Summary
Positive representations for [P (λ) n (x)]2−P −1/(λ) n (x)P +1/(λ) n (x) and for analogous expressions involving orthogonal polynomials are obtained.
Article PDF
Similar content being viewed by others
References
G. Szegö,On an inequaliiy of P. Turán concerning Legendre polynomials, « Bulletin of the American Mathematical Society », vol. 54 (1948), pp. 401–405.
Paul Turán,On the zeros of the polynomials of Legendre, « Časopis pro pěstovánf Matematiky a Fysiky », vol. 75 (1950), pp. 113–122.
B. S. Madhava Rao andV. R. Thiruvenkatachar,On an inequality concerning orthogonal polynomials, « Proceedings of the Indian Academy of Sciences », Section A, vol. 29 (1949), pp. 391–393.
Giovanni Sansone,Su una disuguaglianza di P. Turán relativa ai polinomi di Legendre, « Bollettino della Unione Matematica Italiana », (III), Anno IV (1949), pp. 221–223.
Giovanni Sansone,Su una disuguaglianza relativa ai polinomi di Legendre, « Bollettino della Unione Matematica Italiana », (III), Anno IV (1949), pp. 339–344.
Luigi Merli,Su alcune disuguaglianze riguardanti i polinomi ultrasferici, « Rendiconti dell'Accademia Nazionale dei Lincei », (Classe di scienze fisiche, matematiche e naturali) serie VIII, vol. 10, fasc. 5 (1951), pp. 371–374.
E. F. Beckenbach, W. Seidel, andOtto Szász,Recurrent determinants of Legendre and ultraspherical polynomials, « Duke Mathematical Journal », vol. 18, no 1 (1951), pp. 1–10.
Otto Szász,Inequalities concerning ultraspherical polynomials and Bessel functions, « Proceedings of the American Mathematical Society », vol. 1 (1950), pp. 256–267.
Otto Szász,Identities and inequalities concerning orthogonal polynomials and Bessel functions, « Journal d'Analyse Mathématique », vol. 1 (1951), pp. 116–134.
J. L. Burchnall,An algebraic property of the classical polynomials, « Proceedings of the London Mathematical Society », vol. 3, no 1 (1951), pp. 232–240.
B. N. Mukherjee andT. S. Nanjundiah,On an inequality relating to Laguerre and Hermite polynomials, « Math. Student. », vol. 19 (1951), pp. 47–48.
H. Skovgaard,On inequalities of the Turán type, « Mathematica Scandinavica », vol. 2, fasc. 1 (1954), pp. 65–73.
V. R. Thiruvenkatachar andT. S. Nanjundiah,Inequalities concerning Bessel functions and orthogonal polynomials, « Proceedings of the Indian Academy of Sciences », Section A, vol. 33 (1951), pp. 373–384.
Hsien-yü Hsü,Problem 4125 (1946, 470), « The American Mathematical Monthly », vol. 55, no 1 (1948), pp. 34–35.
Appell etKampé de Fériet,Fonctions hypergéométriques et hypersphériques, Paris, (1926), pp. 360–361.
L. Koschmieder,Das Vorzeichen gewisser aus Hermiteschen Polynomen zweiter Art gebildeter Determinanten, « Anz. Oster. Akad. Wiss. Math. — Nat. Kl. 1951 », (1951), pp. 165–167.
T. S. Nanjundiah,A note on an inequality of P. Turán for Legendre polynomials, « The Half-Yearly Journal of the Mysore University », New Series, Section B, vol. 11, no 1 (1950), pp. 57–58.
Gabor Szegö,Orthogonal Polynomials, « American Mathematical Society Colloquium Publications », vol. 23, New Jork, (1939).
Bateman Manuscript Project,Higher Transcendental Functions, vol. II, New York, (1953).
G. Vitali eG. Sansone,Moderna Teoria delle Funzioni di Variabile Reale, Terza Edizione, Parte II, Sviluppi in Serie di Funzioni Ortogonali, Bologna, (1952).
Wilhelm Magnus andFritz Oberhettinger,Formulas and Theorems for the Special Functions of Mathematical Physics, New York, (1949).
G. N. Watson,A Treatise on the Theory of Bessel Functions, 2d. Ed., Cambridge, (1948).
Author information
Authors and Affiliations
Additional information
This is an excerpt from the author's doctoral dissertation, written under the direction of ProfessorW. Seidel, to whom the author is grateful for his encouragement and assistance.
Rights and permissions
About this article
Cite this article
Danese, A.E. Explicit evaluations of Turán expressions. Annali di Matematica 38, 339–348 (1955). https://doi.org/10.1007/BF02413526
Issue Date:
DOI: https://doi.org/10.1007/BF02413526