W. Lang, 20 Aug 2009
Link from A157165-8.

Lebesgue's constants L(n), n>=0.
G. Szeg\H o (Szego): \"Uber die Lebesgueschen Konstanten bei den Fourierschen Reihen, 
Math. Z. 9 (1921) 163-166. L(n) is called \rho_n in this reference.
See also  Mathword: Lebesgue Constants.
Definition: L(n) = (2/Pi)*int(|sin((2*n+1)*x)|/sin(x),x=0..Pi/2).
Series representation due to Szego:
L(n) = (16/(Pi^2))*sum(Theta(1,(2*n+1)*k)/(4*k^2-1),k=1..infty) with 
Theta(1,m):=sum(1/(2*j-1),j=1..m) = int(((sin(k*x))^2)/sin(x),x=0..Pi/2) 
(See Szeg\H o (Sego) reference formula (R), p.165 and the line before this).
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Rationals r(n;N):= sum(Theta(1,(2*n+1)*k)/(4*k^2-1),k=1..N) 
(N-th partial sum of ((Pi^2)/16)*L(n)).


n=0:  L(0) = 1. 
r(0;N), N=1..30, (r(0;N) in lowest terms).

Numerator sequence A157165:

[1, 19, 734, 16294, 557407, 81759221, 1083213812, 3737624804, 1221606572113, 4687376819963, 543445163726882, 314646975551939566, 4747879086124761619, 415213127253949396153, 374983405094739446762072, 11671625151617599366571432, 11713911632041456348356827, 62118115376291869131554237, 2305143176230889865087842294, 94764086171764978476939512854,...].
  
Denominator sequence:

[3, 45, 1575, 33075, 1091475, 156080925, 2029052025, 6898776885, 2228304933855, 8467558748649, 973769256094635, 559917322254415125, 8398759833816226875, 730692105542011738125, 656892202882268552574375, 20363658289350325129805625, 20363658289350325129805625, 107636479529423147114686875, 3982549742588656443243414375, 163284539446134914172979989375,...].

The denominator sequence for 3*r(0;N) yields A157166:

[1, 15, 525, 11025, 363825, 52026975, 676350675, 2299592295, 742768311285, 2822519582883, 324589752031545, 186639107418138375, 2799586611272075625, 243564035180670579375, 218964067627422850858125, 6787886096450108376601875, 6787886096450108376601875, 35878826509807715704895625, 1327516580862885481081138125, 54428179815378304724326663125,...].

This seems to be the above given denominator sequence divided by 3. 


Rationals r(0;N), N=1..20:

[1/3, 19/45, 734/1575, 16294/33075, 557407/1091475, 81759221/156080925, 1083213812/2029052025, 3737624804/6898776885, 1221606572113/2228304933855, 4687376819963/8467558748649, 543445163726882/973769256094635, 314646975551939566/559917322254415125, 4747879086124761619/8398759833816226875, 415213127253949396153/730692105542011738125, 374983405094739446762072/656892202882268552574375, 11671625151617599366571432/20363658289350325129805625, 11713911632041456348356827/20363658289350325129805625, 62118115376291869131554237/107636479529423147114686875, 2305143176230889865087842294/3982549742588656443243414375, 94764086171764978476939512854/163284539446134914172979989375,...].


3*r(0;N), N=1..20:

[1, 19/15, 734/525, 16294/11025, 557407/363825, 81759221/52026975, 1083213812/676350675, 3737624804/2299592295, 1221606572113/742768311285, 4687376819963/2822519582883, 543445163726882/324589752031545, 314646975551939566/186639107418138375, 4747879086124761619/2799586611272075625, 415213127253949396153/243564035180670579375, 374983405094739446762072/218964067627422850858125, 11671625151617599366571432/6787886096450108376601875, 11713911632041456348356827/6787886096450108376601875, 62118115376291869131554237/35878826509807715704895625, 2305143176230889865087842294/1327516580862885481081138125, 94764086171764978476939512854/54428179815378304724326663125,...].


The sequence of rationals r(0,N) converges to (Pi^2)/16 approximately 0.6168502752 (Maple12 10 digits) 
because L(0) = 1.

The values  r(0,10^k), k=0..4 are:

[.3333333333, .5535688572, .6074302800, .6156169206, .6166981121, .6168323563]

This shows the slow convergence.
  
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n=1:  L(1) = (1 + 6*sqrt(3)/Pi)/3, approximately 1.435991124 (Maple12 10 digits) (from the defining integral).r(1;N), N=1..30, (r(1;N) in lowest terms).

Numerator sequence A157167:

[23, 33073, 55943738, 77064019958, 15226093370063, 31370562345762421, 241905492960111168964, 1683591136668277300660676, 48935652383592600478507247, 713289082617826259771761324613, 143961819529547244077111055694498, 2460282354560331257420364974778935366, 3790786840731003921840152645904014459389, 2172041935768934848601975221239259534857643451, 6015898478763334252390789632019813330057930993384,...].


Denominator sequence:

[45, 51975, 80405325, 105411381075, 20175738337755, 40654112750576325, 308361445213121425125, 2118874834637442560603925, 60963055956568704529375785, 881220973852200623972126972175, 176621860902091067918984877423075, 3000679258254454321752182363934170625, 4600041302904078475246095563911083568125, 2624134961613357702910412429293152821189019375, 7239988359091253902329827892419808633660504455625,...]:
Denominator sequence for R(1;N)=45*r(1;N) is A157168: 

[1, 1155, 1786785, 2342475135, 448349740839, 903424727790585, 6852476560291587225, 47086107436387612457865, 1354734576812637878430573, 19582688307826680532713932715, 3924930242268690398199663942735, 66681761294543429372270719198537125, 102223140064535077227691012531357412625, 58314110258074615620231387317625618248644875, 160888630202027864496218397609329080748011210125,...]. 

This seems to be the above given denominator sequence divided by 45. 

Rationals r(1;N), N=1..20:

[23/45, 33073/51975, 55943738/80405325, 77064019958/105411381075, 15226093370063/20175738337755, 31370562345762421/40654112750576325, 241905492960111168964/308361445213121425125, 1683591136668277300660676/2118874834637442560603925, 48935652383592600478507247/60963055956568704529375785, 713289082617826259771761324613/881220973852200623972126972175, 143961819529547244077111055694498/176621860902091067918984877423075, 2460282354560331257420364974778935366/3000679258254454321752182363934170625, 3790786840731003921840152645904014459389/4600041302904078475246095563911083568125, 2172041935768934848601975221239259534857643451/2624134961613357702910412429293152821189019375, 6015898478763334252390789632019813330057930993384/7239988359091253902329827892419808633660504455625,...].


R(1,N) = 45*r(1,N), N=1..20, is given by A157167(N)/A157168(N):

[23, 33073/1155, 55943738/1786785, 77064019958/2342475135, 15226093370063/448349740839, 31370562345762421/903424727790585, 241905492960111168964/6852476560291587225, 1683591136668277300660676/47086107436387612457865, 48935652383592600478507247/1354734576812637878430573, 713289082617826259771761324613/19582688307826680532713932715, 143961819529547244077111055694498/3924930242268690398199663942735, 2460282354560331257420364974778935366/66681761294543429372270719198537125, 3790786840731003921840152645904014459389/102223140064535077227691012531357412625, 2172041935768934848601975221239259534857643451/58314110258074615620231387317625618248644875, 6015898478763334252390789632019813330057930993384/160888630202027864496218397609329080748011210125, ...].


The sequence of rationals r(1,N) converges to  ((Pi^2)/48)*(1 + 6*sqrt(3)/Pi) approximately 0.8857915201 (Maple12 10 digits) because L(1) = (1 + 6*sqrt(3)/Pi)/3.

The values  r(1,10^k), k=0..4 are:

[.5111111111, .8094327119, .8750050936, .8844209075, .8856256196, .8857722136].

This shows the slow convergence.

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