AA summation by Gen č ev Donal F. Connon [email protected] 5 May 2008
Abstract
At a conference at the University of Ostrava in September 2007, Gen č ev reported that nn nH nn π δ −∞= ⎛ ⎞ = − −⎜ ⎟+ ⎝ ⎠ ∑ where δ is defined by i ii δ ⎡ ⎤⎛ ⎞ ⎛− += −⎢ ⎥⎜ ⎟ ⎜⎜ ⎟ ⎜⎢ ⎥⎝ ⎠ ⎝⎣ ⎦⎞⎟⎟⎠ (1 )dilog( ) (1 ) nn xx Li x n ∞= −= − = ∑ In the above is the harmonic number (1) n H (1) 1 nn k H k = = ∑ and i = − . This note indicates a possible approach to the proof. Proof
We have the Beta function ( , )
B u v defined for Re ( > 0 and Re > 0 by the Eulerian integral ) u ( ) v ( ) ( , ) 1 vu B u v t t dt −− = − ∫ and it is well known that ( ) ( )( , ) ( ) u vB u v u v Γ Γ= Γ + where is the gamma function. We then obtain ( ) u Γ ( 1) ( 1(1 ) (2 2) n n n nv v dv n Γ + Γ +− = Γ + ∫ ) which gives us (1)
21 (1 )2 1 n n n v v dvnn − ⎛ ⎞ = −⎜ ⎟+ ⎝ ⎠ ∫ We have the familiar integral for the digamma function [2, p.15]
1( 1) 1 u tu dt t ψ γ −+ + = − ∫ where integration by parts gives us
11 10 0 (1 ) log(1 ) log(1 ) u u t t u t t − = − − − − ∫ dtt dt and thus we see that ( 1) log(1 ) u u u t ψ γ − + + = − − ∫ In the case where is an integer we have the known result u (2) ( ) ( 1) log 1 nn H n n t γ ψ − = + + = − − ∫ t dt Combining (1) and (2) and making the summation gives us n nnn n nH un u du v v dvnn u −∞ ∞= = ⎛ ⎞ = − − −⎜ ⎟+ −⎝ ⎠ ∑ ∑ ∫ ∫ n We have the geometric series nn ww w ∞= = − ∑ and differentiation results in nn d w wn w w dw w w ∞= = =− − ∑ ) and we end up with the double integral representation [ ] nn nH v v u du dvnn u v v −∞= ⎛ ⎞ −= −⎜ ⎟+ − − −⎝ ⎠ ∑ ∫ ∫ It is easily seen that [ ] [ ] [ ][ ] a u a u auu du a a a ua u − − − − −= − − − −− − ∫ u and we obtain the definite integral [ ] log log(1 )(1 )1 (1 ) u adu a aa u −= −− − ∫ or equivalently [ ] [ ] log 1 (1 )(1 ) log 1 (1 )1 (1 ) (1 ) v vv v u du v vu v v − −− = − −− − − ∫ The problem is then reduced to the single integral (3) [ ] nn n v vH dvnn v −∞= − −⎛ ⎞ = −⎜ ⎟+ − −⎝ ⎠ ∑ ∫ v The Wolfram Integrator gives us in a scintilla temporis [ ]
3( 1) log ( 1)log 1 (1 )6 3 log 11 (1 ) 1 1 vv v dv i vv v ⎡ ⎤− + −− − ⎢ ⎥⎣ ⎦⎡ ⎤= − − − +⎣ ⎦− − + − ∫
21 23 3 v v v v − −⎛ ⎞ ⎡ ⎤⎡ ⎤⎡ ⎤− − − + + − −⎜ ⎟ ⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦⎝ ⎠ v − +
6( 1) 1 ( 1) ( 1)log ( 1) log1 1 1 1 3 v vv Li ⎡ ⎤⎛ ⎞ ⎛− + −⎡ ⎤⎢ ⎥⎜ ⎟ ⎜− + − + −⎢ ⎥⎣ ⎦ ⎜ ⎟ ⎜+ − ⎢ + − ⎥⎝ ⎠ ⎝⎣ ⎦ i ⎞+ − ⎟⎟⎠
6( 1) 1 ( 1) ( 1)log 1 log ( 1)1 1 1 1 1 1 v vv L ⎡ ⎤⎛ ⎞ ⎛− + −⎡ ⎤⎢ ⎥⎜ ⎟ ⎜+ − − − − +⎣ ⎦ ⎜ ⎟ ⎜+ − ⎢ + − + − ⎥⎝ ⎠ ⎝⎣ ⎦ i ⎞+ − ⎟⎟⎠ At we find for the right-hand side v =
22 3 31 3 23 33 23 3 i Li i Li − ⎡ ⎤= − − − +⎣ ⎦ + −⎛ ⎞ ⎡ ⎤⎡ ⎤− − − +⎜ ⎟ ⎣ ⎦⎣ ⎦⎝ ⎠⎡ ⎤⎛ ⎞⎛ ⎞− + −⎢ ⎥⎜ ⎟⎜ ⎟+ − + −⎝ ⎠ ⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞ ⎛⎡ ⎤+ − − − +⎢ ⎥⎜ ⎟ ⎜⎣ ⎦+ − + − + −⎝ ⎠ ⎝⎣ ⎦ ⎞⎟⎠ where we have assumed that, of the three possible cubic roots, we are to use ( 1) 1 − = . We have [ ] log 1 log 2 cos( / 2) / 2 ix e x ⎡ ⎤+ = +⎣ ⎦ ix [ ]
13 3 i i π ππ π ⎛ ⎞ ⎛ ⎞− = + = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ and therefore we have log 1 1 log 2 cos 6 6 i π π ⎡ ⎤⎛ ⎞⎡ ⎤+ − = +⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦ and 3cos 6 2 π ⎛ ⎞ =⎜ ⎟⎝ ⎠ At we find for the right-hand side v = i L − ⎡ ⎤−⎛ ⎞ ⎛ ⎞⎡ ⎤⎡ ⎤ ⎡ ⎤= − − − − − − − −⎢ ⎥⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎣ ⎦⎣ ⎦ + −⎝ ⎠ ⎝ ⎠⎣ ⎦ i i Li ⎡ ⎤⎛ ⎞ ⎛⎡ ⎤+ − − − +⎢ ⎥⎜ ⎟ ⎜⎣ ⎦+ − + − + −⎝ ⎠ ⎝⎣ ⎦ ⎞⎟⎠ In a personal communication, Jonathan Sondow has kindly informed me that Mathematica produces the approximate value of 0.234163… for both sides of equation (3).
REFERENCES [1] M.
Gen č ev, Infinite series involving harmonic numbers and products of the binomial type. 2007. [2] H.M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht, the Netherlands, 2001. [3] S. Wolfram, The Integrator. http://integrals.wolfram.com/ Donal F. Connon Elmhurst Dundle Road Matfield Kent TN12 7HDev, Infinite series involving harmonic numbers and products of the binomial type. 2007. [2] H.M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht, the Netherlands, 2001. [3] S. Wolfram, The Integrator. http://integrals.wolfram.com/ Donal F. Connon Elmhurst Dundle Road Matfield Kent TN12 7HD