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2020 On transcendental entire functions with infinitely many derivatives taking integer values at several points
Michel Waldschmidt
Mosc. J. Comb. Number Theory 9(4): 371-388 (2020). DOI: 10.2140/moscow.2020.9.371

Abstract

Let s0,s1,,sm1 be complex numbers and r0,,rm1 rational integers in the range 0rjm1. Our first goal is to prove that if an entire function f of sufficiently small exponential type satisfies f(mn+rj)(sj) for 0jm1 and all sufficiently large n, then f is a polynomial. Under suitable assumptions on s0,s1,,sm1 and r0,,rm1, we introduce interpolation polynomials Λnj (n0, 0jm1) satisfying

Λ n j ( m k + r ) ( s ) = δ j δ n k f o r n , k 0 a n d 0 j , m 1 ,

and we show that any entire function f of sufficiently small exponential type has a convergent expansion

f ( z ) = n 0 j = 0 m 1 f ( m n + r j ) ( s j ) Λ n j ( z ) .

The case rj=j for 0jm1 involves successive derivatives f(n)(wn) of f evaluated at points of a periodic sequence w=(wn)n0 of complex numbers, where wmh+j=sj (h0, 0jm). More generally, given a bounded (not necessarily periodic) sequence w=(wn)n0 of complex numbers, we consider similar interpolation formulae

f ( z ) = n 0 f ( n ) ( w n ) Ω w , n ( z )

involving polynomials Ωw,n(z) which were introduced by W. Gontcharoff in 1930. Under suitable assumptions, we show that the hypothesis f(n)(wn) for all sufficiently large n implies that f is a polynomial.

Citation

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Michel Waldschmidt. "On transcendental entire functions with infinitely many derivatives taking integer values at several points." Mosc. J. Comb. Number Theory 9 (4) 371 - 388, 2020. https://doi.org/10.2140/moscow.2020.9.371

Information

Received: 30 November 2019; Revised: 1 May 2020; Accepted: 15 May 2020; Published: 2020
First available in Project Euclid: 12 November 2020

zbMATH: 07272356
MathSciNet: MR4170703
Digital Object Identifier: 10.2140/moscow.2020.9.371

Subjects:
Primary: 30D15
Secondary: 41A58

Keywords: entire functions , exponential type , interpolation , Laplace transform , Lidstone series , method of the kernel , transcendental functions

Rights: Copyright © 2020 Mathematical Sciences Publishers

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