Abstract
We survey in this paper characterization theorems dealing with polynomial sets which are orthogonal on the real line.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
. 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.
. J. Aczél, Eine Bemerkung über Charakterisierung der “klassische” orthogonalpolynome. rActa. Math. Acad. Sci. Hung., 4 (1953), 315 – 321.
. J. Aczél, Sur l’equation différentielle des polynomes orthogonaux classiques, Annales Univ. Scient. Budapest, 2 (1959), 27 – 29.
. N.A. Al-Salam, Orthogonal polynomials of hypergeometric type, Duke Math. J., 33 (1966), 109 – 122.
. N.A. Al-Salam and W.A. Al-Salam, Some characterizations of the ultraspherical polynomials,Canad. Math. Bulletin, 11 (1968), 457 – 464.
. 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.
. W.A. Al-Salam, On a characterization of orthogonality, Math. Mag., 31 (1957), 41 – 44.
. W.A. Al-Salam, On a characterization of Meixner’s polynomials, The Quart. J. of Mathematics (Oxf)(2), 17 (1966), 7 – 10.
. W.A. Al-Salam, On a characterization of a certain set of orthogonal polynomials, Boll. Unione Mat. Ital.(3), 19 (1964), 448 – 450.
. 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.
. W.A. Al-Salam, W.R Allaway, and R. Askey, Sieved ultrashperical orthogonal polynomials, Trans. Amer. Math. Soc., 284 (1984), 39 – 55.
. W.A. Al-Salam and L. Carlitz, Some orthogonal q-polynomials, Math. Nachr., 30 (1965), 47 – 61.
. W.A. Al-Salam and T.S. Chihara, Another characterization of the classical orthogonal polynomials, SIAM J. Math. Anal., 3 (1972), 65 – 70.
. W.A. Al-Salam and T.S. Chihara, Convolution of orthogonal polynomials, SIAM J. Math. Anal., 7 (1976), 16 – 28.
. 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.
. W.A. Al-Salam and M. Ismail, Orthogonal polynomials associated with the Rogers- Ramanujan continued fractions, Pacific J. of Math., 104 (1983), 269 – 283.
. W.A. Al-Salam and A. Verma, Some orthogonality preserving operators, Proc. Amer. Math. Soc., 23 (1969), 136 – 139.
W.A. Al-Salam and A. Verma, Orthogonality preserving operators I, Rendiconti Acad. Naz. dei Lincei(8), LVIII(1975), 833–838.
W.A. Al-Salam and A. Verma, Orthogonality preserving operators II, Rendiconti Acad. Naz, dei Lincei(8), LIX(1976), 26–31.
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).
. W.A. Al-Salam and A. Verma, On an orthogonal polynomial set, Indagationes Math- ematicae, 44 (1982), 335 – 340.
. W.A Al-Salam and A. Verma, Some sets of orthogonal polynomials, Rev. Téc. Ing., Univ. Zulia, 9 (1986), 83 – 88.
. W.R. Allaway, The identification of a class of orthogonal polynomial sets, Ph.D. thesis, University of Alberta, Edmonton, Canada., 1972.
. 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
. Angelesco, Sur les polynomes orthogonaux en rapport avec d’autre polynomes, Buletinul Societâtii Stiite din Cluj, 1 (1921), 44 – 59.
. R. Askey, Divided difference operators and classical orthogonal polynomials, Rocky Mountain J. Math, (1989) To appear.
. R. Askey and M. Ismail, A generalization of the ultrasphericai polynomials, Studies in Pure Mathematics, edited by P. Erdös, Birkhauser, Basel, 1983, 55 – 78.
. R. Askey and M. Ismail, Recurrence relations, continued fractions and orthogonal polynomials, Memoirs of the Amer. Math. Soc. #300, 1984.
. R. Askey and J. Wilson, Some basic hypergeometrie orthogonal polynomials that generalize Jacobi polynomials, Memoirs AMS #319, 1985
. F.S. Beale, On a certain class of orthogonal polynomials, Annals of Math. Statistics, 12 (1941), 97 – 103.
S. Bochner, Über Sturm-Liouvillesche Polynomsysteme, Math. Zeit., 29(1929), 730– 736.
. S. Bonan and P. Nevai, Orthogonal polynomials and their derivatives, I, J. Approximation Theory, 40 (1984), 134 – 147.
. S. Bonan, D. Lubinsky, and P. Nevai, Orthogonal polynomials and their derivatives, II, SIAM J. Math. Anal., 18 (1987), 1163 – 1176.
. 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.
. 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.
. L. Carlitz, Characterization of certain sequences of orthogonal polynomials, Portugaliae Math., 20 (1961), 43 – 46.
. L. Carlitz, Characterization of the Krawtchouk polynomials, Revista Mat. Hisp-Amer. (4), 21 (1961), 79 – 84.
. L. Carlitz, Characterization of certain sequences of orthogonal polynomials, Portugaliae Math., 20 (1961), 43 – 46.
. L. Carlitz, Note on Legendre polynomials, Bull. Calcutta Math. Society, 46 (1954), 93 – 95.
. L. Carlitz, Characterization of the Laguerre polynomials, Monatshefte für Mathematik, 66 (1962), 389 – 392.
. T.S. Chihara, Orthogonal polynomials with Brenke type generating function, Duke Math. J., 35 (1968), 505 – 518.
. T.S. Chihara, Orthogonality relations for a class of Brenke polynomials, Duke Math. J., 38 (1971), 599 – 603.
. T.S. Chihara, A characterization of a class of distribution functions for the Stieltjes- Wigert polynomials, Canadian Math. Bull., 13 (1970), 529 – 532.
44.T.S. Chihara, On generalized Stieltjes-Wigert and related orthogonal polynomials, Journal of Computational and Applied Mathematics, 5 (1979), 291 – 297.
T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach Pub., 1976.
. T.S. Chihara and M. Ismail, Orthogonal polynomials suggested by queueing model, Advances in Mathematics, 3 (1982), 441 – 462.
. C.W. Cryer, Rodriques’ formulas and the classical orthogonal polynomials, Boll. Unione Mat. Ital. (3), 25 (1970), 1 – 11.
. Á Császár, Sur les polynômes orthogonaux classiques. Annales Univ. Sci. Budapest sec. Math., 1 (1958), 33 – 39.
. Danese, On a characterization of ultraspherical polynomials, Boll. U.M.I. (3), 21 (1966), 1 – 3.
. D. Dickinson, On quasi-orthogonal polynomials, Proc. Amer. Math. Soc., 12 (1961), 185 – 194.
. G.K. Eagleson, A characterization theorem for positive definite sequences on the Kraw- tchouk polynomials, Australian J. Statistics, 2 (1969), 29 – 38.
. G. Ebert, Über Polynomsysteme mit Rodriquessher Darstellung. Dissertation, Cologne, 1964.
K. Endl, On the involutory property Laguerre polynomials, Colloquia Math. Soc. János Bolyai, 19(1976) Budapest.
. Erdélyi et al., Higher Transcendental Functions, Vol. 2, McGraw-Hill Pub., 1953.
. L. Fejér, Absch. Für die Legendreschen und verwandte Polynome, Math. Zeit., 24 (1925), 285 – 298.
. E. Feldheim, Sur les polynomes généralisés de Legendre, Bull, de l‘Academie des Science de l’URSS, 5 (1941), 241 – 248.
. E. Feldheim, Sur une propriété des polynomes orthogonaux, J. London Math. Soc., 13 (1938), 44 – 53.
. E. Feldheim, Une propriété caractéristique des polynomes de Laguerre, Comment. Math. Helv., 13 (1940), 6 – 10.
. L. Feldmann, On a characterization of the classical orthogonal polynomials, Acta Sc. Math., 17 (1956), 129 – 133.
. L. Feldmann, Über durch Sturm-Liouvillesche Differentialgleichungen charakterisierte orthogonale Polynomsysteme. Pub. Math. (Debrecen), 3 (1954), 297 – 304.
. 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.
. B. Gabutti, Some characteristic property of Meixner polynomials, J. Math. Analysis and Applications, 95 (1983), 265 – 277.
. Ja. L. Geronimus, On polynomials orthogonal with respect to numerical sequences and on Hahn’s theorem, Izv. Akad. Nauk, 4 (1940), 215 – 228.
. Ja. L. Geronimus, The orthogonality of some systems of polynomials, Duke Math. J., 14 (1947), 503 – 510.
Ja. L. Geronimus, Orthogonal polynomials, Appendix, Amer. Math. Soc. translations (2), 108, 37–130.
. 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.
. W. Hahn, Über Orthogonal polynome, die q- Differenzengleichungen genugen, Math Nach., 2 (1949), 4 – 34.
. W. Hahn, Über die Jacobischen Polynome und zwei verwandte Polynomklassen, Math. Zeit., 39 (1935), 634 – 638.
. W. Hahn, Über höhere Ableitungen von Orthogonalpolynomen, Math. Zeit., 43 (1937), 101.
. W. Hahn, Über Polynome, die gleichzweitig zwei verschiedenen Orthogonalsystemen angehoren, Math. Nach., 2 (1949), 263 – 278.
. W. Hahn, Über Orthogonalitätserhaltende Operatoren, Math. Ves., 12 (1975), 337 – 339.
. E.H. Hildebrandt, Systems of polynomials connected with the Charlier expamsion and the Pearson differential equation, Ann. Math. Statistics, 2 (1931), 379 – 439.
. M. Ismail, Orthogonal polynomials in a certain class of polynomials, Bull.Inst. Polit din Iasi, 20 (1974), 45 – 50
. S. Karlin and G. Szegö, On certain determinants, Journal d’Analyse Math. 8 (1960), 1 – 157.
. A.M. Krall, Chebyshev sets of polynomials which satisfy an ordinary differential equation, SIAM Review, 22 (1980), 436 – 441.
. H.L. Krall, On orthogonal polynomials satisfying certain fourth order differential equations, Pennsylvania State College Studies #6, State College,Pa., 1940.
. H.L. Krall, On derivatives of orthogonal polynomials, Amer. Math. Soc. Bull., 42 (1936), 423 – 428.
. H.L. Krall, On higher derivatives of orthogonal polynomials, Amer. Math. Soc. Bull., 42 (1936), 867 – 870.
. H.L. Krall and I.M. Sheffer, A characterization of orthogonal polynomials, J. Math. Anal. Appl., 8 (1964), 232 – 244.
. H.L. Krall and I.M. Sheffer, Differential equations of infinite order for orthogonal polynomials, Annali di Mat. (4), 74 (1966), 136 – 172
. H.L Krall and I.M. Sheffer, On pairs of related orthogonal polynomial sets, Math. Zeit., 86 (1965), 425 – 450
. O.E. Lancaster, Orthogonal polynomials defined by difference equations, American Journal of Mathematics, 63 (1941), 185 – 207.
. 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.
. D.A. Leonard, Orthogonal polynomials, duality and association schemes, SIAM J. Math. Anal., 13 (1982), 656 – 663.
. P. Lesky, Über Polynomsysteme, die Sturm-Liouvilleschen Differenzengleichungen genügen, Math. Zeit., 78 (1962), 439 – 445.
. P. Lesky, Orthogonale Polynomsysteme als Lösungen Sturm-Liouvilleschen Differenzengleichungen, Monatshefte Für Math., 66 (1962), 203 – 214.
. P. Lesky, Polinomi ortogonali classici caratterizzati mediante equazioni differenziali del tipo Sturm-Liouville, Simposio di didatties della matematica, 1964.
. P. Lesky, Die Charakteisierung der klassischen orthogonalen Polynome durch Sturm- Liouvillesche Differentialgleichungen, Archiv for Rational Mechanics and Analysis, 10 (1962), 341 – 351.
. D.C. Lewis, Orthogonal functions whose derivatives are also orthogonal, Rend. Circ. Mat. Palermo (2), 2 (1953), 159 – 168.
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.
P. Maroni, Une caracterisation des polynomes orthogonaux semi-classique, C.R. Acad. Sci. Paris, 301, ser.l (1985), 269 – 272.
. P. Maroni, Prologomenes a l’etude des polynomes orthogonaux semi- classique, Ann. Mat. pura ed Appl. (4), 149 (1987), 165 – 184.
. P. J. McCarthy, Characterization of the classical orthogonal polynomials, Portugaliae Mathematica, 20 (1961), 47 – 52.
. J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugen den Funktionen, J. London Math. Soc., 9 (1934), 6 – 13.
. J. Meixner, Symmetric systems of orthogonal polynomials, Arch. Rat. Mech. Analysis, 44 (1972), 69 – 75.
. J.C. Merlo, On orthogonal polynomials and second order linear difference operators, Annales Polonici Math., 19 (1967), 69 – 79.
. M. Mikolás, Common characterization of the Jacobi, Laguerre and Hermite-like polynomials (in Hungarian), Matematikai Lapok, 7 (1956), 238 – 248.
Nikiforov and V. Uvarov, Special Functions of Mathematical Physics, Birkhauser Verlag, 1988 (transl. by R.P. Boas)
. G. Peebles, Some characterizations of the theory of orthogonal polynomials, Duke Math J., 6 (1940), 89 – 100.
. M. Perlstadt, A property of orthogonal polynomial families with polynomial duals, SIAM J. Math Anal., 15 (1984), 1043 – 1054.
. Ronveaux, Polynôme orthogonaux dont les polynômes derives sont quasi orthogonaux, Comptes Rendus Acad. Sci. Paris, Ser. A, 289 (1979), 433 – 436.
. 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.
. I.M. Sheffer, Some properties of polynomials of type zero, Duke Math. J., 5 (1939), 590 – 622
. J. Shohat, The relation of the classical orthogonal polynomials to the polynomials of Appell, Amer. J. Math., 58 (1936), 453 – 464.
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.
. G. Szegö, Orthogonal Polynomials, 4th ed., Amer. Math Soc Colloqu. Pub., vol. 23, Providence, R.I. 1975.
. M.S. Šun, On a characteristic property of classical orthogonal polynomials, Trudy Harkov Aviction Inst. 15 (1954), 25 – 26.
. L. Toscano, Polinomi ortogonali o reciproci di ortogonali nella classe di Appell, Le Matematica 11 (1956), 168 – 174
. F. Tricomi, Equazioni differenziali, Torino, 1948.
. A.K. Varma, A new characterization of Hermite polynomials, Acta Math. Hung., 49 (1987), 169 – 172
. Verma and J. Prasad, Characterization of some orthogonal polynomials, Pub. de l’Institut Math. (Beograd), 9 (1969), 177 – 180.
. H.S. Wall, A continued fraction related to some partition formulas of Euler, Amer. Math. Monthly, 48 (1941), 102 – 108.
. M. Weber and A. Erdélyi, On the finite difference analog of Rodrigues’ formula, Amer. Math. Monthly, 59 (1952), 163 – 168.
. M. Webster, Orthogonal polynomials with orthogonal derivatives, Bull. Amer. Math. Soc., 44 (1938), 880 – 888.
. K.P. Williams, A uniqueness theorem for the Legendre and Hermite polynomials, Trans Amer. Math. Soc., 26 (1924), 441 – 445.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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