Integer-valued entire functions
Author:
Raphael M. Robinson
Journal:
Trans. Amer. Math. Soc. 153 (1971), 451-468
MSC:
Primary 30.57
DOI:
https://doi.org/10.1090/S0002-9947-1971-0274762-7
MathSciNet review:
0274762
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: The theory of integer-valued entire functions is organized in an improved fashion. Detailed results are proved when the indicator diagram is a line segment. For the first time, a method is developed for treating completely integer-valued functions with an unsymmetrical growth pattern.
- [1] R. Creighton Buck, Integral valued entire functions, Duke Math. J. 15 (1948), 879–891. MR 29984
- [2] G. Doetsch, Theorie und Anwendung der Laplace-Transformation, Springer, Berlin, 1937.
- [3] M. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z. 17 (1923), no. 1, 228–249 (German). MR 1544613, https://doi.org/10.1007/BF01504345
- [4] M. Fekete and G. Szegö, On algebraic equations with integral coefficients whose roots belong to a given point set, Math. Z. 63 (1955), 158–172. MR 72941, https://doi.org/10.1007/BF01187931
- [5] L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 53 (1857), 173-175.
- [6] C. Pisot, Über ganzwertige ganze Funktionen, Jber. Deutsch. Math.-Verein. 52 (1942), 95–102 (German). MR 8256
- [7] Charles Pisot, Sur les fonctions arithmétiques analytiques à croissance exponentielle, C. R. Acad. Sci. Paris 222 (1946), 988–990 (French). MR 16482
- [8] Charles Pisot, Sur les fonctions analytiques arithmétiques et presque arithmétiques, C. R. Acad. Sci. Paris 222 (1946), 1027–1028 (French). MR 16483
- [9] G. Pólya, Über ganzwertige ganze Funktionen, Rend. Circ. Mat. Palermo 40 (1915), 1-16.
- [10] -, Sur les séries entières à coefficients entiers, Proc. London Math. Soc. (2) 21 (1922), 22-38.
- [11] G. Pólya, Über gewisse notwendige Determinantenkriterien für die Fortsetzbarkeit einer Potenzreihe, Math. Ann. 99 (1928), no. 1, 687–706 (German). MR 1512473, https://doi.org/10.1007/BF01459120
- [12] G. Pólya, Untersuchungen über Lücken und Singularitäten von Potenzreihen, Math. Z. 29 (1929), no. 1, 549–640 (German). MR 1545027, https://doi.org/10.1007/BF01180553
- [13] Raphael M. Robinson, Intervals containing infinitely many sets of conjugate algebraic integers, Studies in mathematical analysis and related topics, Stanford Univ. Press, Stanford, Calif., 1962, pp. 305–315. MR 0144892
- [14] Raphael M. Robinson, Algebraic equations with span less than 4, Math. Comp. 18 (1964), 547–559. MR 169374, https://doi.org/10.1090/S0025-5718-1964-0169374-X
- [15] Raphael M. Robinson, Intervals containing infinitely many sets of conjugate algebraic units, Ann. of Math. (2) 80 (1964), 411–428. MR 175881, https://doi.org/10.2307/1970656
- [16] Raphael M. Robinson, An extension of Pólya’s theorem on power series with integer coefficients, Trans. Amer. Math. Soc. 130 (1968), 532–543. MR 219706, https://doi.org/10.1090/S0002-9947-1968-0219706-9
- [17] Raphael M. Robinson, Conjugate algebraic integers on a circle, Math. Z. 110 (1969), 41–51. MR 246826, https://doi.org/10.1007/BF01114639
Retrieve articles in Transactions of the American Mathematical Society with MSC: 30.57
Retrieve articles in all journals with MSC: 30.57
Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1971-0274762-7
Keywords:
Integer-valued,
completely integer-valued,
unsymmetrical growth,
exponential type,
indicator diagram,
Laplace transform,
singularity hull,
generating function,
power series with integer coefficients,
transfinite diameter,
conjugate algebraic integers
Article copyright:
© Copyright 1971
American Mathematical Society