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A General Turán Expression for the Zeta Function

  • H. W. Gould

Extract

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.

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References

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

  • H. W. Gould

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