial. We derive an explicit form of $ G(Z,\\omega)$, which has a resolvent form and resembles the generating function in the classical case, albeit it involves integrals of non-commuting operators. We finally discuss a related problem of the action of the annihilation operators $\\partial_t$, $t\\in T$. In contrast to the classical case, we prove that the operators $\\di_t$ related to the free Gaussian and Poisson processes have a property of globality. This result is genuinely infinite-dimensional, since in one dimension one loses the notion of globality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-03-15T19:20:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-commutative generalized stochastic processes",
      "generating function",
      "freely independent values",
      "meixner class",
      "th orthogonal polynomial"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00220-010-1134-4",
      "journal": "Communications in Mathematical Physics",
      "year": 2011,
      "month": "Mar",
      "volume": 302,
      "number": 2,
      "pages": 425
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011CMaPh.302..425B"
    }
  }
}