January  2010, 13(1): 229-248. doi: 10.3934/dcdsb.2010.13.229

Infinite sum of the product of exponential and logarithmic functions, its analytic continuation, and application

1. 

Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, 91125, United States, United States, United States

2. 

Hawaii Baptist Academy, 2429 Pali Highway, Honolulu, HI, 96817, United States

Received  June 2009 Revised  September 2009 Published  October 2009

We show that the function $S_1(x)=\sum_{k=1}^\infty e^{-2\pi kx} \log k$ can be expressed as the sum of a simple function and an infinite series, whose coefficients are related to the Riemann zeta function. Analytic continuation to the imaginary argument $S_1(ix)=K_0(x) - iK_1(x)$ is made. For $x=\frac{p}{q}$ where $p$ and $q$ are integers with $p$<$q$, closed finite sum expressions for $K_0 (\frac{p}{q} )$ and $K_1 ( \frac{p}{q}) $ are derived. The latter results enable us to evaluate Ramanujan's function $\varphi (x)=\sum_{k=1}^\infty (\frac{\log k}{k}-\frac{\log(k+x)}{k+x})$ for $x=-\frac{2}{3}, -\frac{3}{4},$ and $-\frac{5}{6},$ confirming what Ramanujan claimed but did not explicitly reveal in his Notebooks. The interpretation of a pair of apparently inscrutable divergent series in the notebooks is discussed. They reveal hitherto unsuspected connections between Ramanujan's $\varphi(x), K_0(x), K_1(x),$ and the classical formulas of Gauss and Kummer for the digamma function.
Citation: Yuk L. Yung, Cameron Taketa, Ross Cheung, Run-Lie Shia. Infinite sum of the product of exponential and logarithmic functions, its analytic continuation, and application. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 229-248. doi: 10.3934/dcdsb.2010.13.229
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