New proof of the duplicatio ad multiplicatio formulae for the gamma ad the Bare double gamma fuctio Abtract Doal F. Coo dcoo@btopeworld.com 6 March 9 New proof of the duplicatio formulae for the gamma ad the Bare double gamma fuctio are derived uig the Hurwitz zeta fuctio. Cocie derivatio of Gau multiplicatio theorem for the gamma fuctio ad a correpodig oe for the double gamma fuctio are alo reported. Thi paper alo refer to ome coectio with the Stieltje cotat.. Legedre duplicatio formula for the gamma fuctio Hae ad Patric [] howed i 96 that the Hurwitz zeta fuctio could be writte a (.) ς(, ) = ς(,) ς, + ad, by aalytic cotiuatio, thi hold for all. Differetiatio reult i (.) ad with (.3) = ς (, ) = ς (,) + log ς(,) ς, + we have ς (, ) = ς (, ) + log ς(, ) ς, + We recall Lerch idetity for Re () > (.4) log Γ ( ) = ς (, ) ς () = ς (, ) + log( π ) The above relatiohip betwee the gamma fuctio ad the Hurwitz zeta fuctio wa etablihed by Lerch i 894 (ee, for eample, Berdt paper [6]). A differet proof i cotaied i [9].
We have the well ow relatiohip betwee the Hurwitz zeta fuctio ad the Beroulli polyomial B ( u ) (for eample, ee Apotol boo [4, pp. 64-66]). (.5) (, ) B ( ) m+ ς m = m + for m N o which give u the well-ow formula ς (, ) = Therefore we have from (.3) ad (.4) log Γ ( ) + log Γ + = log Γ ( ) + log + log π (.6) ( ) ad hece we obtai Legedre duplicatio formula [6, p.4] for the gamma fuctio (.7) Γ( ) Γ + = π Γ() Hae ad Patric [] alo howed that (.8) r ς, = ( ) ς ( ) Differetiatio reult i (.9) r ς, = ( ) ς () + ς()log ad with = we have (.) r ς, = log Subtitutig Lerch idetity (.4) we get (.) r log Γ = log( π ) log ad with = thi immediately give u the well-ow reult [5, p.3]
(.) log Γ = logπ It hould be oted that the proof of the above idetity i depedet o Lerch idetity which may be derived without aumig ay prior owledge of (.). I the author view, thi i ai to the marvel eperieced whe firt cofroted with a derivatio of Euler itegral π π logi d= log With ad thu = 4 i (.) we ee that 3 3 log Γ + log Γ + log Γ = log( π ) log 4 4 4 (.3) 3 log Γ + log Γ = logπ + log 4 4 which of coure may alo be eaily obtaied directly from Euler reflectio formula for the gamma fuctio. With = i (.9), ad uig ς ( ) =, we obtai (.4) r ς =, ( ) ς ( ) log ad with = we have (.5) ς, = ς ( ) log 4 which we hall alo ee below i (3.).. Gau multiplicatio theorem for the gamma fuctio The geeral Kubert idetity i derived i [3, p.69] (.) Φ (, z, ) = Φ,, z 3
where Φ( z,, ) i the Hurwitz-Lerch zeta fuctio (.) Φ (, z, ) = = z ( + ) We ee that Φ (,,) = ς (, ) ad therefore we have (.3) ς(, ) = ς, which correpod with (.8) whe =. Differetiatio reult i (.4) ς (, ) + ς(, )log = ς, ad lettig = ad ubtitutig Lerch idetity (.4) we get (.5) r+ ( ) log Γ ( ) = log Γ log( π ) log or ( )/ (/) (.6) ( π ) Γ ( ) = Γ (( r+ )/ ) which i Gau multiplicatio theorem for the gamma fuctio [3, p.3]. I ubeuetly dicovered that a imilar procedure wa employed i Milor paper [4]. Lettig = i (.3) give u ς(, ) = ς, ad uig (.5) reult i r+ B( ) = B where the ubtitutio give u the multiplicatio formula for the Beroulli polyomial [5, p.6] 4
r B( ) = B + Differetiatio of (.5) give u [5, p.] (.7) ψ ( ) log r+ = + ψ ad further differetiatio give u ψ Sice [5, p.] r+ = ( ) ( ) ( ) ψ + ψ ( ) = ( )! ς ( +, ) ( ) + we ee that thi reult i ς( +, ) = ς, + + which i a particular cae of (.3) for poitive iteger value of. Hae ad Patric [] alo howed that (.9) r ς, b= ς(, b ) ad lettig b = we have ς, = ς(, + ) Notig that (.) ς(, + ) = ς(, ) thi become 5
ς, = ς(, ) which may be writte a ς, ς, + ς, + = ς(, ) Lettig i (.) we the obtai aother derivatio of (.3). 3. Duplicatio formula for the Bare double gamma fuctio With = i (.) we have (3.) ς (, ) ς (, ) log ς(, ) ς = +, + ad uig (.5) we have ς (, ) = 4 + 6 For eample, euatio (3.) alo give u for = / (3.) ς, = log ς ( ) 4 We have the Goper/Vardi fuctioal euatio for the Bare double gamma (3.3) ς (, ) = ς ( ) log G( + ) + log Γ ( ) which wa derived by Vardi i 988 ad alo by Goper i 997 (ee []). A differet derivatio i give i euatio (4.3.6) of [9]. Uig thi ad (3.3) we may eaily deduce that (3.4) 3 log G = logπ + log + ς ( ) 4 4 a origially determied by Bare [5] i 899. Combiig (3.) ad (3.3) reult i 6
log G( + ) + log Γ ( ) = log G( + ) + log Γ ( ) 3 3 G 4 6 4 + log ς ( ) log + + + log Γ + Sice G( + ) = G( ) Γ( ) thi may be writte a log G ( ) log Γ ( ) + log Γ ( ) = log G( ) log Γ ( ) + log Γ ( ) 3 G 4 6 4 + log ς ( ) log + log Γ + + + log Γ + ad uig (.6) we thereby obtai the duplicatio formula for the Bare double gamma fuctio. I 899 Bare developed a multiplicatio formula for G ( ) (ee [5, p.3]) ad a particular cae i et out below [5, p.9] (3.5) G ( ) G + Γ ( ) = J( ) G( ) where for coveiece J( ) i defied by J = A+ + + 4 log ( ) 3log 3 log log A differet derivatio of thi duplicatio formula wa give by Choi [7] i 996 where he ued the double Hurwitz zeta fuctio defied by π ς (, a) = ( a+ + ), 4. A multiplicatio formula for the Bare double gamma fuctio With = i (.4) we have ς (, ) B ( )log= ς, ad with the Goper/Vardi fuctioal euatio (3.3) thi become 7
ς ( ) log G( + ) + log Γ( ) B ( )log r+ r+ = ς ( ) log G + + log Γ However, it i ot immediately clear how thi may be epreed i the form of the multiplicatio formula origially derived by Bare [5, p.9]. Subtitutig = t we have ς ( ) log G( + t) + tlog Γ( t) B ( t)log r r r = ς ( ) log G + t+ + t+ log Γ t+ 5. Other multiple gamma fuctio Adamchi [] ha how that for Re ( ) >, = ( ) Q! ( )logγ ( ) (5.) ς ( ) ς ( ), + = where the polyomial Q, ( ) are defied by ad j j j Q, ( ) = ( ) j= j are the Stirlig ubet umber defied by j = +,, = =, We have [5, p.39] + = ) Γ ( ) = [ G ( ) ] (5.) G( ) G( ) G ( ad it i eaily ee that ( ) 8
log G ( + ) = log G ( ) + log G ( ) log Γ ( ) = ( ) log G( ) ad from thi we obtai (5.3) Γ Γ ( + ) = Γ ( ) ( ) Particular cae of (5.) are (5.4) ς ( ) ς ( ), = log Γ( ) log G( + ), = log Γ ( ) + (3 )log ( ) ( ) log Γ( ) (5.5) ς ( ) ς ( ) G 3 Hece, uig (.3) we may obtai multiplicatio formulae for the higher order multiple gamma fuctio. 6. Some coectio with the Stieltje cotat The geeralied Euler-Macheroi cotat γ (or Stieltje cotat) are the coefficiet of the Lauret epaio of the Riema zeta fuctio ς ( ) about = (6.) ( ) ς() = + γ( )! = The Stieltje cotat γ ( ) are the coefficiet i the Lauret epaio of the Hurwitz zeta fuctio ς (, u) about = (6.) ( ) ς(, ) = = + γ()( ) ( + )! = = ad γ ( ) ψ ( ) =, where ψ ( ) i the digamma fuctio which i the logarithmic d derivative of the gamma fuctio ψ ( ) = log Γ ( ). It i eaily ee from the defiitio d of the Hurwitz zeta fuctio that ς (,) = ς ( ) ad accordigly that γ () = γ. Sice lim ς ( ) γ = it i clear that γ = γ. It may be how, a i [, p.4], that 9
(6.3) γ N + N N log log N log log t = lim = lim dt + t N N = = where, throughout thi paper, we defie It wa previouly how i [] that log =. (6.4) γ i + + i j + ( ) = ( ) log ( + j) i= i j= j We ee from (6.) that for (6.5) d d + + [( ) ς(, )] = ( ) ( + ) γ ( ) = We multiply (.3) by ( ) ( ) ς(, ) = ( ) ς, ad, uig the Leibiz rule to differetiate thi + time, we obtai + d d d [( ) (, )] = ( ) ς, d d d + + + ς + + = Evaluatig thi at = reult i + + r+ r+ log + ( ) ( + ) γ log = ( ) ( + ) γ = (where we have iolated the ( + )th term uig lim[( ) ς (, )] = ) Uig the biomial idetity + + = + thi may be epreed a (6.6) + log = ( ) + ( ) γ ( )log + r γ + = ad otig that
r+ m + m + + f = f = f + f m= m= we ee that for iteger ad = (6.7) ( ) ( ) log + γ r log j r γ = + + γ = + = which wa previouly derived by Coffey [8] uig the relatio (.3). With = i (6.7) we have (6.8) r γ = γ + log + γ Sice ψ ( ) = γ ( ) we ee from (.7) that (6.9) γ( ) = log+ γ ad thi cocur with (6.6) whe =. With = thi become r+ r+ γ = log + γ = log + γ + ad therefore we obtai (6.8) agai. γ Lettig + i (6.9) we obtai γ + + + = + r ( ) log γ ad we the have m+ = log + γ m=
m+ + γ( + ) = log + γ γ + γ m= Comparig thi with (6.9) we obtai (6.) γ( + ) = γ( ) γ + γ + For eample, lettig = we ee that γ( + ) = γ( ) γ + γ + Sice ψ ( ) = γ ( ) we may epre (6.) a ( ) ( ) ψ + ψ = ψ + ψ ad thi may be eaily verified by otig that [5, p.4] ψ ( + ) ψ ( ) = = ( / ) Lettig + i (6.6) we obtai + + log = ( ) + ( ) γ ( + )log + r γ + = ad otig that we deduce that r+ + r+ f = f + f + f ( ) ( )log ( ) ( )log γ γ γ γ + + = + = = or euivaletly
( ) [ ( ) ( )]log γ γ + = γ γ + = With the reideig = m we have = ( ) [ γ ( + ) γ ( )]log REFERENCES m m = ( ) [ γm( + ) γm( )]log m= m = ( ) log ( ) [ γm( + ) γm( )]log m= m m m [] V.S.Adamchi, Cotributio to the Theory of the Bare Fuctio. Computer Phyic Commuicatio, 3. http://www-.c.cmu.edu/~adamchi/article/bare.pdf [] V.S.Adamchi, The multiple gamma fuctio ad it applicatio to computatio of erie. The Ramauja Joural, 9, 7-88, 5. [3] G.E. Adrew, R. Aey ad R. Roy, Special Fuctio. Cambridge Uiverity Pre, Cambridge, 999. [4] T.M. Apotol, Itroductio to Aalytic Number Theory. Spriger-Verlag, New Yor, Heidelberg ad Berli, 976. [5] E.W. Bare, The theory of the G-fuctio. Quart. J. Math., 3, 64-34, 899. [6] B.C. Berdt, The Gamma Fuctio ad the Hurwitz Zeta Fuctio. Amer. Math. Mothly, 9,6-3, 985. [7] J. Choi, A duplicatio formula for the double gamma fuctio Γ. Bull. Korea Math. Soc. 33 (996), No., 89-94. http://www.mathet.or.r/mathet/m_te/54.pdf [8] M.W. Coffey, New reult o the Stieltje cotat: Aymptotic ad eact evaluatio. J. Math. Aal. Appl., 37 (6) 63-6. arxiv:math-ph/566 [p, pdf, other] [9] D.F. Coo, Some erie ad itegral ivolvig the Riema zeta fuctio, biomial coefficiet ad the harmoic umber. Volume II(a), 7. 3
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