Zeros of successive derivatives of entire functions of the form

Author:
Albert Edrei

Journal:
Trans. Amer. Math. Soc. **259** (1980), 207-226

MSC:
Primary 30D20; Secondary 30D15

DOI:
https://doi.org/10.1090/S0002-9947-1980-0561833-6

MathSciNet review:
561833

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Abstract: Consider where is a real entire function of finite order having no zeros in some strip . The author studies the power series (1) ( real) and the number of real zeros of which lie in the interval . He proves (2) . With regard to the expansion (1) he determines a positive, strictly increasing, unbounded sequence such that and having the following properties: (i) if , then and all the have the same sign; (ii) if in addition , tnen . It is possible to deduce from (2) the complete characterization of the final set (in the sense of Pólya) of .

**[1]**A. Edrei and G. R. MacLane,*On the zeros of the derivatives of an entire function*, Proc. Amer. Math. Soc.**8**(1957), 702-706. MR**0087741 (19:403a)****[2]**S. Hellerstein and J. Williamson,*Derivatives of entire functions and a question of Pólya*, Trans. Amer. Math. Soc.**227**(1977), 227-249. MR**0435393 (55:8353)****[3]**-,*Derivatives of entire functions and a question of Pólya*. II, Trans. Amer. Math. Soc.**234**(1977), 497-503. MR**0481004 (58:1151)****[4]**L. Moser and M. Wyman,*An asymptotic formula for the Bell numbers*, Trans. Roy. Soc. Canada**49**(1955), 49-54. MR**0078489 (17:1201c)****[5]**G. Pólya,*On the zeros of the derivatives of a function and its analytic character*, Bull. Amer. Math. Soc.**49**(1943), 178-191. MR**0007781 (4:192d)****[6]**G. Pólya and G. Szegö,*Aufgaben und Lehrsätze aus der Analysis*, vol. II, Springer-Verlag, Berlin and New York, 1925.**[7]**C. Prather,*On the zeros of successive derivatives of some entire functions*(unpublished).**[8]**G. Szegö,*Orthogonal polynomials*, 3rd ed., Amer. Math. Soc. Colloq. Publ., no. 23, Amer. Math. Soc., Providence, R. I., 1967.**[9]**E. C. Titchmarsh,*The theory of functions*, 2nd ed., Oxford Univ. Press, London, 1952. MR**0197687 (33:5850)**

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DOI:
https://doi.org/10.1090/S0002-9947-1980-0561833-6

Article copyright:
© Copyright 1980
American Mathematical Society