منابع مشابه
Series representations for the Stieltjes constants
The Stieltjes constants γk(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s, a) about s = 1. We present series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives an addition formula for the Stieltjes constants. As a byproduct, expressions for derivatives of a...
متن کاملSeries of zeta values , the Stieltjes constants , and a sum
We present a variety of series representations of the Stieltjes and related constants, the Stieltjes constants being the coefficients of the Laurent expansion of the Hurwitz zeta function ζ(s, a) about s = 1. Additionally we obtain series and integral representations of a sum Sγ(n) formed as an alternating binomial series from the Stieltjes constants. The slowly varying sum Sγ(n) + n is an impo...
متن کاملSummability of a Class of Fourier-stieltjes Series
If 0 « (t)-)-s then, for any /3 > oc, Op (t)->-s. It is sufficient to assume that s = 0. By (3), since F(t) is increasing, op(t) <-a q « (t) and the required result follows immediately. t LEMMA 6. Let y(t) be an increasing function. If g(t) =10 y(u) du-sta as t-*0, then y(t)~sct a-1 The proof is similar to that of a result due to Hardy and Littlewood*, to which the lemma reduces when y(t) is a ...
متن کاملOn the Distributional Stieltjes Transformation
This paper is concerned with some general theorems on the distributional Stieltjes transformation. Some Abelian theorems are proved. Throughout the paper, r will denote a positive continuous function on an interval (X,=), X > 0, such that the limit lim r(pt) r(t) exists for' every p > 0. Such functions are called regularly varying fu.Lc-tions (r.v.f.) at infinity and it is well known ([ 7] that...
متن کاملOn Higher Heine-stieltjes Polynomials
Take a linear ordinary differential operator d(z) = Pk i=1 Qi(z) d dzi with polynomial coefficients and set r = maxi=1,...,k(degQi(z) − i). If d(z) satisfies the conditions: i) r ≥ 0 and ii) degQk(z) = k + r we call it a nondegenerate higher Lamé operator. Following the classical examples of E. Heine and T. Stieltjes we initiated in [6] the study of the following multiparameter spectral problem...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1929
ISSN: 0002-9947
DOI: 10.2307/1989560