Taylor-Dirichlet series and algebraic differential-difference equations

Wadleigh, Frank

doi:10.1090/S0002-9939-1980-0574513-3

Taylor-Dirichlet series and algebraic differential-difference equations
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by Frank Wadleigh PDF

Proc. Amer. Math. Soc. 80 (1980), 83-89 Request permission

Abstract:

It is proved that if a convergent Taylor-Dirichlet series \[ \sum \limits _{k = 0}^\infty {{P_k}(s){e^{ - {\lambda _k}s}},\quad s = \sigma + it,{\lambda _k} \in {\mathbf {C}},{P_k}} (s) \in {\mathbf {C}}[s],\mathcal {R}({\lambda _k}) \uparrow \infty ,\] satisfies an algebraic differential-difference equation then the set of its exponents $\{ {\lambda _k}\} _{k = 0}^\infty$ has a finite, linear, integral basis. This generalizes a theorem of A. Ostrowski. An application of the theorem to a problem of oscillation in neuro-muscular systems is indicated.

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Verallgemeinerung des Begriffes der Dirichletschen Reihen

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