Abstract
We analyze co-recursivity for indeterminate Hamburger moment problems and the duality transformation of Karlin and McGregor for indeterminate Stieltjes moment problems. In both cases the transformed Nevanlinna matrix is given and the Nevanlinna extremal measures are discussed. An example involving associated polynomials, relevant for a quartic birth and death process, is worked out.
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References
N. I. Akhiezer (1965):The Classical Moment Problem. Edinburgh: Oliver and Boyd.
C. Berg (1994):Markov’s theorem revisited. J. Approx. Theory,78:260–275.
C. Berg, H. L. Pedersen (1994):On the order and type of the entire functions associated with an indeterminate Hamburger problem. Arkiv. Math.,32:1–11.
C. Berg, G. Valent (1994):The Nevanlinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes. Methods Appl. Anal.,1:169–209.
T. S. Chihara (1957):On co-recursive orthogonal polynomials. Proc. Amer. Math. Soc.,8:899–905.
T. S. Chihara (1982):Indeterminate symmetric moment problems. J. Math. Anal. Appl.,85:331–346.
T. S. Chihara, M. E. H. Ismail (1993):Extremal measures for a system of orthogonal polynomials. Constr. Approx.9:111–119.
M. E. H. Ismail, D. Masson (1994):q-Hermite polynomials, biorthogonal functions and q-beta integrals. Trans. Amer. Math. Soc.,346:63–116.
S. Karlin, J. McGregor (1957):The classification of birth and death processes. Trans. Amer. Math. Soc.86:366–401.
S. Karlin, J. McGregor (1958):The differential equations of birth and death processes and the Stieltjes moment problem. Trans. Amer. Math. Soc.,85:489–546.
W. Magnus, F. Oberhettinger, R. P. Soni (1966): Formulas and Theorems of the Special Functions of Mathematical Physics, New York, Springer-Verlag.
A. I. Markushevich (1965): Theory of Functions of a Complex Variable, volume 2. New York: Chelsea.
H. L. Pedersen (1995):Stieltjes moment problems and the Friedrichs extension of a positive definite operator. J. Approx. Theory83:289–307.
L.J. Slater (1966): Generalized Hypergeometric Series. Cambridge: Cambridge University Press.
J. A. Shohat, J. D. Tamarkin (1950): The Problem of Moments. Mathematical Surveys, Volume 1, Amer. Math. Soc., Providence, RI.
W. Van Assche (1991):Orthogonal polynomials, associated polynomials and functions of the second kind. J. Comput. Appl. Math.37:237–249.
G. Valent (1993):Orthogonal polynomials for a quartic birth and death process. J. Comp. Appl. Math.49:281–288.
G. Valent (1994):Asymptotic analysis of some associated orthogonal polynomials connected with elliptic functions. SIAM J. Math. Anal.,25:749–775.
G. Valent (1996):Exact solutions of a quartic birth and death process and related orthogonal polynomials. J. Comput. Appl. Math.67:103–127.
E. T. Whittaker, G. N. Watson (1965): A Course of Modern Analysis. Cambridge: Cambridge University Press.
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Communicated by Mourad Ismail.
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Valent, G. Co-Recursivity and karlin-McGregor duality for indeterminate moment problems. Constr. Approx 12, 531–553 (1996). https://doi.org/10.1007/BF02437507
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DOI: https://doi.org/10.1007/BF02437507