He showed that just like , for nonnegative integers , though is a different function from . (red) and (blue) for , showing their coincidence at positive integers.
Euler, L. "Methodus generalis summandi progressiones." Commentarii academiae scientiarum imperialis Petropolitanae6, pp. 68-97,
(1732/33) 1738. Reprinted in Opera omnia I. 14, pp. 42-72.Euler,
L. "Consideratio quarundam serierum quae singularibus proprietatibus sunt praeditae."
Novi commentarii academiae scientiarum imperialis Petropolitanae3,
10-12, (1750/51) 1753. Reprinted in Opera omnia I. 14, pp. 516-541.Sandifer,
E. "How Euler Did It: A False Logarithmic Series." Dec. 2007. http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2050%20false%20log%20series.pdf.
![s_a(x)=sum_(n=1)^infty[1/(1-a^n)product_(k=0)^(n-1)(1-xa^(-k))].](/images/equations/FalseLogarithmicSeries/NumberedEquation1.svg)
, 
for nonnegative integers 
, though 
is a different function from 
. 
(red) and 
(blue) for 
, showing their coincidence at positive integers.
is the q-polygamma function.
