Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula

Hans Weber

Open Mathematics (2007)

  • Volume: 5, Issue: 2, page 415-427
  • ISSN: 2391-5455

Abstract

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Starting from the Rodrigues representation of polynomial solutions of the general hypergeometric-type differential equation complementary polynomials are constructed using a natural method. Among the key results is a generating function in closed form leading to short and transparent derivations of recursion relations and addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and obey Rodrigues formulas. Applications to the classical polynomials are given.

How to cite

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Hans Weber. "Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula." Open Mathematics 5.2 (2007): 415-427. <http://eudml.org/doc/269025>.

@article{HansWeber2007,
abstract = {Starting from the Rodrigues representation of polynomial solutions of the general hypergeometric-type differential equation complementary polynomials are constructed using a natural method. Among the key results is a generating function in closed form leading to short and transparent derivations of recursion relations and addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and obey Rodrigues formulas. Applications to the classical polynomials are given.},
author = {Hans Weber},
journal = {Open Mathematics},
keywords = {Polynomials with Rodrigues formula; solutions of hypergeometric-type differential equation; generating function in closed form; recursion relations; addition theorem; polynomials with Rodrigues formula},
language = {eng},
number = {2},
pages = {415-427},
title = {Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula},
url = {http://eudml.org/doc/269025},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Hans Weber
TI - Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula
JO - Open Mathematics
PY - 2007
VL - 5
IS - 2
SP - 415
EP - 427
AB - Starting from the Rodrigues representation of polynomial solutions of the general hypergeometric-type differential equation complementary polynomials are constructed using a natural method. Among the key results is a generating function in closed form leading to short and transparent derivations of recursion relations and addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and obey Rodrigues formulas. Applications to the classical polynomials are given.
LA - eng
KW - Polynomials with Rodrigues formula; solutions of hypergeometric-type differential equation; generating function in closed form; recursion relations; addition theorem; polynomials with Rodrigues formula
UR - http://eudml.org/doc/269025
ER -

References

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  1. [1] A.F. Nikiforov and V.B. Uvarov: Special Functions of Mathematical Physics, Birkhäuser Verlag, Basilea, 1988. 
  2. [2] M.E.H. Ismail: Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge Univ. Press, Cambridge, 2005. Zbl1082.42016
  3. [3] P.A. Lesky: “Endliche und unendliche Systeme von kontinuierlichen klassischen Orthogonalpoynomen”, Z. Angew. Math. Mech., Vol. 76(3), (1996), pp. 181–184. 
  4. [4] G. Szegö: Orthogonal Polynomials, Vol. XXIII, American Math. Soc., Providence, RI, 1939. 
  5. [5] G.B. Arfken and H.J. Weber: Mathematical Methods for Physicists, 6th ed., Elsevier-Academic Press, Amsterdam, 2005. Zbl1066.00001
  6. [6] P. Dennery and A. Krzywicki: Mathematics for Physicists, Dover, New York, 1996. Zbl1141.00003
  7. [7] M. Abramowitz and I.A. Stegun: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, 2nd edition, New York, 1972. 
  8. [8] I.S. Gradshteyn and I.M. Ryzhik: Table of Integrals, Series and Products, ed. A. Jeffrey, Acad. Press, San Diego, 2000. Zbl0981.65001

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