Geodesic
Symmetric moment problems and a conjecture of Valent
Sbornik. Mathematics, Tome 208 (2017) no. 3, pp. 335-359

Voir la notice de l'article provenant de la source Math-Net.Ru

In 1998 Valent made conjectures about the order and type of certain indeterminate Stieltjes moment problems associated with birth and death processes which have polynomial birth and death rates of degree $p\geqslant 3$. Romanov recently proved that the order is $1/p$ as conjectured. We prove that the type with respect to the order is related to certain multi-zeta values and that this type belongs to the interval $$ \biggl[\frac{\pi}{p\sin(\pi/p)},\,\frac{\pi}{p\sin(\pi/p)\cos(\pi/p)}\biggr], $$ which also contains the conjectured value. This proves that the conjecture about type is asymptotically correct as $p\to\infty$. The main idea is to obtain estimates for order and type of symmetric indeterminate Hamburger moment problems when the orthonormal polynomials $P_n$ and those of the second kind $Q_n$ satisfy $P_{2n}^2(0)\sim c_1n^{-1/\beta}$ and $Q_{2n-1}^2(0)\sim c_2 n^{-1/\alpha}$, where $0<\alpha,\beta<1$ may be different, and $c_1$ and $c_2$ are positive constants. In this case the order of the moment problem is majorized by the harmonic mean of $\alpha$ and $\beta$. Here $\alpha_n\sim \beta_n$ means that $\alpha_n/\beta_n\to 1$. This also leads to a new proof of Romanov's Theorem that the order is $1/p$. Bibliography: 19 titles.
Keywords: indeterminate moment problem, birth and death process with polynomial rates, multi-zeta values. 
Ch. Berg; R. Szwarc. Symmetric moment problems and a conjecture of Valent. Sbornik. Mathematics, Tome 208 (2017) no. 3, pp. 335-359. http://geodesic.mathdoc.fr/item/SM_2017_208_3_a2/
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[1] N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965, x+253 pp. | MR | MR | Zbl | Zbl

[2] E. Artin, The gamma function, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York–Toronto–London, 1964, vii+39 pp. | MR | Zbl

[3] C. Berg, “Indeterminate moment problems and the theory of entire functions”, J. Comput. Appl. Math., 65:1–3 (1995), 27–55 | DOI | MR | Zbl

[4] C. Berg, H. L. Pedersen, “On the order and type of the entire functions associated with an indeterminate Hamburger moment problem”, Ark. Mat., 32:1 (1994), 1–11 | DOI | MR | Zbl

[5] C. Berg, R. Szwarc, “On the order of indeterminate moment problems”, Adv. Math., 250 (2014), 105–143 | DOI | MR | Zbl

[6] C. Berg, G. Valent, “The Nevanlinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes”, Methods Appl. Anal., 1:2 (1994), 169–209 | DOI | MR | Zbl

[7] R. P. Boas, Jr., Entire functions, Academic Press Inc., New York, 1954, x+276 pp. | MR | Zbl

[8] T. S. Chihara, “Indeterminate symmetric moment problems”, J. Math. Anal. Appl., 85:2 (1982), 331–346 | DOI | MR | Zbl

[9] J. Gilewicz, E. Leopold, G. Valent, “New Nevanlinna matrices for orthogonal polynomials related to cubic birth and death processes”, J. Comput. Appl. Math., 178:1-2 (2005), 235–245 | DOI | MR | Zbl

[10] J. Gilewicz, E. Leopold, A. Ruffing, G. Valent, “Some cubic birth and death processes and their related orthogonal polynomials”, Constr. Approx., 24 (2006), 71–89 | DOI | MR | Zbl

[11] M. E. H. Ismail, D. R. Masson, J. Letessier, G. Valent, “Birth and death processes and orthogonal polynomials”, Orthogonal polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 294, Kluwer Acad. Publ., Dordrecht, 1990, 229–255 | DOI | MR | Zbl

[12] S. Karlin, J. L. McGregor, “The differential equations of birth-and-death processes and the Stieltjes moment problem”, Trans. Amer. Math. Soc., 85:2 (1957), 489–546 | DOI | MR | Zbl

[13] H. L. Pedersen, “The Nevanlinna matrix of entire functions associated with a shifted indeterminate Hamburger moment problem”, Math. Scand., 74:1 (1994), 152–160 | DOI | MR | Zbl

[14] R. Romanov, “Order problem for canonical systems and a conjecture of Valent”, Trans. Amer. Math. Soc. (to appear); 2015, arXiv: 1502.04402

[15] J. Shohat, J. D. Tamarkin, The problem of moments, Amer. Math. Soc. Math. Surv., 1, rev. ed., Amer. Math. Soc., Providence, RI, 1950, xiv+144 pp. | MR | Zbl

[16] B. Simon, “The classical moment problem as a self-adjoint finite difference operator”, Adv. Math., 137:1 (1998), 82–203 | DOI | MR | Zbl

[17] G. Valent, “Indeterminate moment problems and a conjecture on the growth of the entire functions of the Nevanlinna parametrization”, Applications and computation of orthogonal polynomials (Oberwolfach, 1998), Internat. Ser. Numer. Math., 131, Birkhäuser, Basel, 1999, 227–237 | MR | Zbl

[18] G. Valent, “From asymptotics to spectral measures: determinate versus indeterminate moment problems”, Mediterr. J. Math., 3:2 (2006), 327–345 | DOI | MR | Zbl

[19] W. V. Zudilin, “Algebraic relations for multiple zeta values”, Russian Math. Surveys, 58:1 (2003), 1–29 | DOI | DOI | MR | Zbl