On a class of differential-difference equations arising in number theory

Hildebrand, Adolf; Tenenbaum, Gérald

doi:10.1007/bf02788841

Cited by 33 publications

(28 citation statements)

References 13 publications

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“…[12]. (This formula corrects the one stated in [12] where there is a typo that, as Prof. Tenenbaum pointed out to us, was introduced by the printer after the proofcorrections had taken place. )…”

Section: Proof Of Theoremsupporting

confidence: 72%

Multivariate Diophantine equations with many solutions

et al. 2003

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Among other things we show that for each n-tuple of positive rational numbers (a 1 , . . . , a n ) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a 1 x 1 +• • •+a n x n = 1 with x 1 , . . . , x n S-units are not contained in fewer than exp((4 + o(1))s 1/2 (log s) −1/2 ) proper linear subspaces of C n . This generalizes a result of Erdős, Stewart and Tijdeman [7] for S-unit equations in two variables. Further, we prove that for any algebraic number field K of degree n, any integer m with 1 ≤ m < n, and any sufficiently large s there are integers

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Section: Proof Of Theoremsupporting

confidence: 72%

Multivariate Diophantine equations with many solutions

et al. 2003

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“…The fourth type focuses on estimating incomplete sums of multiplicative functions [3,13,29]. Despite the extensive literature on the subject, apart from the treatment of a few isolated cases [7,8,21,22,23,31,32,39,41,42] the main focus has been on the application to problems in number theory and combinatorics as opposed to the study of the underlying difference differential equations as a problem in its own right. Although a systematic study of equation (1.1) in full generality was initiated in [9], there is still additional work that remains to be done with regard to providing a comprehensive treatment.…”

Section: Historical Overviewmentioning

confidence: 99%

A pair of difference differential equations of Euler-Cauchy type

Bradley

2003

Trans. Amer. Math. Soc.

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Abstract. We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory and combinatorics. They are also of use in linear control theory. Here, we study these equations in a general setting. Building on previous work going back to de Bruijn, we show how adjoint equations arise naturally in the problem of uniqueness of solutions. Exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results. Specifically, we prove here that the general Euler-Cauchy difference differential equation with advanced arguments has a unique solution (up to a multiplicative constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with retarded arguments, we focus on a corresponding class of solutions, locating and classifying the points of discontinuity. We also provide an explicit asymptotic expansion at infinity.

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“…Recently, Hildebrand [9] and Hildebrand and Tenenbaum [11] investigated the asymptotic behavior of the solution of more general a differential-difference equation. In their argument, the complex integration and the Laplace transform were used.…”

Section: Corollary 3 For Any Positive Integer K and U > U O We Havementioning

confidence: 99%