The Problem of Moments

Abstract

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𝑞-Hermite polynomials, biorthogonal rational functions, and 𝑞-beta integrals

“…Furthermore the sequences {xn{ax)} and {x"{a2)} interlace when ax±a2. This is part of Theorem 2.13, page 60 in [47]. A proof is in [49], see Theorem 10.41, pp 584-589.…”

“…This easily follows from the asymptotics established in (2.9) of Section 2 which yields \hn(x)\2 < const.\q\n/2. Thus the /j"'s are orthogonal with respect to infinitely many probability measures [1], [12], [47]. Among these measures there is a family of extremal measures {d\f/(x, a)} indexed by a parameter a for every real a including a = ±oo .…”

“…Akhiezer [1] uses iV-extremal to describe what Shohat and Tamarkin [47] refer to as extremal measures. We decided to follow the notation of Shohat and Tamarkin [47] throughout this work. In Section 5 we compute A(z), B(z), C(z), and D(z).…”

A note on recovering the distributions from exponential moments

“…Let us denote the pdf of the Beta (c, d) distribution by (3) where the shape parameters c, d > 0. Also for the simplicity of notations, let us write The following statement is true:…”

“…Here we suggest the modified, scaled version of the MR-Laplace transform inversion that enables us to apply it in the case of the Stieltjes moment problem as well, i.e., when T = ∞. The reader is referred to [3][4][5][6], where the questions on the momentdeterminacy of probability distributions and their approximations in the framework of inverse moment problem are investigated. See also Tagliani and Velasquez [7], where the fractional moments are used to approximate the Laplace transform inversion.…”