Abstract
The concept of generalized classical polyorthogonal polynomials and, in particular, that of generalized Laguerre polynomials, corresponding to a collection of measures with supports on infinite rays in the complex plane, is introduced. The asymptotic behavior of these polynomials and of their corresponding functions of the second kind is investigated. Moreover, generalizations of the Bessel functions and of the Euler integral of the second kind are defined and investigated. The convergence of the simultaneous Pade approximants to certain Stieltjes type functions is proved.
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Literature cited
H. S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, New York (1948).
O. Perron, Die Lehre von den Kettenbruchen, Teubner, Stuttgart (1957).
P. L. Chebyshev, Collected Works [in Russian], Vol. 3, Izd. Akad. Nauk SSSR, Moscow (1948).
A. A. Markov, Selected Papers on Continued Fractions and the Theory of Functions Deviating Least from Zero [in Russian], Gostekhizdat, Moscow (1948).
T. Stieltjes, “Recherches sur les fractions continues,” Ann. Fac. Sci. Univ. Toulouse,8, J1-J122 (1894);9, A1–A47 (1895). [Reprinted in his Oeuvres Completes, Tome 2, Noordhoff (1918), pp. 402–566.]
N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Oliver and Boyd, Edinburgh (1965).
P. K. Suetin, Classical Orthogonal Polynomials [in Russian], Nauka, Moscow (1976).
K. Mahler, “Perfect systems,” Compositio Math.,19, No. 2, 95–166 (1968).
M. G. de Bruin, Three new examples of generalized Pade tables which are partly normal. Report 76-11, Dept. of Math., Univ. of Amsterdam, Amsterdam (1976).
V. A. Kalyagin, “Normality and convergence questions for simultaneous Pade approximations,” Author's Abstracts of Candidate's Dissertation, Moscow State Univ. (1980).
C. Hermite, “Sur la fonction exponentielle,” C. R. Acad. Sci. Paris,77, 18–24, 74–79, 226–233, 285–293 (1873). [Reprinted in his Oeuvres, Tome III, Gauthier-Villars, Paris (1912), pp. 150–181.]
V. I. Parusnikov, “The Jacobi-Perron algorithm and simultaneous approximation of functions,” Mat. Sb.,114 (156), No. 2, 322–333 (1982).
A. Angelesco, “Sur deux extensions des fractions continues algebriques,” C. R. Acad. Sci. Paris,168, 262–265 (1919).
E. M. Nikishin, “On a system of Markov functions,” Vestn. Mosk. Univ., Ser. I Mat. Mekh., No. 4, 60–63 (1979).
E. M. Nikishin, “On simultaneous Pade approximations,” Mat. Sb.,113 (155), No. 4, 499–519 (1980).
V. N. Sorokin, “On the simultaneous approximation of several linear forms,” Vestn. Mosk. Univ., Ser. Mat. Mekh., No. 1, 44–47 (1983).
P. Appell, “Sur une suite des polynomes ayant toutes leurs racines reeles,” Archiv. Math. Physik (3), b. 1, 69–71 (1901).
N. Abramesco, Sulle serie di polinomi di una variabile complessa. Le serie di Darboux,” Ann. Mat. Pura Appl., Ser. 3,31, 207–249 (1922).
V. A. Kalyagin, “On a certian class of polynomials defined by two orthogonality relations,” Mat. Sb.,110 (1952), No. 4, 609–627 (1979).
V. N. Sorokin, “The asymptotic behavior of linear functional forms of two logarithms,” Usp. Mat. Nauk,38, No. 1, 193–194 (1983).
V. N. Sorokin, The asymptotic behavior of linear functional forms for a class of Markov functions. Manuscript deposited at VINITI, Dep. No. 4526-82 (1982).
M. G. Krein and A. A. Nudel'man, The Markov Moment Problem and Extremal Problems, Am. Math. Soc., Providence (1977).
S. J. Karlin and W. J. Studden, Tchebycheff Systems: with Applications in Analysis and Statistics, Interscience, Wiley (1966).
V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).
G. Szego, Orthogonal Polynomials, Am. Math. Soc., Providence (1959).
B. Ya. Levin, Entire Functions [in Russian], Moscow State Univ. (1971).
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press (1962).
H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vols. I and II, McGraw-Hill, New York (1953).
A. G. Sveshnikov and A. N. Tikhonov, Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow (1970).
Yu. V. Sidorov, M. V. Fedoryuk, and M. N. Shabunin, Lectures in the Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow (1976).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1976).
L. Fejer, “On the determination of asymptotic values,” Mat. Term. Ertesito,27, 1–33 (1909) [reprinted in Gesammelte Arbeiten, Vol. I, Birkhauser, Basel (1970), pp. 445–473 (in Hungarian); pp. 474–503 (in German).
Y. L. Luke, Mathematical Functions and Their Applications, Academic Press, New York (1975).
Additional information
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 125–165, 1986.
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Sorokin, V.N. A generalization of classical orthogonal polynomials and the convergence of simultaneous Pade approximants. J Math Sci 45, 1461–1499 (1989). https://doi.org/10.1007/BF01097274
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DOI: https://doi.org/10.1007/BF01097274