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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

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

Pages: 229 - 248, Volume 13, Issue 1, January 2010      doi:10.3934/dcdsb.2010.13.229

 
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Yuk L. Yung - Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, 91125, United States (email)
Cameron Taketa - Hawaii Baptist Academy, 2429 Pali Highway, Honolulu, HI, 96817, United States (email)
Ross Cheung - Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, 91125, United States (email)
Run-Lie Shia - Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, 91125, United States (email)

Abstract: 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.

Keywords:  special function, radiation, exponential integral, series.
Mathematics Subject Classification:  Primary: 11Yxx; Secondary: 44A15, 44A20.

Received: June 2009;      Revised: September 2009;      Available Online: October 2009.