Rogers L-Function

Rogers L-Function:

\( L(x) = 6/(pi^2)[Li_2(x)+1/2lnxln(1-x)]

= 6/(pi^2)[sum_(n=1)^(infty)(x^n)/(n^2)+1/2lnxln(1-x)], \)

where \( Li_2(x) \) is the dilogarithm.

\( L(x)+L(1-x)=1 \)

\( L(x)+L(y)=L(xy)+L((x(1-y))/(1-xy))+L((y(1-x))/(1-xy)) \)

\( 1/2L(x^2)=L(x)-L(x/(1+x)). \)

\( Sum_(k=2)^inftyL(1/(k^2))=1 \)

Rogers, L. J. "On Function Sum Theorems Connected with the Series \( sum_1^(infty)x^n/n^2 \) ." Proc. London Math. Soc. 4, 169-189, 1907.

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