Lerch-harmonic numbers related to Lerch transcendent

Figures & data

Figure 1. Contour plot of the generalized harmonic functions H_n(x,r) on the region (x,r)∈[0.1,1.0]² by varying n = 10,20,40.

Figure 2. Contour plot of Lerch-harmonic numbers H₂₀(β,a,α) on the region (α,β)∈[0.2,1.0]² by varying a = 0.5,1.0,1.5.

Table 1. H_n(β,a,α).

$n\to$	1	2	3	4	5	6	7	8	9	10
$H_n(\frac{1}{2},1,1)$	$\frac{5}{4}$	$\frac{4}{3}$	$\frac{131}{96}$	$\frac{661}{480}$	$\frac{1327}{960}$	$\frac{1163}{840}$	$\frac{148969}{107520}$	$\frac{447047}{322560}$	$\frac{44711}{32256}$	$\frac{983705}{709632}$
$H_n(\frac{1}{2},1,\frac{1}{2})$	$\frac{7}{3}$	$\frac{73}{30}$	$\frac{1037}{420}$	$\frac{6257}{2520}$	$\frac{137969}{55440}$	$\frac{3590659}{1441440}$	$\frac{7184321}{2882880}$	$\frac{244311959}{98017920}$	$\frac{9284620207}{3724680960}$	$\frac{2652847607}{1064194560}$
$H_n(\frac{1}{2},1,\frac{1}{3})$	$\frac{27}{8}$	$\frac{195}{56}$	$\frac{1971}{560}$	$\frac{1608}{455}$	$\frac{824661}{232960}$	$\frac{15679479}{4426240}$	$\frac{172526139}{48688640}$	$\frac{862744809}{243443200}$	$\frac{3451183011}{973772800}$	$\frac{106989526191}{30186956800}$
$H_n(\frac{1}{2},1,\frac{1}{4})$	$\frac{22}{5}$	$\frac{203}{45}$	$\frac{5323}{1170}$	$\frac{181567}{39780}$	$\frac{2545253}{556920}$	$\frac{25466453}{5569200}$	$\frac{1477402349}{323013600}$	$\frac{32506216403}{7106299200}$	$\frac{2405571049747}{525866140800}$	$\frac{197260934408479}{43121023545600}$
$H_n(\frac{1}{3},1,1)$	$\frac{7}{6}$	$\frac{65}{54}$	$\frac{131}{108}$	$\frac{1969}{1620}$	$\frac{17731}{14580}$	$\frac{13793}{11340}$	$\frac{744857}{612360}$	$\frac{20111419}{16533720}$	$\frac{20111503}{16533720}$	$\frac{24580757}{20207880}$
$H_n(\frac{1}{3},1,\frac{1}{2})$	$\frac{20}{9}$	$\frac{34}{15}$	$\frac{2152}{945}$	$\frac{58174}{25515}$	$\frac{640124}{280665}$	$\frac{8322382}{3648645}$	$\frac{14980688}{6567561}$	$\frac{254673698}{111648537}$	$\frac{14516434820}{6363966609}$	$\frac{130648005758}{57275699481}$
$H_n(\frac{1}{3},1,\frac{1}{3})$	$\frac{13}{4}$	$\frac{277}{84}$	$\frac{4169}{1260}$	$\frac{162731}{49140}$	$\frac{1953227}{589680}$	$\frac{111341219}{33611760}$	$\frac{3674329387}{1109188080}$	$\frac{55115245109}{16637821200}$	$\frac{165346007027}{49913463600}$	$\frac{15377186261111}{4641952114800}$
$H_n(\frac{1}{3},1,\frac{1}{4})$	$\frac{64}{15}$	$\frac{1748}{405}$	$\frac{22784}{5265}$	$\frac{129196}{29835}$	$\frac{24422464}{5638815}$	$\frac{122118508}{28194075}$	$\frac{10624464896}{2452884525}$	$\frac{1051826511004}{242835567975}$	$\frac{12972543418816}{2994972005025}$	$\frac{4786870347467204}{1105144669854225}$