Progresivní sumační metody nekonečných řad

Mgr. Marian GENČEV

Doctoral thesis

ADVANCED SUMMATION METHODS

Anotácia:

Tato disertaˇcn´? pr´ace pojedn´av´a o progresivn´?ch sumaˇcn´?ch metod´ach nekone ˇcn´ych ˇrad re´aln´ych ˇc´?sel ve speci´aln´?m tvaru, jeˇz doposud nebyly prozkoum´any. Prvn´? kapitola zav´ad´? pojem nekoneˇcn´e ˇrady, nekoneˇcn´eho souˇcinu, jejich konvergence, obecn´e pozn´amky o sumaˇcn´?ch metod´ach nekoneˇcn´ych ˇrad a v´yznaˇcn´e vˇety a pˇr´?klady, kter´e jsou pouˇzity ve druh´e kapitole. Druh´a kapitola pˇredstavuje autorovy vlastn´? v´ysledky obsahuj´?c´? metody sˇc´?t´an´? nekoneˇcn´ych ˇrad speci´aln´?ch racion´aln´?ch nebo re´aln´ych ˇc´?sel. Tato kapitola se dˇel´? na ˇctyˇri ˇc´asti. Prvn´? ˇc´ast obsahuje obecnou sumaˇcn´? metodu nˇekter´ych nekoneˇcn´ych ˇrad obsahuj´?c´?ch zobecnˇen´e binomick´e koeficienty. Ve druh´e ˇc´asti jsou vyˇsetˇrov´any sumaˇcn´? techniky nekoneˇcn´ych ˇrad, jejichˇz ˇcleny obsahuj´? zobecnˇen´e binomick´e koeficienty a zobecnˇen´a harmonick´a ˇc´?sla prvn´?ho ˇr´adu. Tyto v´ysledky jsou aplikov´any k nalezen´? speci´aln´?ch hodnot dvojn´ych a trojn´ych nekoneˇcn´ych ˇrad, kter´e vedou k zaj´?mav´ym vyj´adˇren´?m ˇc´?sla . Ve tˇret´? ˇc´asti autor prezentuje techniku pro nalezen´? speci´aln´?ho nekoneˇcn´eho souˇcinu komplexn´?ch ˇc´?sel pro konstantu e6 (3) bez pouˇzit´? vˇety o hodnot´ach funkce (x), jeˇz se uv´ad´? v literatuˇre jako Gauss? digamma theorem. V´ysledky t´eto ˇc´asti obsahuj´? tak´e jednoduchou nerovnost pro diference funkce (x) odvozenou element´arn´?mi metodami. Posledn´? sekce popisuje nov´y pojem tzv. ˇretˇezov´ych ˇrad, kter´y byl definov´an autorem t´eto pr´ace. Obecn´e v´ysledky t´eto sekce poskytuj´? informaci o jist´e pˇrirozenˇe vznikaj´?c´? posloupnosti nekoneˇcn´ych ˇrad. Ukazuje se pˇrekvapiv´y vztah mezi limitou vzpomenut´e posloupnosti a nekoneˇcn´ymi souˇciny. Jako d°usledek v´ysledk°u prezentovan´ych v t´eto ˇc´asti uvedeme vyˇc´?slen´? speci´aln´? hodnoty Gaussovy zobecnˇen´e hypergeometrick´e funkce s iracion´aln´?mi hodnotami parametr°u pomoc ´? element´arn´?ch matematick´ych konstant. Aplikujeme teorii ˇretˇezov´ych ˇrad k jist´emu zrychlen´? konvergence Wallisova souˇcinu a k nalezen´? nov´ych pˇr´?klad°u re´aln´ych ˇc´?sel jeˇz obvykle naz´yv´ame almost integers. Vˇsechny v´ysledky byly, de facto, nalezeny v roce 2007. Tedy v´ysledky uveden´e v t´eto disertaˇcn´? pr´aci budou publikov´any v r°uzn´ych matematick´ych ˇcasopisech.

Abstract:

This thesis deals with advanced summation methods of infinite series of real numbers in a special form which were not discussed at this time. The thesis dividies into two chapters. The first chapter introduces the concept of infinite series, infinite products, the convergence of an infinite series or infinite product, general remarks to the summation of infinite series and significant theorems and examples which are used in Chapter 2. The second one presents author?s main results concerning summation of infinite series which summands are special rational or real numbers. This chapter is divided into four sections. The first section presents a general summation method of infinite series involving generalized binomial coefficients. In the second section are investigated summation techniques of infinite series which summands involve generalized binomial coefficients and generalized harmonic numbers of order 1. As a consequence of this results we apply them to deduce certain nontrivial evaluations of double and triple infinite series which lead to interesting representations of the number . In the third section the author explores a technique for finding a special infinite product of complex numbers for the constant e6 (3) without using Gauss? digamma theorem. The results in this section include a simple inequality for the differences of the digamma function (x) which proof uses elementary methods. The last section describes a new concept of the so called continued series which was defined by the author. General results of this section give an information about a naturally arising sequence of infinite series. There is found a surprising connection of the limit of the mentioned sequence of infinite series to special infinite products. As a consequence of the presented results we give one special evaluation of Gauss? generalized hypergeometric function with a special nonrational values of its paramaters. We apply the theory of continued series to accelerate Wallis? product and to find a new type of almost integers. All the results were found, in fact, in the year 2007. Hence the results described in this thesis are going to be published in mathematical journals.

Kľúčové slová

Nekoneˇcn´e ˇrady, nekoneˇcn´e souˇciny, posloupnosti re´aln´ych ˇc´?sel, sumaˇcn´?

Jazyk práce: Czech

Datum vytvoření / odevzdání či podání práce: 7. 11. 2007

Zverejniť od: 7. 11. 2007

Obhajoba závěrečné práce

• Obhajoba proběhla 7. 10. 2008
• Vedúci: prof. RNDr. Jaroslav Hančl, CSc.

University of Ostrava

Faculty of Science

Doctoral programme / odbor:
Applied Mathematics / Applied Mathematics