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A General Turán Expression for the Zeta Function

Published online by Cambridge University Press:  20 November 2018

H. W. Gould
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In 1948 Gabor Szegő [9] gave four proofs of a remarkable inequality communicated to him by Paul Turán, who later published an original proof [10]. The Turán theorem states that if Pn(x) is the Le gendre polynomial, then

1.1

with equality holding only when |x| = 1.

Since then many similar inequalities have been found for various special functions, particularly for the Legendre and Hermite polynomials. Reference may be had to the recent work of Danese [2] and Chatterjea [1]. Danese gives an extensive bibliography.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 6 , Issue 3 , 01 September 1963 , pp. 359 - 366
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Chatterjea, S.K., On an associated function of Hermite polynomials, Math, Z., 78(1962), 116-121.Google Scholar
2. Danese, A.E., Explicit evaluations of Turán expressions, Ann, Mat. Pura Appl., 38(1955), 339-348.Google Scholar
3. Gould, H. W., Ageneralization of a problem of L, Lorch and L. Moser, Canad, Math, Bull., 4(1961), 303-305.Google Scholar
4. Gould, H. W., The operator Tf(x) = f(x+a)f(x+b)-f(x)f(x+a+b), to appear in the Math. Mag.Google Scholar
5. Gould, H. W., Notes on a calculus of Turán operators, Mathematica Monongaliae, No. 6, May, 1962.Google Scholar
6. Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers (3rd Ed.), Oxford, 1954.Google Scholar
7. Nőrlund, N. E., Vorle sungen űber Differenzenrechnung, New York, 1954.Google Scholar
8. Ramanujan, S., Some formulae in the analytic theory of numbers, Messenger of Math., 45(1916), 81-84.Google Scholar
9. Szegő, G., On an inequality of P, Turán concerning Le gendre polynomials, Bull. Amer. Math, Soc., 54(1948), 401-405.Google Scholar
10. Paul, Turán, On the zeros of the polynomials of Legendre, Časopis pěst, mat., 75(1950), 113-122.Google Scholar
11. Wilson, B. M., Proofs of some formulae enunciated by Ramanujan, Proc. London Math. Soc., (2)21(1923), 235-255.Google Scholar
12. Problem E 1396, Amer. Math. Monthly, 67(1960), 81-82; solution, 67(1960), 694.Google Scholar