Ogled vira Stieltjesove konstante

[[Slika:gamma-area.svg|thumb|right|250px|[[Površina]] modrega območja konvergira k [[Euler-Mascheronijeva konstanta|Euler-Mascheronijevi konstanti]], ki je ničta Stieltjesova konstanta.]]

'''Stieltjesove konstante''' (ali '''posplošene Eulerjeve konstante'''<ref name="adell_2011">{{sktxt|Adell|2011}}.</ref>) so v [[matematika|matematki]] [[število|števila]] <math>\gamma_{n}\,</math>, ki se pojavljajo v [[Laurentova vrsta|Laurentovi vrsti]] za [[Riemannova funkcija zeta|Riemannovo funkcijo ζ]]:

: <math> \zeta (s) = \frac{1}{s-1}+\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} \gamma_{n} \; (s-1)^{n} \!\, . </math>

Ničta Stieltjesova konstanta <math>\gamma_{0} \equiv \gamma = {0},577215664901\ldots\, </math> je znana kot [[Euler-Mascheronijeva konstanta]]. [[matematična konstanta|Konstante]] se imenujejo po nizozemskem matematiku [[Thomas Joannes Stieltjes|Thomasu Joannesu Stieltjesu]] in redkeje po švicarskem matematiku, fiziku in astronomu [[Leonhard Euler|Leonhardu Eulerju]].

== Izrazi ==

Stieltjes je pokazal, da so konstante dane z [[limita (matematika)|limito]]:<ref name="adell_2011" /><ref>{{sktxt|Stieltjes|1905}}.</ref>

: <math> \gamma_{n} = \lim_{m \rightarrow \infty}
{ \left\{ \sum_{k = 1}^{m} \frac{\ln^{n} k}{k} - \frac{\ln^{n+1} \! m}{n+1} \right\} } \!\, . </math>{{efn|V primeru <math>n = 0\, </math> prvi sumandd zahteva računanje [[potenciranje|0<sup>0</sup>]], kar je zaradi [[prazni produkt|praznega produkta]] po dogovoru enako [[1 (število)|1]].}}

[[Cauchyjeva integralska formula|Cauchyjeva formula za odvod]] vodi do integralskega izraza:

: <math> \gamma_{n} = \frac{(-1)^{n} n!}{2\pi} \int_{0}^{2\pi} e^{-nix} \zeta \left( e^{ix}+1 \right) \mathrm{d} x \!\, . </math>

Več [[integral]]skih izrazov in [[neskončna vrsta|neskončnih vrst]] so v svojem delu podali [[Johan Ludvig William Valdemar Jensen|Jensen]], Franel, [[Charles Hermite|Hermite]], [[Godfrey Harold Hardy|Hardy]], [[Srinivasa Ajangar Ramanudžan|Ramanudžan]], Ainsworth, Howell, Coppo, Connon, Coffey, Choi, Blagouchine in drugi avtorji.<ref name="cpo">{{sktxt|Coppo|1999}}.</ref><ref name="cf1">{{sktxt|Coffey|2009}}.</ref><ref name="cf2">{{sktxt|Coffey|2010}}.</ref><ref>{{sktxt|Choi|2013}}.</ref><ref name="blg_02">{{sktxt|Blagouchine|2015a}}.</ref><ref name="blg_03">{{sktxt|Blagouchine|2015b}}.</ref> Še posebej Jensen-Franelova integralska formula, večkrat napačno pripisana Ainsworthu in Howellu, pravi, da velja:

: <math> \gamma_{n} \,=\,\frac{1}{2}\delta_{n,0}+\,\frac{1}{i}\!\int_{0}^{\infty} \! \frac{\mathrm{d} x}{e^{2\pi x}-1} \left\{ 
\frac{\ln^{n}(1-ix)}{1-ix} - \frac{\ln^{n}(1+ix)}{1+ix} \right\} \,, \qquad ( n=0, 1, 2,\ldots) \!\, , </math>

kjer je <math>\delta_{n,k}\, </math> [[Kroneckerjeva delta]].<ref name="blg_02"/><ref name="blg_03"/> Med drugimi formulami so (glej: <ref name="cpo"/><ref name="blg_02"/><ref>{{navedi splet|url=http://math.stackexchange.com/questions/1056962/|title=Math StackExchange: A couple of definite integrals related to Stieltjes constants|date= |accessdate= |language= en}}</ref>):

: <math> \gamma_{n} \,=\,-\frac{\pi}{2(n+1)}\! \int_{-\infty}^{+\infty} \frac{\ln^{n+1}\!\big(\frac{1}{2}\pm ix\big)}{\cosh^{2}\!\pi x}\, \mathrm{d} x \, , \qquad\qquad\qquad\qquad\qquad ( n=0, 1, 2,\ldots) \!\, , </math>
 
: <math> \begin{array}{l}
\displaystyle 
\gamma_{1} =-\left[\gamma -\frac{\ln2}{2}\right]\ln2+\,i\!\int_{0}^{\infty} \! \frac{\mathrm{d} x}{e^{\pi x}+1} \left\{
\frac{\ln(1-ix)}{1-ix} - \frac{\ln(1+ix)}{1+ix} \right\} \, \\[6mm]
\displaystyle
\gamma_{1} = -\gamma^{2} - \int_{0}^{\infty} \left[ \frac{1}{1-e^{-x}}-\frac{1}{x} \right] e^{-x}\ln x \, \mathrm{d} x \!\, .
\end{array} </math>

Znano vrsto, ki vsebuje [[celi del]] [[logaritem|logaritma]], je podal Hardy leta 1912:<ref>{{sktxt|Hardy|2012}}.</ref>

: <math> \gamma_{1} \,=\, \frac{\ln2}{2}\sum_{k=2}^{\infty} \frac{(-1)^{k}}{k} \,\lfloor \operatorname{lb} k \rfloor\cdot
\big(2\operatorname{lb} k - \lfloor \operatorname{lb}(2k)\rfloor\big) \!\, . </math>

Tu je <math>\operatorname{lb}\, </math> [[dvojiški logaritem]].

Israilov<ref>{{sktxt|Israilov|1981}}.</ref> je podal delno konvergentno vrsto z [[Bernoullijevo število|Bernoullijevimi števili]] <math>B_{2k}\, </math>:

: <math> \gamma_{m} \,=\,\sum_{k=1}^{n} \frac{\,\ln^{m}\! k\,}{k} - \frac{\,\ln^{m+1}\! n\,}{m+1} - \frac{\,\ln^{m}\! n\,}{2n} - \sum_{k=1}^{N-1} \frac{\,B_{2k}\,}{(2k)!} \left[ \frac{\ln^{m}\! x}{x} \right] ^{(2k-1)}_{x=n} - \theta\cdot\frac{\,B_{2N}\,}{(2N)!} \left[ \frac{\ln^{m}\! x}{x} \right] ^{(2N-1)}_{x=n} \,,\qquad ( 0 < \theta < 1 ) \!\, . </math>

Oloa in Tauraso<ref>{{navedi splet|url=http://math.stackexchange.com/questions/866382/|title=Math StackExchange: A closed form for the series ...|date= |accessdate= |language= en}}</ref> sta pokazala, da vrsta s [[harmonično število|harmoničnimi števili]] <math>H_{n}\, </math> lahko vodi do Stieltjesovih konstant:

: <math> \begin{array}{l}
\displaystyle 
\sum_{n=1}^{\infty} \frac{\,H_{n} - (\gamma+\ln n)\,}{n} \,=\, \,-\gamma_1 -\frac{1}{2}\gamma^{2}+\frac{1}{12}\pi^{2} \\[6mm]
\displaystyle 
\sum_{n=1}^{\infty} \frac{\,H^{2}_n - (\gamma+\ln n)^{2}\,}{n} \,=\,\,-\gamma_{2} -2\gamma\gamma_{1} -\frac{2}{3}\gamma^{3}+\frac{5}{3}\zeta(3) \!\, . \end{array} </math>

Blagouchine<ref name="blg_03" /> je našel počasi konvergentno vrsto, ki vsebuje [[predznak|nepredznačena]] [[Stirlingova števila prve vrste]] <math>\left[{\cdot \atop \cdot}\right]\, </math>:

: <math> \gamma_{m} \,=\,\frac{1}{2}\delta_{m,0} + \frac{\,(-1)^{m} m!\,}{\pi} \sum_{n=1}^{\infty}\frac{1}{\,n\cdot n!\,} 
\sum_{k=0}^{\lfloor\!\frac{1}{2}n\!\rfloor}\frac{\,(-1)^{k}\cdot\left[{2k+2\atop m+1}\right] \cdot\left[{n\atop 2k+1}\right]\,}
{\,(2\pi)^{2k+1}\,}\,,\qquad ( m=0,1,2,\ldots ) \!\, , </math>

kot tudi delno konvergentno vrsto s samimi [[racionalno število|racionalnimi]] členi:

: <math> \gamma_{m} \,=\,\frac{1}{2}\delta_{m,0}+(-1)^{m} m!\cdot\!\sum_{k=1}^{N}\frac{\,\left[{2k\atop m+1}\right]\cdot B_{2k}\,}{(2k)!} \,+\, \theta\cdot\frac{\,(-1)^{m} m!\!\cdot \left[{2N+2\atop m+1}\right]\cdot B_{2N+2}\,}{(2N+2)!}\,,\qquad ( 0 < \theta < 1, m = 0,1,2,\ldots ) \!\, . </math>

Več drugih vrst je danih v Coffeyjevemu delu.<ref name="cf1"/><ref name="cf2"/>

== Asimptotična rast ==

Za Stieltjesove konstante velja meja:

: <math> \big|\gamma_{n} \big| \, \le \,
\begin{cases}
\displaystyle \frac{2\,(n-1)!}{\pi^n} \, ; &  n=1, 3, 5,\ldots \\[3mm]
\displaystyle \frac{4\,(n-1)!}{\pi^n} \, ; &  n=2, 4, 6,\ldots \!\, , 
\end{cases} </math>

ki jo je podal [[Bruce Carl Berndt|Berndt]] leta 1972.<ref>{{sktxt|Berndt|1972}}.</ref> Boljše meje so našli Lavrik, Israilov, Matsuoka, Nan-You, Williams, Knessl, Coffey, Adell, Saad-Eddin, Fekih-Ahmed in Blagouchine.{{efn|Glej seznam virov dan v <ref name="blg_03" />.}} Eno od najboljših ocen z [[elementarna funkcija|elementarnimi funkcijami]] je podal Matsuoka leta 1985:<ref>{{sktxt|Matsuoka|1985}}.</ref>

: <math> |\gamma_n| < 10^{-4} e^{n \ln \ln n} \, , \qquad ( n \ge 5) \!\, . </math>

Dokaj točne ocene z neelementarnimi funkcijami so podali Knessl, Coffey<ref name="kns">{{sktxt|Knessl|Coffey|2011}}.</ref> in Fekih-Ahmed.<ref name="ahm">{{sktxt|Fekih-Ahmed|2014}}.</ref> Knessl in Coffey sta na primer dala naslednjo formulo, ki relativno dobro aproksimira Stieltjesove konstante za velike <math>n\, </math>.<ref name="kns" /> Če je <math>v\, </math> enolična rešitev enačbe:

: <math> 2 \pi \exp(v \operatorname{tg}\, v) = n \frac{\cos v}{v} \!\, , </math>

z <math>0 < v < \pi/2\, </math>, in, če je <math>u = v \operatorname{tg}\, v\, </math>, potem velja:

: <math> \gamma_{n} \sim \frac{B}{\sqrt{n}} e^{nA} \cos(an+b) \!\, , </math>

kjer je:

: <math> A = \frac{1}{2} \ln(u^{2}+v^{2}) - \frac{u}{u^{2}+v^{2}} \!\, , </math>

: <math> B = \frac{2 \sqrt{2\pi} \sqrt{u^{2}+v^{2}}}{[(u+1)^{2}+v^{2}]^{1/4}} \!\, , </math>

: <math> a = \operatorname{tg}^{-1} \left( \frac{v}{u} \right) + \frac{v}{u^{2}+v^{2}} \!\, , </math>

: <math> b = \operatorname{tg}^{-1} \left( \frac{v}{u} \right) - \frac{1}{2} \left( \frac{v}{u+1} \right) \!\, . </math>

Vse do <math>n = 100000\, </math> Knessl-Coffeyjev približek trenutno predvideva predznak <math>\gamma_{n}\, </math> z eno izjemo za <math>n = 137\, </math>.<ref name="kns" />

== Številske vrednosti ==

Prve [[desetiški številski sistem|desetiške]] vrednosti Stieltjesovih konstant podaja razpredelnica:

:{| class="wikitable"
| <math>n\, </math> || [[desetiški številski sistem|desetiške]] vrednosti <math>\gamma_{n}\, </math> || [[Spletna enciklopedija celoštevilskih zaporedij|OEIS]]
|-
| 0 || +0,5772156649015328606065120900824024310421593359 || {{OEIS2|id=A001620}}
|-
| 1 || &minus;0,0728158454836767248605863758749013191377363383 || {{OEIS2|id=A082633}}
|-
| 2 || &minus;0,0096903631928723184845303860352125293590658061 || {{OEIS2|id=A086279}}
|-
| 3 || +0,0020538344203033458661600465427533842857158044 || {{OEIS2|id=A086280}}
|-
| 4 || +0,0023253700654673000574681701775260680009044694 || {{OEIS2|id=A086281}}
|-
| 5 || +0,0007933238173010627017533348774444448307315394 || {{OEIS2|id=A086282}}
|-
| 6 || &minus;0,0002387693454301996098724218419080042777837151 || {{OEIS2|id=A183141}}
|-
| 7 || &minus;0,0005272895670577510460740975054788582819962534 || {{OEIS2|id=A183167}}
|-
| 8 || &minus;0,0003521233538030395096020521650012087417291805 || {{OEIS2|id=A183206}}
|-
| 9 || &minus;0,0000343947744180880481779146237982273906207895 || {{OEIS2|id=A184853}}
|-
| 10 || +0,0002053328149090647946837222892370653029598537 || {{OEIS2|id=A184854}}
|-
| 100 || &minus;4,2534015717080269623144385197278358247028931053 · 10<sup>17</sup> || &nbsp;
|-
| 1000 || &minus;1,5709538442047449345494023425120825242380299554 · 10<sup>486</sup> || &nbsp;
|-
| 10000 || &minus;2,2104970567221060862971082857536501900234397174 · 10<sup>6883</sup> || &nbsp;
|-
| 100000 || +1,9919273063125410956582272431568589205211659777 · 10<sup>83432</sup> || &nbsp;
|}

Za velike <math>n\, </math> [[absolutna vrednost|absolutne vrednosti]] Stieltjesovih konstant naraščajo hitro, predznak pa se spreminja v zapletenem vzorcu.

Dodatne informacije o numeričnem določevanju Stieltjesovih konstant se lahko najde v delu avtorjev: [[Jerry B. Keiper|Keiper]],<ref>{{sktxt|Keiper|1992}}.</ref> Kreminski,<ref>{{sktxt|Kreminski|2003}}.</ref> [[Simon Plouffe|Plouffe]]<ref>{{sktxt|Plouffe|1986}}.</ref> in Johansson.<ref>{{sktxt|Johansson|2013}}.</ref> Johansson je podal vrednosti Stieltjesovih konstant do <math>n = 100000\, </math>, vsaka točna na več kot 10000 števk. Številske vrednosti se lahko dobijo v podatkovni bazi [[LMFDB]].<ref>{{navedi splet|url= http://beta.lmfdb.org/riemann/stieltjes/|title= Stieltjes Constants|work= [[LMFDB]]|date= 2015-08-05|accessdate= 2015-08-07|language= en}}</ref>

== Posplošene Stieltjesove konstante ==

=== Splošna informacija ===

Bolj splošno se lahko definirajo Stieltjesove konstante <math>\gamma_{n} (a)\, </math>, ki se pojavljajo v Laurentovi vrsti za [[Hurwitzeva funkcija zeta|Hurwitzevo funkcijo ζ]]:

: <math> \zeta (s,a) = \frac{1}{s-1}+\sum_{n=0}^{\infty} \frac{(-1)^n}{n!} \gamma_{n} (a) \; (s-1)^{n} \!\, . </math>

Tu je <math>a\, </math> [[kompleksno število]] z <math>\Re (a) > 0\, </math>. Ker je Hurwitzeva funkcija ζ posplošitev Riemannove funkcije ζ, velja <math>\gamma_{n} (1) = \gamma_{n}\, </math>. Ničta konstanta je preprosto [[funkcija digama]] <math>\gamma_{0} (a) = - \digamma (a)\, </math>.<ref name="mse">{{navedi splet|url= http://math.stackexchange.com/questions/703923/|title=Math StackExchange: Definite integral|date= |accessdate= |language= en}}</ref> Za druge konstante ni znana [[razčlenjevanje|razčlenitev]] na elementarne ali klasične funkcije iz analize. Ne glede na to obstaja več izrazov zanje. Na primer naslednji asimptotični izraz:

: <math> \gamma_{n} (a) \,=\, \lim_{m\to\infty} \left\{ \sum_{k=0}^{m} \frac{\ln^{n} (k+a)}{k+a} - \frac{\ln^{n+1} (m+a)}{n+1} 
\right\} \, , \qquad \;
\begin{array}{l}
n=0, 1, 2,\ldots\, \\[1mm]
a\neq0, -1, -2, \ldots \!\, , 
\end{array} </math>
ki sta jo podala Berndt in Wilton. Analogon Jensen-Franelove formule za posplošeno Stieltjesovo konstanto je [[Charles Hermite|Hermitova]] formula:<ref name="blg_02"/>

: <math> \gamma_{n} (a) \,=\, \left[ \frac{1}{2a}-\frac{\ln{a}}{n+1} \right] \ln^{n}\!{a}-i\!\int_{0}^{\infty} \! \frac{\mathrm{d} x}{e^{2\pi x}-1} \left\{ \frac{\ln^{n} (a-ix)}{a-ix} - \frac{\ln^{n} (a+ix)}{a+ix} \right\} \, , \qquad\;
\begin{array}{l}
n=0, 1, 2,\ldots\, \\[1mm]
a\neq0, -1, -2, \ldots
\end{array} </math>

Za posplošene Stieltjesove konstante velja naslednja rekurenčna zveza:

: <math> \gamma_{n} (a+1) \, =\, \gamma_{n} (a) - \frac{\,\ln^{n}\! a\,}{a} \, , \qquad\;
\begin{array}{l}
n=0, 1, 2,\ldots\, \\[1mm]
a\neq0, -1, -2, \ldots \!\, , 
\end{array} </math>

kakor tudi multiplikacijski izrek:

: <math> \sum_{l=0}^{n-1} \gamma_{p}\! \left( \! a+\frac{l}{\,n\,} \right) =\, (-1)^{p} n \! \left[ \frac{\ln n}{\,p+1\,} - \digamma (an) \right] \!\ln^{p}\! n \,+\, n\sum_{r=0}^{p-1}(-1)^{r} \binom{p}{r} \gamma_{p-r} (an) \cdot \ln^{r}\!{n} \, , \qquad ( n=2, 3, 4,\ldots) \!\, , </math>

kjer <math>\binom{p}{r}</math> označuje [[binomski koeficient]].<ref>{{sktxt|Connon|2009a}}.</ref><ref name="blg_01">{{sktxt|Blagouchine|2014}}.</ref>{{rp|101–102}}

=== Prva posplošena Stieltjesova konstanta ===

Prva posplošena Stieltjesova konstanta ima več pomembnih značilnosti. 
* '''Malmstenova enakost (refleksijska formula za prve posplošene Stieltjesove konstante):''' [[refleksijska formula]] za prvo posplošeno Stieltjesovo konstanto ima obliko:

: <math> \gamma_{1} \biggl( \frac{m}{n} \biggr) - \gamma_{1} \biggl( 1-\frac{m}{n} \biggr) = 2\pi\sum_{l=1}^{n-1} \sin\frac{2\pi m l}{n} \cdot\ln\Gamma \biggl( \frac{l}{n} \biggr) - \pi(\gamma+\ln2\pi n)\cot\frac{m\pi}{n} \!\, , </math>

kjer sta <math>m\, </math> in <math>n\,</math> takšni pozitivni celi števii, da velja <math>m < n\, </math>, <math>\Gamma\, </math> pa je [[funkcija gama|funkcija Γ]]. Formulo so dolgo časa pripisovali Almkvistu in Meurmanu, ki sta jo izpeljala v 1990-ih.<ref name="adm">{{sktxt|Adamchik|1997}}.</ref> Vendar je nedavno Blagouchine odkril, da je to enakost, sicer v malo drugačni obliki, našel [[Carl Johan Malmsten|Malmsten]] leta 1846.<ref name="blg_02" /><ref>{{navedi splet|url=http://math.stackexchange.com/questions/121545/|title=Math StackExchange: evaluation of a particular integral|date= |accessdate= |language= en}}</ref>

* '''Izrek o racionalnih argumentih:''' prva posplošena Stieltjesova konstanta z racionalnim [[argument (kompleksna analiza)|argumentom]] se lahko izračuna iz delno sklenjene oblike s formulo:<ref name="blg_02" /><ref name="mse" />

: <math>\begin{array}{ll}
\displaystyle 
\gamma_{1} \biggl( \frac{r}{m} \biggr) =& \displaystyle
\gamma_{1} +\gamma^{2} + \gamma\ln2\pi m + \ln2\pi\cdot\ln{m}+\frac{1}{2}\ln^{2}\!{m} + (\gamma+\ln2\pi m)\cdot\digamma\! \left( \!\frac{r}{m}\! \right) \\[5mm]
\displaystyle & \displaystyle\qquad
+\pi\sum_{l=1}^{m-1} \sin\frac{2\pi r l}{m} \cdot\ln\Gamma \biggl( \frac{l}{m} \biggr) + \sum_{l=1}^{m-1} \cos\frac{2\pi rl}{m}\cdot\zeta''\! \left( \! 0,\,\frac{l}{m}\! \right) \end{array} \, ,\qquad\quad ( r=1, 2, 3,\ldots, m-1 ) \!\, . </math>

Alternativni dokaz je kasneje predložil Coffey.<ref name="coff">{{sktxt|Coffey|2014}}.</ref>

* '''Končne vsote:''' za prve posplošene Stieltjesove konstante obstaje veliko sumacijskih formul. Na primer:{{efn|Za podrobnosti in druge sumacijske formule glej <ref name="blg_02" /><ref name="blg_01" />.}}

: <math> \begin{array}{ll}
\displaystyle 
\sum_{r=0}^{m-1} \gamma_{1}\! \left( \! a+\frac{r}{\,m\,} \right) =\, m\ln{m}\cdot\digamma (am) - \frac{m}{2}\ln^{2}\!m + m\gamma_{1}(am) \, , \qquad ( a\in\Complex ) \\[6mm]
\displaystyle
\sum_{r=1}^{m-1} \gamma_{1}\! \left( \!\frac{r}{\,m\,} \right) =\, (m-1)\gamma_{1} - m\gamma\ln{m} - \frac{m}{2}\ln^{2}\!m \\[6mm]
\displaystyle
\sum_{r=1}^{2m-1} (-1)^{r} \gamma_{1} \biggl( \!\frac{r}{2m} \!\biggr) \,=\, -\gamma_{1}+m(2\gamma+\ln2+2\ln m)\ln2\\[6mm]
\displaystyle
\sum_{r=0}^{2m-1} (-1)^{r} \gamma_1 \biggl( \!\frac{2r+1}{4m} \!\biggr) \,=\, m \left\{ 4\pi\ln\Gamma \biggl( \frac{1}{4} \biggr) - \pi \big( 4\ln2+3\ln\pi+\ln m+\gamma \big) \! \right\} \\[6mm]
\displaystyle
\sum_{r=1}^{m-1} \gamma_{1} \biggl(\!\frac{r}{m}\! \biggr) \!\cdot\cos\dfrac{2\pi rk}{m} \,=\, -\gamma_{1} + m(\gamma+\ln2\pi m)
\ln\! \left( \!2\sin\frac{\,k\pi\,}{m}\! \right) +\frac{m}{2} \left\{ \zeta''\!\left(\! 0,\,\frac{k}{m}\! \right) + \, \zeta''\! \left( \! 0,\,1-\frac{k}{m}\! \right) \! \right\} \, , \qquad ( k=1,2,\ldots,m-1 )  \\[6mm] \displaystyle
\sum_{r=1}^{m-1} \gamma_{1} \biggl( \!\frac{r}{m} \! \biggr) \!\cdot\sin\dfrac{2\pi rk}{m} \,=\,\frac{\pi}{2} (\gamma+\ln2\pi m)(2k-m) - \frac{\pi m}{2} \left\{ \ln\pi -\ln\sin\frac{k\pi}{m} \right\} + m\pi\ln\Gamma \biggl( \frac{k}{m} \biggr) \, , \qquad ( k=1,2,\ldots,m-1 ) \\[6mm]
\displaystyle
\sum_{r=1}^{m-1} \gamma_{1} \biggl( \!\frac{r}{m} \! \biggr) \cdot\cot\frac{\pi r}{m}  =\, \displaystyle
\frac{\pi }{6} \Big\{ \!(1-m)(m-2)\gamma + 2(m^2-1)\ln2\pi - (m^2+2)\ln{m} \Big\} -2\pi\!\sum_{l=1}^{m-1} l\!\cdot\!\ln\Gamma\! \left( \! \frac{l}{m}\! \right) \\[6mm]
\displaystyle
\sum_{r=1}^{m-1} \frac{r}{m} \cdot\gamma_{1} \biggl( \!\frac{r}{m}  \! \biggr) =\, \frac{1}{2} \left\{ \!(m-1)\gamma_{1} - m\gamma\ln{m} - \frac{m}{2}\ln^2\!{m}\!  \right\} - \frac{\pi}{2m}(\gamma+\ln2\pi m) \!\sum_{l=1}^{m-1} l\!\cdot\! \cot\frac{\pi l}{m} 
-\frac{\pi}{2} \!\sum_{l=1}^{m-1} \cot\frac{\pi l}{m} \cdot\ln\Gamma \biggl( \!\frac{l}{m} \! \biggr) \!\, . \end{array} </math>

* '''Nekatere posebne vrednosti:''' nekatere posebne vrednosti prve Stieltjesove konstante z racionalnimi argumenti se lahko zreducirajo na funkcijo Γ, prvo Stieltjesovo konstanto <math>\gamma_{1}\, </math> in elementarne funkcije. Na primer:

: <math> \gamma_{1}\! \left( \!\frac{1}{\,2\,}\! \right) = - 2\gamma\ln2 - \ln^{2}\!2 + \gamma_{1} \, =
\,-1,353459680804 \ldots \!\, , </math> {{OEIS|id=A254327}},

Vrednosti prvih posplošenih Stieltjesovih konstant v točkah 1/4, 3/4 in 1/3 sta prva neodvisno izračunala Connon<ref>{{sktxt|Connon|2009b}}.</ref> in Blagouchine<ref name="blg_01" />

: <math> \displaystyle
\gamma_{1}\! \left( \!\frac{1}{\,4\,}\! \right) =\, 2\pi\ln\Gamma\! \left( \!\frac{1}{\,4\,} \! \right) - \frac{3\pi}{2}\ln\pi - \frac{7}{2}\ln^{2}\!2 - (3\gamma+2\pi)\ln2 - \frac{\gamma\pi}{2}+\gamma_{1} \,=\,-5,518076350199 \ldots  \!\, , </math> {{OEIS|id=A254347}},

: <math> \displaystyle
\gamma_{1}\! \left( \!\frac{3}{\,4\,} \! \right) =\, -2\pi\ln\Gamma\! \left( \!\frac{1}{\,4\,}\! \right) + \frac{3\pi}{2}\ln\pi - \frac{7}{2}\ln^{2}\!2 - (3\gamma-2\pi)\ln2 + \frac{\gamma\pi}{2}+\gamma_1 \,=\,-0,391298902404 \ldots \!\, , </math> {{OEIS|id=A254348}},

: <math> \displaystyle
\gamma_{1}\! \left( \!\frac{1}{\,3\,}  \! \right) = \, - \frac{3\gamma}{2}\ln3 - \frac{3}{4}\ln^{2}\!3 + \frac{\pi}{4\sqrt{3\,}} \left\{ \ln3 - 8\ln2\pi -2\gamma + 12 \ln\Gamma\!\left(\!\frac{1}{\,3\,} \! \right) \! \right\} + \,\gamma_1 \, =
\,-3,259557515917 \ldots \!\, , </math> {{OEIS|id=A254331}}.

Vrednosti v točkah 2/3, 1/6 in 5/6 je izračunal Blagouchine.<ref name="blg_01" />:

: <math> \displaystyle
\gamma_{1}\! \left( \!\frac{2}{\,3\,} \! \right) = \, - \frac{3\gamma}{2}\ln3 - \frac{3}{4}\ln^{2}\!3 - \frac{\pi}{4\sqrt{3\,}} \left\{ \ln3 - 8\ln2\pi -2\gamma + 12 \ln\Gamma\! \left( \!\frac{1}{\,3\,} \! \right)  \!\right\} + \,\gamma_1 \, =
\,-0,5989062842859 \ldots \!\, , </math> {{OEIS|id=A254345}},

: <math> \begin{align} \displaystyle
\gamma_{1}\! \left( \!\frac{1}{\,6\,} \! \right) = &- \frac{3\gamma}{2}\ln3 - \frac{3}{4}\ln^{2}\!3 - \ln^{2}\!2 - (3\ln3+2\gamma)\ln2 + \frac{3\pi\sqrt{3\,}}{2}\ln\Gamma\! \left( \!\frac{1}{\,6\,}\! \right) \\
\displaystyle
 &- \frac{\pi}{2\sqrt{3\,}} \left\{ 3\ln3 + 11\ln2 + \frac{15}{2}\ln\pi + 3\gamma \right\} +\, \gamma_1 \, =\,-10,742582529547 \ldots \!\, , \end{align} </math> {{OEIS|id=A254349}},

: <math> \begin{align} \displaystyle
\gamma_{1}\! \left( \!\frac{5}{\,6\,} \! \right) = &- \frac{3\gamma}{2}\ln3 - \frac{3}{4}\ln^{2}\!3 - \ln^{2}\!2 - (3\ln3+2\gamma)\ln2 - \frac{3\pi\sqrt{3\,}}{2}\ln\Gamma\! \left( \!\frac{1}{\,6\,}\! \right) \\
\displaystyle
 &+ \frac{\pi}{2\sqrt{3\,}} \left\{ 3\ln3 + 11\ln2 + \frac{15}{2}\ln\pi + 3\gamma \right\} +\, \gamma_1 \, =\, -0,246169003811 \ldots \!\, , \end{align} </math> {{OEIS|id=A254350}},

Podal je tudi vrednosti v točkah 1/5, 1/8 in 1/12:
 
: <math> \begin{align} \displaystyle 
\gamma_{1} \biggl( \!\frac{1}{5} \!\biggr)=& \displaystyle \,\,\, \gamma_{1} + \frac{\sqrt{5}}{2}\! \left\{ \zeta''\! \left( \! 0,\,\frac{1}{5}\! \right) + \zeta''\! \left( \! 0,\,\frac{4}{5}\! \right)\! \right\} + \frac{\pi\sqrt{10+2\sqrt5}}{2} \ln\Gamma \biggl( \!\frac{1}{5} \! \biggr)
\\[5mm]
& \displaystyle 
+ \frac{\pi\sqrt{10-2\sqrt5}}{2} \ln\Gamma \biggl( \!\frac{2}{5} \!\biggr) + \left\{\!\frac{\sqrt{5}}{2} \ln{2} -\frac{\sqrt{5}}{2} \ln\! \big( 1+\sqrt{5} \big) -\frac{5}{4}\ln5 - \frac{\pi\sqrt{25+10\sqrt5}}{10} \right\} \!\cdot\gamma \\[5mm]
& \displaystyle 
- \frac{\sqrt{5}}{2} \left\{ \ln2+\ln5+\ln\pi+\frac{\pi\sqrt{25-10\sqrt5}}{10} \right\} \!\cdot\ln\! \big( 1+\sqrt{5}) + \frac{\sqrt{5}}{2}\ln^{2}\!2 + \frac{\sqrt{5} \big( 1-\sqrt{5} \big) }{8}\ln^{2}\!5 \\[5mm]
& \displaystyle 
 +\frac{3\sqrt{5}}{4}\ln2\cdot\ln5 + \frac{\sqrt{5}}{2}\ln2\cdot\ln\pi+\frac{\sqrt{5}}{4}\ln5\cdot\ln\pi - \frac{\pi\big(2\sqrt{25+10\sqrt5}+5\sqrt{25+2\sqrt5} \big) }{20}\ln2\\[5mm]
& \displaystyle 
- \frac{\pi \big( 4\sqrt{25+10\sqrt5}-5\sqrt{5+2\sqrt5} \big) }{40}\ln5 - \frac{\pi \big( 5\sqrt{5+2\sqrt5}+\sqrt{25+10\sqrt5} \big) }{10}\ln\pi\\[5mm]
& \displaystyle = -8,030205511035 \ldots \!\, , \end{align} </math> {{OEIS|id=A251866}},

: <math> \begin{align} \displaystyle 
\gamma_{1} \biggl( \!\frac{1}{8} \!\biggr) =& \displaystyle \,\,\, \gamma_{1}  + \sqrt{2}\left\{\zeta''\! \left( \! 0,\,\frac{1}{8}\! \right) + \zeta''\! \left( \! 0,\,\frac{7}{8} \right) \! \right\} + 2\pi\sqrt{2}\ln\Gamma \biggl( \!\frac{1}{8} \! \biggr) - \pi \sqrt{2} \big( 1-\sqrt2 \big) \ln\Gamma \biggl( \!\frac{1}{4} \!\biggr) \\[5mm]
& \displaystyle 
- \left\{ \!\frac{1+\sqrt2}{2}\pi+4\ln{2} +\sqrt{2}\ln\! \big( 1+\sqrt{2} \big) \! \right\} \!\cdot\gamma - \frac{1}{\sqrt{2}} \big( \pi+8\ln2+2\ln\pi \big) \!\cdot\ln\! \big( 1+\sqrt{2}) \\[5mm]
& \displaystyle 
 - \frac{7 \big( 4-\sqrt2 \big) }{4}\ln^{2}\!2 + \frac{1}{\sqrt{2}}\ln2\cdot\ln\pi - \frac{\pi \big( 10+11\sqrt2 \big) }{4}\ln2 - \frac{\pi \big( 3+2\sqrt2 \big) }{2}\ln\pi\\[5mm]
& \displaystyle = -16,641719763609 \ldots \end{align} </math> {{OEIS|id=A255188}},

: <math> \begin{align} \displaystyle 
\gamma_{1} \biggl( \!\frac{1}{12} \!\biggr) =& \displaystyle \,\,\, \gamma_{1} + \sqrt{3} \left\{ \zeta''\! \left( \! 0,\,\frac{1}{12}\! \right) + \zeta''\! \left( \! 0,\,\frac{11}{12} \right) \! \right\} + 4\pi\ln\Gamma \biggl( \!\frac{1}{4} \! \biggr) + 3\pi \sqrt{3}\ln\Gamma \biggl( \!\frac{1}{3} \! \biggr) \\[5mm]
& \displaystyle 
- \left\{ \!\frac{2+\sqrt3}{2}\pi+\frac{3}{2}\ln3 -\sqrt3(1-\sqrt3)\ln{2} +2\sqrt{3}\ln\! \big( 1+\sqrt{3} \big) \! \right\} \!\cdot\gamma \\[5mm]
& \displaystyle 
- 2\sqrt3 \big( 3\ln2+\ln3 +\ln\pi \big) \!\cdot\ln\! \big( 1+\sqrt{3}) - \frac{7-6\sqrt3}{2}\ln^{2}\!2 - \frac{3}{4}\ln^{2}\!3  \\[5mm]
& \displaystyle 
+ \frac{3\sqrt3(1-\sqrt3)}{2}\ln3\cdot\ln2 + \sqrt3\ln2\cdot\ln\pi - \frac{\pi \big( 17+8\sqrt3 \big) }{2\sqrt3}\ln2 \\[5mm]
& \displaystyle 
 + \frac{\pi \big( 1-\sqrt3 \big) \sqrt3}{4}\ln3 - \pi\sqrt3(2+\sqrt3)\ln\pi = -29,842878232041 \ldots \!\, , \end{align} </math> {{OEIS|id=A255189}},

kakor tudi nekatere druge vrednosti.

=== Druga posplošena Stieltjesova konstanta ===

Drugo posplošeno Stieltjesovo konstanto so manj raziskovali od prve. Blagouchine je pokazal, da se lahko podobno kot prva posplošena Stieltjesova konstanta druga posplošena Stieltjesova konstanta z racionalnim argumentom izračuna s pomočjo formule:

: <math> \begin{array}{rl}
\displaystyle
\gamma_{2} \biggl( \frac{r}{m} \biggr) = \, \gamma_{2} + \frac{2}{3}\!\sum_{l=1}^{m-1} \cos\frac{2\pi r l}{m} \cdot\zeta'''\! \left( \!0,\,\frac{l}{m}\! \right) - 2 (\gamma+\ln2\pi m)\! \sum_{l=1}^{m-1} \cos\frac{2\pi r l}{m} \cdot\zeta''\! \left( \!0,\,\frac{l}{m}\! \right) \\[6mm]
\displaystyle \quad
+ \pi\!\sum_{l=1}^{m-1} \sin\frac{2\pi r l}{m} \cdot\zeta''\! \left( \!0,\,\frac{l}{m}\! \right) - 2\pi(\gamma+\ln2\pi m)\!
\sum_{l=1}^{m-1} \sin\frac{2\pi r l}{m} \cdot\ln\Gamma \biggl( \frac{l}{m} \biggr) - 2\gamma_1 \ln{m} \\[6mm]
\displaystyle\quad
- \gamma^{3} - \left[ (\gamma+\ln2\pi m)^2-\frac{\pi^2}{12} \right] \!\cdot\! \digamma \!\biggl( \frac{r}{m} \biggr) + 
\frac{\pi^3}{12}\cot\frac{\pi r}{m} - \gamma^{2}\ln\big(4\pi^{2} m^{3} \big) + \frac{\pi^{2}}{12}(\gamma+\ln{m}) \\[6mm]
\displaystyle\quad
 - \gamma \big( \ln^{2}\!{2\pi} + 4 \ln{m}\cdot\ln{2\pi}+2\ln^{2}\!{m} \big) - \left\{\!\ln^{2}\!{2\pi}+2\ln{2\pi}\cdot\ln{m}+\frac{2}{3}\ln^{2}\!{m}\! \right\} \!\ln{m} \end{array} \, , \qquad \quad ( r=1, 2, 3,\ldots, m-1 ) \!\, . </math>

Podobni rezultat je kasneje dobil Coffey z drugo metodo.<ref name="coff" />

== Opombe ==

{{notelist}}

== Sklici ==

{{sklici|3}}

== Viri ==

{{refbegin|2}}
* {{citat|last1= Adamchik|first1= V.|title= A class of logarithmic integrals|journal= Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation|date= 1997|pages= 1–8|ref= harv}}
* {{citat|last1= Adell|first1= José Antonio|title= Asymptotic estimates for Stieltjes constants: a probabilistic approach|journal= [[Proceedings of the Royal Society|Proc. R. Soc. A]]|date= 2011|volume= 467|issue= 2128|pages= 954–963|url= http://www.unizar.es/analisis_matematico/preprints/Adell-2010-PRSL.pdf|doi= 10.1098/rspa.2010.0397|ref= harv}}
* {{citat|last1= Berndt|first1= Bruce Carl|authorlink1= Bruce Carl Berndt|title= On the Hurwitz Zeta-function|journal= [[Rocky Mountain Journal of Mathematics]]|date= 1972|volume= 2|issue= 1|pages= 151–157|ref= harv}}
* {{citat||last1= Blagouchine|first1= Iaroslav V.|title= Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results|journal= [[The Ramanujan Journal]]|date= 2014|volume= 35|issue= 1|pages= 21–110|url= http://link.springer.com/article/10.1007/s11139-013-9528-5|ref= harv}} [https://www.researchgate.net/publication/257381156_Rediscovery_of_Malmsten's_integrals_their_evaluation_by_contour_integration_methods_and_some_related_results PDF]
* {{citat|last1= Blagouchine|first1= Iaroslav V.|title= A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations|journal= [[Journal of Number Theory|J. Number Theory]] (Elsevier)|date= 2015a|volume= 148, 151|issue= |pages=537–592, 276-277|url= http://www.sciencedirect.com/science/article/pii/S0022314X14002820|arxiv=1401.3724|ref= harv}}
* {{citat|last1= Blagouchine|first1= Iaroslav V.|title= Expansions of generalized Euler's constants into the series of polynomials in π<sup>&minus;2</sup> and into the formal enveloping series with rational coefficients only|journal= [[Journal of Number Theory|J. Number Theory]] (Elsevier)|date= 2015b|volume= 158|issue= |pages=365–396|url= http://www.sciencedirect.com/science/article/pii/S0022314X15002255|arxiv=1501.00740|ref= harv}}
* {{citat|last1= Choi|first1= Junesang|title= Certain integral representations of Stieltjes constants|journal= [[Journal of Inequalities and Applications]]|date= 2013|volume= 532|issue= |pages= 1–10|ref= harv}}
* {{citat|last1= Coffey|first1= Mark W.|title= Series representations for the Stieltjes constants|date= 2009|arxiv= 0905.1111|ref= harv}}
* {{citat|last1= Coffey|first1= Mark W.|title= Addison-type series representation for the Stieltjes constants|journal= [[Journal of Number Theory|J. Number Theory]]|date= 2010|volume= 130|issue= |pages= 2049–2064|url= http://www.sciencedirect.com/science/article/pii/S0022314X10000399|ref= harv}}
* {{citat|last1= Coffey|first1= Mark W.|title= Functional equations for the Stieltjes constants|date= 2014|arxiv= 1402.3746|ref= harv}}
* {{citat|last1= Connon|first1= Donal F.|title= New proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions|date= 2009a|arxiv= 0903.4539|ref= harv}}
* {{citat|last1= Connon|first1= Donal F.|title= The difference between two Stieltjes constants|date= 2009b|arxiv= 0906.0277|ref= harv}}
* {{citat|last1= Coppo|first1= Marc-Antoine|title= Nouvelles expressions des constantes de Stieltjes|journal= [[Expositiones Mathematicae]]|date= 1999|volume= 17|issue= |pages= 349–358|ref= harv}}
* {{citat|last1= Fekih-Ahmed|first1= Lazhar|title= A New Effective Asymptotic Formula for the Stieltjes Constants|date= 2014|arxiv=1407.5567|ref= harv}}
* {{citat|last1= Hardy|first1= Godfrey Harold|authorlink1= Godfrey Harold Hardy|title= Note on Dr. Vacca's series for γ''|journal= [[Q. J. Pure Appl. Math.]]|date= 2012|volume= 43|issue= |pages= 215–216|ref= harv}}
* {{citat|last1= Israilov|first1= Maruf Israilovič|title= On the Laurent decomposition of Riemann's zeta function [v ruščini]|journal= [[Trudy Mat. Inst. Akad. Nauk. SSSR]]|date= 1981|volume= 158|issue= |pages= 98–103|MR= 662837|zbl= 0477.10031|url= http://mi.mathnet.ru/eng/tm2377|ref= harv}}
* {{citat|last1= Johansson|first1= Fredrik|title= Rigorous high-precision computation of the Hurwitz zeta function and its derivatives|date= 2013|arxiv= 1309.2877|ref= harv}}
* {{citat|last1= Keiper|first1= Jerry B.|authorlink1= Jerry B. Keiper|title= Power series expansions of Riemann ζ-function|journal= [[Mathematics of Computation|Math. Comp.]]|date= 1992|volume= 58|issue= 198|pages= 765–773|doi= 10.1090/S0025-5718-1992-1122072-5|bibcode= 1992MaCom..58..765K|ref= harv}}
* {{citat|last1= Knessl|first1= Charles|last2= Coffey|first2= Mark W.|title= An effective asymptotic formula for the Stieltjes constants|journal= [[Mathematics of Computation|Math. Comp.]]|date= 2011|volume= 80|issue= 273|pages= 379–386|ref= harv}}
* {{citat|last1= Kreminski|first1= Rick|title= Newton-Cotes integration for approximating Stieltjes generalized Euler constants|journal= [[Mathematics of Computation|Math. Comp.]]|date= 2003|volume= 72|issue= 243|pages= 1379–1397|ref= harv}}
* {{citat|last1= Matsuoka|first1= Yasushi|chapter= Generalized Euler Constants Associated with the Riemann Zeta Function|title= Number Theory and Combinatorics|date= 1985|location= Singapur|publisher= World Scientific|pages= 279-295|MR= 827790}} 
* {{citat|last1= Plouffe|first1= Simon|authorlink1= Simon Plouffe|url= http://www.plouffe.fr/simon/constants/stieltjesgamma.txt|title= Stieltjes Constants, from 0 to 78, 256 digits each|date= 1986|accessdate= 2015-08-07|language= en|ref= harv}}
* {{citat|last1= Stieltjes|first1= Thomas Joannes|authorlink1= Thomas Joannes Stieltjes|title= Correspondance d'Hermite et de Stieltjes, volumes 1, 2|date= 1905|publisher= Gauthier-Villars|location= Pariz|ref= harv}}
{{refend}}

== Zunanje povezave ==

* {{MathWorld|id=StieltjesConstants|title=Stieltjes Constants}}

[[Kategorija:Funkcije zeta in L-funkcije]]
[[Kategorija:Matematične konstante]]