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Question Number 114146 by Eric002 last updated on 17/Sep/20

prove  ∫_0 ^1 ((t^(n+2) φ(t,1,n+2)+ln(1−t)+t H_(n+1) )/(t(t−1)))dt  =((H_(n+1) ^((2)) −(H_n )^2 )/2)

$${prove} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{t}^{{n}+\mathrm{2}} \phi\left({t},\mathrm{1},{n}+\mathrm{2}\right)+{ln}\left(\mathrm{1}−{t}\right)+{t}\:{H}_{{n}+\mathrm{1}} }{{t}\left({t}−\mathrm{1}\right)}{dt} \\ $$$$=\frac{{H}_{{n}+\mathrm{1}} ^{\left(\mathrm{2}\right)} −\left({H}_{{n}} \right)^{\mathrm{2}} }{\mathrm{2}} \\ $$

Commented by mindispower last updated on 18/Sep/20

φ(t,1,n+2)  is what function ?

$$\phi\left({t},\mathrm{1},{n}+\mathrm{2}\right)\:\:{is}\:{what}\:{function}\:? \\ $$

Commented by Eric002 last updated on 18/Sep/20

lerch transcendent

$${lerch}\:{transcendent}\: \\ $$

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