The moment problem associated with the Stieltjes–Wigert polynomials

Abstract We consider the indeterminate Stieltjes moment problem associated with the Stieltjes–Wigert polynomials. After a presentation of the well-known solutions, we study a transformation  T of the set of solutions. All the classical solutions turn out to be fixed under this transformation but this is not the case for the so-called canonical solutions. Based on generating functions for the Stieltjes–Wigert polynomials, expressions for the entire functions  A , B , C , and  D from the Nevanlinna parametrization are obtained. We describe T ( n ) ( μ ) for n∈ N when μ = μ 0 is a particular N -extremal solution and explain in detail what happens when n →∞.

[1]  Limits of some q -Laguerre polynomials , 1986 .

[2]  Mizan Rahman,et al.  Basic Hypergeometric Series , 1990 .

[3]  T. Stieltjes.  Recherches sur les fractions continues , 1995 .

[4]  C. Berg.  On some indeterminate moment problems for measures on a geometric progression , 1998 .

[5]  Richard Askey,et al.  More q-beta integrals , 1986 .

[6]  On generalized Stieltjes-Wigert and related orthogonal polynomials , 1979 .

[7]  Henrik L. Pedersen,et al.  On the order and type of the entire functions associated with an indeterminate Hamburger moment problem , 1994 .

[8]  Christiansen.  The Moment Problem Associated with the q -Laguerre Polynomials , 2003 .

[9]  L. J. Rogers.  Second Memoir on the Expansion of certain Infinite Products , 1893 .

[10]  Henrik L. Pedersen.  Stieltjes moment problems and the Friedrichs extension of a positive definite operator , 1995 .

[11]  George E. Andrews,et al.  Enumeration of Partitions: The Role of Eulerian Series and q-Orthogonal Polynomials , 1977 .

[12]  T. Chihara,et al.  An Introduction to Orthogonal Polynomials , 1979 .

[13]  N. Akhiezer,et al.  The Classical Moment Problem. , 1968 .

[14]  Rene F. Swarttouw,et al.  The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue Report Fac , 1996, math/9602214.

[15]  A Characterization and a Class of Distribution Functions for the Stieltjes-Wigert Polynomials , 1970, Canadian Mathematical Bulletin.

[16]  A. Pakes.  Length Biasing and Laws Equivalent to the Log-Normal , 1996 .