Skip to main content

Characterization Theorems for Orthogonal Polynomials

  • Chapter
Orthogonal Polynomials

Part of the book series: NATO ASI Series ((ASIC,volume 294))

Abstract

We survey in this paper characterization theorems dealing with polynomial sets which are orthogonal on the real line.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. . N.A. Abdul-Halim and W.A. Al-Salam, A characterization of the Laguerre polynomials, Rend, del Seminario Mat. Univ. Padova, 34 (1964), 176 – 179.

    MATH  MathSciNet  Google Scholar 

  2. . J. Aczél, Eine Bemerkung über Charakterisierung der “klassische” orthogonalpolynome. rActa. Math. Acad. Sci. Hung., 4 (1953), 315 – 321.

    Article  MATH  Google Scholar 

  3. . J. Aczél, Sur l’equation différentielle des polynomes orthogonaux classiques, Annales Univ. Scient. Budapest, 2 (1959), 27 – 29.

    MATH  Google Scholar 

  4. . N.A. Al-Salam, Orthogonal polynomials of hypergeometric type, Duke Math. J., 33 (1966), 109 – 122.

    MATH  MathSciNet  Google Scholar 

  5. . N.A. Al-Salam and W.A. Al-Salam, Some characterizations of the ultraspherical polynomials,Canad. Math. Bulletin, 11 (1968), 457 – 464.

    Article  MATH  MathSciNet  Google Scholar 

  6. . W.A. Al-Salam, Characterization of certain classes of orthogonal polynomials related to elliptic functions, Annali di Matematica pura ed applicata (IV), LXVII(1965), 75 – 94.

    Google Scholar 

  7. . W.A. Al-Salam, On a characterization of orthogonality, Math. Mag., 31 (1957), 41 – 44.

    Article  MATH  MathSciNet  Google Scholar 

  8. . W.A. Al-Salam, On a characterization of Meixner’s polynomials, The Quart. J. of Mathematics (Oxf)(2), 17 (1966), 7 – 10.

    Article  MathSciNet  Google Scholar 

  9. . W.A. Al-Salam, On a characterization of a certain set of orthogonal polynomials, Boll. Unione Mat. Ital.(3), 19 (1964), 448 – 450.

    MATH  MathSciNet  Google Scholar 

  10. . W.A. Al-Salam,W. Allaway and R. Askey, A characterization of the continuous q- ultraspherical polynomials, Canad. Math. Bull., 27 (3) (1984), 329 – 336.

    MathSciNet  Google Scholar 

  11. . W.A. Al-Salam, W.R Allaway, and R. Askey, Sieved ultrashperical orthogonal polynomials, Trans. Amer. Math. Soc., 284 (1984), 39 – 55.

    Article  MATH  MathSciNet  Google Scholar 

  12. . W.A. Al-Salam and L. Carlitz, Some orthogonal q-polynomials, Math. Nachr., 30 (1965), 47 – 61.

    Article  MATH  MathSciNet  Google Scholar 

  13. . W.A. Al-Salam and T.S. Chihara, Another characterization of the classical orthogonal polynomials, SIAM J. Math. Anal., 3 (1972), 65 – 70.

    MATH  MathSciNet  Google Scholar 

  14. . W.A. Al-Salam and T.S. Chihara, Convolution of orthogonal polynomials, SIAM J. Math. Anal., 7 (1976), 16 – 28.

    MATH  MathSciNet  Google Scholar 

  15. . W.A. Al-Salam and T.S. Chihara, q-Pollaczek polynomials and a conjecture of Andrews and Askey, SIAM J. Math. Anal., 18 (1987), 228 – 242.

    MATH  MathSciNet  Google Scholar 

  16. . W.A. Al-Salam and M. Ismail, Orthogonal polynomials associated with the Rogers- Ramanujan continued fractions, Pacific J. of Math., 104 (1983), 269 – 283.

    MATH  MathSciNet  Google Scholar 

  17. . W.A. Al-Salam and A. Verma, Some orthogonality preserving operators, Proc. Amer. Math. Soc., 23 (1969), 136 – 139.

    Article  MATH  MathSciNet  Google Scholar 

  18. W.A. Al-Salam and A. Verma, Orthogonality preserving operators I, Rendiconti Acad. Naz. dei Lincei(8), LVIII(1975), 833–838.

    Google Scholar 

  19. W.A. Al-Salam and A. Verma, Orthogonality preserving operators II, Rendiconti Acad. Naz, dei Lincei(8), LIX(1976), 26–31.

    Google Scholar 

  20. W.A. Al-Salam and A. Verma, On the Geronimus polynomial sets, Proc. Orthogonal Polynomials and Their Applications, Segovia 1986. Lecture Notes in Mathematics #1329 Springer-Verlag (pp. 193–202).

    Google Scholar 

  21. . W.A. Al-Salam and A. Verma, On an orthogonal polynomial set, Indagationes Math- ematicae, 44 (1982), 335 – 340.

    MATH  MathSciNet  Google Scholar 

  22. . W.A Al-Salam and A. Verma, Some sets of orthogonal polynomials, Rev. Téc. Ing., Univ. Zulia, 9 (1986), 83 – 88.

    MathSciNet  Google Scholar 

  23. . W.R. Allaway, The identification of a class of orthogonal polynomial sets, Ph.D. thesis, University of Alberta, Edmonton, Canada., 1972.

    Google Scholar 

  24. . G.E. Andrews and R. Askey, Classical orthogonal polynomials, Polynômes Orthogonaux et Applications- Proc Bar-le-Duc 1984, Lecture Notes in Math. # 1171, Springer- Verlag

    Google Scholar 

  25. . Angelesco, Sur les polynomes orthogonaux en rapport avec d’autre polynomes, Buletinul Societâtii Stiite din Cluj, 1 (1921), 44 – 59.

    Google Scholar 

  26. . R. Askey, Divided difference operators and classical orthogonal polynomials, Rocky Mountain J. Math, (1989) To appear.

    Google Scholar 

  27. . R. Askey and M. Ismail, A generalization of the ultrasphericai polynomials, Studies in Pure Mathematics, edited by P. Erdös, Birkhauser, Basel, 1983, 55 – 78.

    Google Scholar 

  28. . R. Askey and M. Ismail, Recurrence relations, continued fractions and orthogonal polynomials, Memoirs of the Amer. Math. Soc. #300, 1984.

    Google Scholar 

  29. . R. Askey and J. Wilson, Some basic hypergeometrie orthogonal polynomials that generalize Jacobi polynomials, Memoirs AMS #319, 1985

    Google Scholar 

  30. . F.S. Beale, On a certain class of orthogonal polynomials, Annals of Math. Statistics, 12 (1941), 97 – 103.

    Article  MATH  MathSciNet  Google Scholar 

  31. S. Bochner, Über Sturm-Liouvillesche Polynomsysteme, Math. Zeit., 29(1929), 730– 736.

    Google Scholar 

  32. . S. Bonan and P. Nevai, Orthogonal polynomials and their derivatives, I, J. Approximation Theory, 40 (1984), 134 – 147.

    Article  MATH  MathSciNet  Google Scholar 

  33. . S. Bonan, D. Lubinsky, and P. Nevai, Orthogonal polynomials and their derivatives, II, SIAM J. Math. Anal., 18 (1987), 1163 – 1176.

    MATH  MathSciNet  Google Scholar 

  34. . Boukhemis et P. Maroni, Une caractérisation des polynomes strictement 1/p orthogonaux de type Sheffer. Etude du cas p = 2. J. of Approximation Theory, 54 (1988), 67 – 91.

    Article  MATH  MathSciNet  Google Scholar 

  35. . W.C. Brenke, On polynomial solutions of a class of linear differential equations of the second order, Bull. Amer. Math. Soc., 36 (1930), 77 – 84.

    Article  MATH  MathSciNet  Google Scholar 

  36. . L. Carlitz, Characterization of certain sequences of orthogonal polynomials, Portugaliae Math., 20 (1961), 43 – 46.

    MATH  Google Scholar 

  37. . L. Carlitz, Characterization of the Krawtchouk polynomials, Revista Mat. Hisp-Amer. (4), 21 (1961), 79 – 84.

    MATH  Google Scholar 

  38. . L. Carlitz, Characterization of certain sequences of orthogonal polynomials, Portugaliae Math., 20 (1961), 43 – 46.

    MATH  Google Scholar 

  39. . L. Carlitz, Note on Legendre polynomials, Bull. Calcutta Math. Society, 46 (1954), 93 – 95.

    MATH  Google Scholar 

  40. . L. Carlitz, Characterization of the Laguerre polynomials, Monatshefte für Mathematik, 66 (1962), 389 – 392.

    Article  MATH  Google Scholar 

  41. . T.S. Chihara, Orthogonal polynomials with Brenke type generating function, Duke Math. J., 35 (1968), 505 – 518.

    Article  MATH  MathSciNet  Google Scholar 

  42. . T.S. Chihara, Orthogonality relations for a class of Brenke polynomials, Duke Math. J., 38 (1971), 599 – 603.

    MATH  MathSciNet  Google Scholar 

  43. . T.S. Chihara, A characterization of a class of distribution functions for the Stieltjes- Wigert polynomials, Canadian Math. Bull., 13 (1970), 529 – 532.

    Article  MathSciNet  Google Scholar 

  44. 44.T.S. Chihara, On generalized Stieltjes-Wigert and related orthogonal polynomials, Journal of Computational and Applied Mathematics, 5 (1979), 291 – 297.

    Article  MATH  MathSciNet  Google Scholar 

  45. T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach Pub., 1976.

    Google Scholar 

  46. . T.S. Chihara and M. Ismail, Orthogonal polynomials suggested by queueing model, Advances in Mathematics, 3 (1982), 441 – 462.

    Article  MATH  MathSciNet  Google Scholar 

  47. . C.W. Cryer, Rodriques’ formulas and the classical orthogonal polynomials, Boll. Unione Mat. Ital. (3), 25 (1970), 1 – 11.

    MathSciNet  Google Scholar 

  48. . Á Császár, Sur les polynômes orthogonaux classiques. Annales Univ. Sci. Budapest sec. Math., 1 (1958), 33 – 39.

    MATH  Google Scholar 

  49. . Danese, On a characterization of ultraspherical polynomials, Boll. U.M.I. (3), 21 (1966), 1 – 3.

    Google Scholar 

  50. . D. Dickinson, On quasi-orthogonal polynomials, Proc. Amer. Math. Soc., 12 (1961), 185 – 194.

    Article  MATH  MathSciNet  Google Scholar 

  51. . G.K. Eagleson, A characterization theorem for positive definite sequences on the Kraw- tchouk polynomials, Australian J. Statistics, 2 (1969), 29 – 38.

    MathSciNet  Google Scholar 

  52. . G. Ebert, Über Polynomsysteme mit Rodriquessher Darstellung. Dissertation, Cologne, 1964.

    Google Scholar 

  53. K. Endl, On the involutory property Laguerre polynomials, Colloquia Math. Soc. János Bolyai, 19(1976) Budapest.

    Google Scholar 

  54. . Erdélyi et al., Higher Transcendental Functions, Vol. 2, McGraw-Hill Pub., 1953.

    Google Scholar 

  55. . L. Fejér, Absch. Für die Legendreschen und verwandte Polynome, Math. Zeit., 24 (1925), 285 – 298.

    Article  MATH  Google Scholar 

  56. . E. Feldheim, Sur les polynomes généralisés de Legendre, Bull, de l‘Academie des Science de l’URSS, 5 (1941), 241 – 248.

    MATH  MathSciNet  Google Scholar 

  57. . E. Feldheim, Sur une propriété des polynomes orthogonaux, J. London Math. Soc., 13 (1938), 44 – 53.

    Article  Google Scholar 

  58. . E. Feldheim, Une propriété caractéristique des polynomes de Laguerre, Comment. Math. Helv., 13 (1940), 6 – 10.

    Article  MathSciNet  Google Scholar 

  59. . L. Feldmann, On a characterization of the classical orthogonal polynomials, Acta Sc. Math., 17 (1956), 129 – 133.

    MATH  MathSciNet  Google Scholar 

  60. . L. Feldmann, Über durch Sturm-Liouvillesche Differentialgleichungen charakterisierte orthogonale Polynomsysteme. Pub. Math. (Debrecen), 3 (1954), 297 – 304.

    MATH  MathSciNet  Google Scholar 

  61. . B.M. Gagaev, Sur quelques classes de fonctions orthogonales (in Russian), Bulletin (Izv.) de l‘Academie des Sciences de l’URSS, 10 (1946), 197 – 206.

    Google Scholar 

  62. . B. Gabutti, Some characteristic property of Meixner polynomials, J. Math. Analysis and Applications, 95 (1983), 265 – 277.

    Article  MATH  MathSciNet  Google Scholar 

  63. . Ja. L. Geronimus, On polynomials orthogonal with respect to numerical sequences and on Hahn’s theorem, Izv. Akad. Nauk, 4 (1940), 215 – 228.

    MathSciNet  Google Scholar 

  64. . Ja. L. Geronimus, The orthogonality of some systems of polynomials, Duke Math. J., 14 (1947), 503 – 510.

    MATH  MathSciNet  Google Scholar 

  65. Ja. L. Geronimus, Orthogonal polynomials, Appendix, Amer. Math. Soc. translations (2), 108, 37–130.

    Google Scholar 

  66. . Gnedenko, Sur l’unicité du système de fonctions orthogonales invariant par rapport à la dérivation, Compt. Rend s (Dokl.) de l’Acad. des Sci. de l’URSS, 14 (1937), 159 – 161.

    Google Scholar 

  67. . W. Hahn, Über Orthogonal polynome, die q- Differenzengleichungen genugen, Math Nach., 2 (1949), 4 – 34.

    Article  MATH  Google Scholar 

  68. . W. Hahn, Über die Jacobischen Polynome und zwei verwandte Polynomklassen, Math. Zeit., 39 (1935), 634 – 638.

    Article  Google Scholar 

  69. . W. Hahn, Über höhere Ableitungen von Orthogonalpolynomen, Math. Zeit., 43 (1937), 101.

    Article  Google Scholar 

  70. . W. Hahn, Über Polynome, die gleichzweitig zwei verschiedenen Orthogonalsystemen angehoren, Math. Nach., 2 (1949), 263 – 278.

    Article  MATH  Google Scholar 

  71. . W. Hahn, Über Orthogonalitätserhaltende Operatoren, Math. Ves., 12 (1975), 337 – 339.

    Google Scholar 

  72. . E.H. Hildebrandt, Systems of polynomials connected with the Charlier expamsion and the Pearson differential equation, Ann. Math. Statistics, 2 (1931), 379 – 439.

    Article  MATH  Google Scholar 

  73. . M. Ismail, Orthogonal polynomials in a certain class of polynomials, Bull.Inst. Polit din Iasi, 20 (1974), 45 – 50

    Google Scholar 

  74. . S. Karlin and G. Szegö, On certain determinants, Journal d’Analyse Math. 8 (1960), 1 – 157.

    Article  Google Scholar 

  75. . A.M. Krall, Chebyshev sets of polynomials which satisfy an ordinary differential equation, SIAM Review, 22 (1980), 436 – 441.

    Article  MATH  MathSciNet  Google Scholar 

  76. . H.L. Krall, On orthogonal polynomials satisfying certain fourth order differential equations, Pennsylvania State College Studies #6, State College,Pa., 1940.

    Google Scholar 

  77. . H.L. Krall, On derivatives of orthogonal polynomials, Amer. Math. Soc. Bull., 42 (1936), 423 – 428.

    Article  MathSciNet  Google Scholar 

  78. . H.L. Krall, On higher derivatives of orthogonal polynomials, Amer. Math. Soc. Bull., 42 (1936), 867 – 870.

    Article  MathSciNet  Google Scholar 

  79. . H.L. Krall and I.M. Sheffer, A characterization of orthogonal polynomials, J. Math. Anal. Appl., 8 (1964), 232 – 244.

    Article  MATH  MathSciNet  Google Scholar 

  80. . H.L. Krall and I.M. Sheffer, Differential equations of infinite order for orthogonal polynomials, Annali di Mat. (4), 74 (1966), 136 – 172

    Article  MathSciNet  Google Scholar 

  81. . H.L Krall and I.M. Sheffer, On pairs of related orthogonal polynomial sets, Math. Zeit., 86 (1965), 425 – 450

    Article  MATH  MathSciNet  Google Scholar 

  82. . O.E. Lancaster, Orthogonal polynomials defined by difference equations, American Journal of Mathematics, 63 (1941), 185 – 207.

    Article  MathSciNet  Google Scholar 

  83. . I.L. Lanzewizky, Über Die Orthogonalität der Fejér-Szegöschen polynome, Comptes Rendus (Doklady) de l‘Académie des Sciences de l’URSS, 31 (1941), 199 – 200.

    MathSciNet  Google Scholar 

  84. . D.A. Leonard, Orthogonal polynomials, duality and association schemes, SIAM J. Math. Anal., 13 (1982), 656 – 663.

    Google Scholar 

  85. . P. Lesky, Über Polynomsysteme, die Sturm-Liouvilleschen Differenzengleichungen genügen, Math. Zeit., 78 (1962), 439 – 445.

    Article  MATH  MathSciNet  Google Scholar 

  86. . P. Lesky, Orthogonale Polynomsysteme als Lösungen Sturm-Liouvilleschen Differenzengleichungen, Monatshefte Für Math., 66 (1962), 203 – 214.

    Article  MATH  MathSciNet  Google Scholar 

  87. . P. Lesky, Polinomi ortogonali classici caratterizzati mediante equazioni differenziali del tipo Sturm-Liouville, Simposio di didatties della matematica, 1964.

    Google Scholar 

  88. . P. Lesky, Die Charakteisierung der klassischen orthogonalen Polynome durch Sturm- Liouvillesche Differentialgleichungen, Archiv for Rational Mechanics and Analysis, 10 (1962), 341 – 351.

    Article  MATH  MathSciNet  Google Scholar 

  89. . D.C. Lewis, Orthogonal functions whose derivatives are also orthogonal, Rend. Circ. Mat. Palermo (2), 2 (1953), 159 – 168.

    Article  MATH  MathSciNet  Google Scholar 

  90. N.N. Luzin, Integnal and Trigonometric series in “Collected Works of N.N. Luzin” vol. 1, 48–212, Acad. Of Sciences of the USRR,1953.

    Google Scholar 

  91. P. Maroni, Une caracterisation des polynomes orthogonaux semi-classique, C.R. Acad. Sci. Paris, 301, ser.l (1985), 269 – 272.

    Google Scholar 

  92. . P. Maroni, Prologomenes a l’etude des polynomes orthogonaux semi- classique, Ann. Mat. pura ed Appl. (4), 149 (1987), 165 – 184.

    MathSciNet  Google Scholar 

  93. . P. J. McCarthy, Characterization of the classical orthogonal polynomials, Portugaliae Mathematica, 20 (1961), 47 – 52.

    MATH  MathSciNet  Google Scholar 

  94. . J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugen den Funktionen, J. London Math. Soc., 9 (1934), 6 – 13.

    Article  Google Scholar 

  95. . J. Meixner, Symmetric systems of orthogonal polynomials, Arch. Rat. Mech. Analysis, 44 (1972), 69 – 75.

    MathSciNet  Google Scholar 

  96. . J.C. Merlo, On orthogonal polynomials and second order linear difference operators, Annales Polonici Math., 19 (1967), 69 – 79.

    MATH  MathSciNet  Google Scholar 

  97. . M. Mikolás, Common characterization of the Jacobi, Laguerre and Hermite-like polynomials (in Hungarian), Matematikai Lapok, 7 (1956), 238 – 248.

    MathSciNet  Google Scholar 

  98. Nikiforov and V. Uvarov, Special Functions of Mathematical Physics, Birkhauser Verlag, 1988 (transl. by R.P. Boas)

    Google Scholar 

  99. . G. Peebles, Some characterizations of the theory of orthogonal polynomials, Duke Math J., 6 (1940), 89 – 100.

    Article  MathSciNet  Google Scholar 

  100. . M. Perlstadt, A property of orthogonal polynomial families with polynomial duals, SIAM J. Math Anal., 15 (1984), 1043 – 1054.

    MATH  MathSciNet  Google Scholar 

  101. . Ronveaux, Polynôme orthogonaux dont les polynômes derives sont quasi orthogonaux, Comptes Rendus Acad. Sci. Paris, Ser. A, 289 (1979), 433 – 436.

    Google Scholar 

  102. . E.J. Routh, On some properties of certain solutions of a differential equation of the second order, Proc London Math Soc., 16 (1885), 245 – 261.

    Article  MATH  Google Scholar 

  103. . I.M. Sheffer, Some properties of polynomials of type zero, Duke Math. J., 5 (1939), 590 – 622

    MathSciNet  Google Scholar 

  104. . J. Shohat, The relation of the classical orthogonal polynomials to the polynomials of Appell, Amer. J. Math., 58 (1936), 453 – 464.

    Article  MathSciNet  Google Scholar 

  105. N. Ja. Sonin, Über die angenäherte Berechnung der bestimmten Integrale und Über die dabei vorkommenden ganzen Functionen. Warsaw Univ. Izv., 18(1887), 1–76. Jbuch. Fortschritte Math. 19, p. 282.

    Google Scholar 

  106. . G. Szegö, Orthogonal Polynomials, 4th ed., Amer. Math Soc Colloqu. Pub., vol. 23, Providence, R.I. 1975.

    Google Scholar 

  107. . M.S. Šun, On a characteristic property of classical orthogonal polynomials, Trudy Harkov Aviction Inst. 15 (1954), 25 – 26.

    Google Scholar 

  108. . L. Toscano, Polinomi ortogonali o reciproci di ortogonali nella classe di Appell, Le Matematica 11 (1956), 168 – 174

    MathSciNet  Google Scholar 

  109. . F. Tricomi, Equazioni differenziali, Torino, 1948.

    Google Scholar 

  110. . A.K. Varma, A new characterization of Hermite polynomials, Acta Math. Hung., 49 (1987), 169 – 172

    Article  MATH  MathSciNet  Google Scholar 

  111. . Verma and J. Prasad, Characterization of some orthogonal polynomials, Pub. de l’Institut Math. (Beograd), 9 (1969), 177 – 180.

    Google Scholar 

  112. . H.S. Wall, A continued fraction related to some partition formulas of Euler, Amer. Math. Monthly, 48 (1941), 102 – 108.

    Article  Google Scholar 

  113. . M. Weber and A. Erdélyi, On the finite difference analog of Rodrigues’ formula, Amer. Math. Monthly, 59 (1952), 163 – 168.

    Article  MATH  Google Scholar 

  114. . M. Webster, Orthogonal polynomials with orthogonal derivatives, Bull. Amer. Math. Soc., 44 (1938), 880 – 888.

    Article  MathSciNet  Google Scholar 

  115. . K.P. Williams, A uniqueness theorem for the Legendre and Hermite polynomials, Trans Amer. Math. Soc., 26 (1924), 441 – 445.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Kluwer Academic Publishers

About this chapter

Cite this chapter

Al-Salam, W.A. (1990). Characterization Theorems for Orthogonal Polynomials. In: Nevai, P. (eds) Orthogonal Polynomials. NATO ASI Series, vol 294. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0501-6_1

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-0501-6_1

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6711-9

  • Online ISBN: 978-94-009-0501-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics