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Some properties of biorthogonal polynomials and their application to Padé approximations

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Abstract

Transformations of biorthogonal polynomials under certain transformations of biorthogonalizable sequences are studied. The obtained result is used to construct Padé approximants of orders [N−1/N],N ε ℕ, for the functions

$$\tilde f(z) = \sum\limits_{m = 0}^M {\alpha _m } \frac{{f(z) - T_{m - 1} [f;z]}}{{z^m }},$$

wheref(z) is a function with known Padé approximants of the indicated orders,T j [f;z] are Taylor polynomials of degreej for the functionf(z), and α m, M =\(\overline {1,M} \) are constants.

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References

  1. V. K. Dzyadyk, “On the generalization of the moment problem,”Dokl. Akad. Nauk Ukr.SSR, Ser. A, No. 6, 8–12 (1981).

    Google Scholar 

  2. A. P. Golub, “Some properties of biorthogonal polynomials,”Ukr. Mat. Zh.,41, No. 10, 1384–1388 (1989).

    Google Scholar 

  3. A. Iserles and S. P. Nørsett, “Biorthogonal polynomials,”Lect. Notes Math.,1171, 92–100 (1985).

    Google Scholar 

  4. A. Iserles and S. P. Nørsett, “On the theory of biorthogonal polynomials,”Math. Comput., No. 1, 42 (1986).

    Google Scholar 

  5. A. P. Golub,Application of the Generalized Moment Problem to the Padé Approximation of Some Functions [in Russian], Preprint No. 81.58, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1981), pp. 16–56.

    Google Scholar 

  6. P. K. Suétin,Classical Orthogonal Polynomials [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  7. A. P. Golub, “On generalized moment representations of special type,”Ukr. Mat. Zh.,41, No. 11, 1455–1460 (1989).

    Google Scholar 

  8. R. Walliser, “Rationale Approximation desq-Analogous der Exponentialfunktion und Irrationalitätsaussagen für diese Funktion,”Arch. Math.,44, No. 1, 59–64 (1985).

    Google Scholar 

  9. H. Bateman and A. Erdélyi,Higher Transcendental Functions, Vol. 1, McGraw Hill, New York, Toronto, London (1953).

    Google Scholar 

  10. F. H. Jackson, “Transformation ofq-series,”Mess. Math.,39, 145–153 (1910).

    Google Scholar 

  11. E. Andrews and R. Askey, “Classical orthogonal polynomials,”Lect. Notes Math.,1171, 36–62 (1985).

    Google Scholar 

  12. A. P. Golub,Generalized Moment Representations and Rational Approximants [in Russian], Preprint No. 87.25, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1987).

    Google Scholar 

  13. G. A. Baker (Jr.) and P. Graves-Morris,Padé Approximants, Addison-Wesley, London, Amsterdam (1981).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    A. P. Golub

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  1. A. P. Golub
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 977–984, August, 1994.

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Golub, A.P. Some properties of biorthogonal polynomials and their application to Padé approximations. Ukr Math J 46, 1070–1078 (1994). https://doi.org/10.1007/BF01056168

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  • DOI: https://doi.org/10.1007/BF01056168

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