ON THE DOMAIN OF THE CONVERGENCE OF TAYLOR-DIRECHLET SERIES WITH COMPLEX EXPONENTS
DOI:
https://doi.org/10.31471/2304-7399-2023-18(68)-25-31Keywords:
Taylor-Dirichlet series; abscissa of the existence of maximal termAbstract
The article deals with Dirichlet series of the form F(z) = Σ+∞ k=0 akezλk , as well as Taylor-Dirichlet series of the form F1(z) =+∞Σk=0 zmkakezλk , where (λk) is some sequence of pairwise different complex numbers, and mk is a sequence of non-negative integers, ak ∈ C. The main statements of the work are Statements 1–3 and relate to the description and relationship between the domains of convergence, absolute convergence, and the domains of existence of the maximum term of the above series F and F1.
References
W. Schnee, Zur Konvegenzproblem der Dirichletschen Reihen, Math. Ann., 66 (1908), no.3, 337–349. https://doi.org/10.1007/BF01450693
W. Schnee, Über Dirichlet’sche reihen, Rend. Circ. Matem. Palermo, 27, 87–116 (1909). https://doi.org/10.1007/BF03019647
W. Schnee, Über irreguläre Potenzreihen und Dirichletsche Reihen, Thesis, Diss. Friedrich Wilhelms Universität zu Berlin, Berlin, 1908.
G.H. Hardy, M. Riesz, The general theory of Dirichlet’s series, Cambridge Tracts in Math, and Math. Phys., 18, 1915, 78 p. https://ia902204.us.archive.org/5/items/cu31924060184441/cu31924060184441.pdf
G. Valiron, Sur l’abscisse de convergence des séries de Dirichlet, Bull. Soc. Math. Fr., 52 (1924), 166–174.
S. Mandelbrojt, Séries de Dirichlet: Principes et Méthodes, Gauthier-Villars, Paris, 1969.
J.F. Ritt, On a class of linear homogeneous differential equations of finite order with constant coefficients, Trans. Amer. Math. Soc., 18 (1917), 27–49.
E. Hille, Note on Dirichlet’s series with complex exponents, Ann. Math., 25 (1924), 261–278.
J. Micusiński, On Dirichlet series with complex exponents, Ann. Pol. Math., 2 (1955), no.2, 254–256.
T.M. Gallie, Region of convergence of Dirichlet series with complex exponents, Proc. Amer. Math. Soc., 7 (1956), no. 4, 627–629.
T.M. Gallie, Mandelbrojt’s inequality and Dirichlet series with complex exponents, Trans. Amer. Math. Soc., 90 (1959), no. 1, 57–72
O.A. Krivosheyeva, The convergence domain for series of exponential monomials, Ufa Math. J., 3 (2011), 42–55.
G. Peyser, On the domain of absolute convergence of Dirichlet series in several variables, Proc. Amer. Math. Soc., 9 (1958), 545–550.
А.Ю. Боднарчук, М.Р. Куриляк, О.Б. Скаскiв, Про абсцису iснування максимального члена випадкових рядiв Дiрiхле з довiльними додатними показниками, Вiсн. Львiв. ун-ту, сер. мех.-мат., 94 (2022), 79–88.
M.R. Kuryliak, O.B. Skaskiv, On the domain of convergence of general Dirichlet series with complex exponents, Carpathian Math. Publ., (2023) (submitted)
М.Р. Куриляк, О.Б. Скаскiв, Випадковi банахово-значнi ряди Дiрiхле зi зростаючою детермiнованою послiдовнiстю додатних показникiв, Прикарпатський вiсник НТШ. Число, 2022, No 17(64), 9–20. https://doi.org/10.31471/2304-7399-2022-17(64)-9-21
