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Duplication Formulae for the Gamma and Double

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Duplication Formulae for the Gamma and Double New proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions Donal F. Connon [email protected] 26 March 2009 Abstract New proofs of the duplication formulae for the gamma and the Barnes double gamma functions are derived using the Hurwitz zeta function. Concise derivations of Gauss’s multiplication theorem for the gamma function and a corresponding one for the double gamma function are also reported. This paper also refers to some connections with the Stieltjes constants. 1. Legendre’s duplication formula for the gamma function Hansen and Patrick [11] showed in 1962 that the Hurwitz zeta function could be written as s ⎛⎞1 (1.1) ςςς(,sx )=− 2 (,2) s x⎜⎟ sx , + ⎝⎠2 and, by analytic continuation, this holds for all s . Differentiation results in ss ⎛⎞1 (1.2) ςς′′(,sx )=+ 2 (,2) sx 2log2(,2) ςς sx −′⎜⎟ sx , + ⎝⎠2 and with s = 0 we have ⎛⎞1 (1.3) ςς′′(0,xx )=+ (0,2 ) log 2 ς (0,2 xx ) −+ ς′⎜⎟ 0, ⎝⎠2 We recall Lerch’s identity for Re ()s > 0 1 (1.4) logΓ= (xx )ς ′′′ (0, ) −ςς (0) = (0, x ) + log(2π ) 2 The above relationship between the gamma function and the Hurwitz zeta function was established by Lerch in 1894 (see, for example, Berndt’s paper [6]). A different proof is contained in [9]. We have the well known relationship between the Hurwitz zeta function and the Bernoulli polynomials Bn ()u (for example, see Apostol’s book [4, pp. 264-266]). B ()x (1.5) ς (,)−=−mx m+1 for m∈ N m +1 o which gives us the well-known formula 1 ς (0,x ) =−x 2 Therefore we have from (1.3) and (1.4) ⎛⎞1 (1.6) logΓ+ (xx ) log Γ+⎜⎟ = log Γ (2 xx ) +−() 1 2 log 2 + log π ⎝⎠2 and hence we obtain Legendre’s duplication formula [16, p.240] for the gamma function 21x− ⎛⎞1 (1.7) 2()ΓΓ+x ⎜⎟xx =π Γ (2) ⎝⎠2 Hansen and Patrick [11] also showed that q−1 ⎛⎞r s (1.8) ∑ς ⎜⎟sq,(1)(=−ς s) r=1 ⎝⎠q Differentiation results in q−1 ⎛⎞r ss (1.9) ∑ςςς′′⎜⎟sq,=− ( 1)() sqs + ()logq r=1 ⎝⎠q and with s = 0 we have q−1 ⎛⎞r 1 (1.10) ∑ς ′⎜⎟0,=− log q r=1 ⎝⎠q 2 Substituting Lerch’s identity (1.4) we get q−1 ⎛⎞rq−11 (1.11) ∑log Γ=⎜⎟ log(2π ) − log q r=1 ⎝⎠q 22 and with q = 2 this immediately gives us the well-known result [15, p.3] 2 ⎛⎞11 (1.12) logΓ=⎜⎟ logπ ⎝⎠22 It should be noted that the proof of the above identity is dependent on Lerch’s identity which may be derived without assuming any prior knowledge of (1.12). In the author’s view, this is akin to the marvel experienced when first confronted with a derivation of Euler’s integral π 2 π ∫ logsinxdx=− log 2 0 2 With q = 4 in (1.10) we see that ⎛⎞11331 ⎛⎞ ⎛⎞ logΓ+Γ+Γ=⎜⎟ log ⎜⎟ log ⎜⎟ log(2π ) − log 4 ⎝⎠42422 ⎝⎠ ⎝⎠ and thus ⎛⎞13 ⎛⎞ 1 (1.13) logΓ+Γ=⎜⎟ log ⎜⎟ logπ + log 2 ⎝⎠44 ⎝⎠ 2 which of course may also be easily obtained directly from Euler’s reflection formula for the gamma function. 1 With s =−1 in (1.9), and using ς (1)− =− , we obtain 12 q−1 ⎛⎞r −1 1 (1.14) ∑ςς′′⎜⎟−=−−−1, (qq 1) ( 1) log r=1 ⎝⎠qq12 and with q = 2 we have ⎛⎞11 1 (1.15) ςς′′⎜⎟−=−−−1, ( 1) log 2 ⎝⎠22 24 which we shall also see below in (3.2). 2. Gauss’s multiplication theorem for the gamma function The general Kubert identity is derived in [13, p.169] q−1 −s ⎛⎞rx+ q (2.1) Φ=Φ(,sxz , ) q∑ ⎜⎟ s , , z r=0 ⎝⎠q 3 where Φ(,,)sxz is the Hurwitz-Lerch zeta function ∞ zn sxz (2.2) Φ=(, , ) ∑ n n=0 ()nx+ We see that Φ=(,sx ,1)ς (, sx ) and therefore we have q−1 s ⎛⎞rx+ (2.3) qsxςς(, )= ∑ ⎜⎟ s , r=0 ⎝⎠q which corresponds with (1.8) when x =1. Differentiation results in q−1 ss ⎛⎞rx+ (2.4) qsxqsxqςς′′(, )+= (, )log∑ ς⎜⎟ s , r=0 ⎝⎠q and letting s = 0 and substituting Lerch’s identity (1.4) we get q−1 ⎛⎞rx+−(1) q ⎛⎞ 1 (2.5) logΓ= (x )∑ log Γ⎜⎟ −log(2π ) −−⎜⎟xq log r=0 ⎝⎠q 22⎝⎠ or q−1 (2.6) (2π )(1)/2(1/2)qx−−nxΓ=Γ+()()/∏ () rxq r=0 which is Gauss’s multiplication theorem for the gamma function [3, p.23]. I subsequently discovered that a similar procedure was employed in Milnor’s paper [14]. Letting s=−1 n in (2.3) gives us 1 q−1 ⎛⎞rx+ ςς(1nx , ) 1n , −=n−1 ∑ ⎜⎟ − qqr=0 ⎝⎠ and using (1.5) results in 1 q−1 ⎛⎞rx+ Bx()= B nnn−1 ∑ ⎜⎟ qqr=0 ⎝⎠ where the substitution x → qx gives us the multiplication formula for the Bernoulli polynomials [15, p.60] 4 1 q−1 ⎛⎞r Bqx()=+ B x nnn−1 ∑ ⎜⎟ qqr=0 ⎝⎠ Differentiation of (2.5) gives us [15, p.12] 1 q−1 ⎛⎞rx+ (2.7) ψ ()xq=+ log ∑ψ ⎜⎟ qqr=0 ⎝⎠ and further differentiations give us 1 q−1 ⎛⎞rx+ ψ ()nn()x ψ () = n+1 ∑ ⎜⎟ qqr=0 ⎝⎠ Since [15, p.22] ψ ()nn()x =− (1)+1nn !(ς + 1,) x we see that this results in 1 q−1 ⎛⎞rx+ ςς(1,)nx n1, +=n+1 ∑ ⎜⎟ + qqr=0 ⎝⎠ which is a particular case of (2.3) for positive integer values of s . Hansen and Patrick [11] also showed that q ⎛⎞r s (2.9) ∑ςς⎜⎟sbqsbq,(,−=1 −) r=1 ⎝⎠q x and letting b =− we have q q ⎛⎞rx+ s ∑ςς⎜⎟sqs,(,=+1)x r=1 ⎝⎠q Noting that (2.10) ςς(sx ,1+= ) ( sxx , ) −−s this becomes 5 q s ⎛⎞rx+ s q sqsx,(,) ∑ςς⎜⎟=−s r=1 ⎝⎠qx which may be written as q−1 s ⎛⎞⎛⎞⎛⎞rx+ x xs q sssqsx,,,1(,) ∑ςςςς⎜⎟⎜⎟⎜⎟−++=−s r=0 ⎝⎠⎝⎠⎝⎠qq q x x Letting x → in (2.10) we then obtain another derivation of (2.3). q 3. Duplication formula for the Barnes double gamma function With s =−1 in (1.2) we have 11 ⎛⎞1 (3.1) ςς′′(1,)−=−+xx (1,2) log2(1,2) ςς −−−+ xx′⎜⎟ 1, 22 ⎝⎠2 and using (1.5) we have 11⎛⎞2 ς (1,2)−=−−+xxx⎜⎟ 4 2 26⎝⎠ For example, equation (3.1) also gives us for x =1/2 ⎛⎞11 1 (3.2) ςς′′⎜⎟−=−−−1, log 2 ( 1) ⎝⎠2242 We have the Gosper/Vardi functional equation for the Barnes double gamma (3.3) ς ′′(1,)−=−−x ς (1)log(1Gxx ++Γ ) log() x which was derived by Vardi in 1988 and also by Gosper in 1997 (see [1]). A different derivation is given in equation (4.3.126) of [9]. Using this and (3.3) we may easily deduce that ⎛⎞11 1 3 (3.4) log G ⎜⎟=−logπς + log 2 +′ ( − 1) ⎝⎠24 242 as originally determined by Barnes [5] in 1899. Combining (3.1) and (3.3) results in 6 1 −++Γ=−++ΓlogGxx (1 ) log ( x ) log G (1 2 xx ) log (2 x ) 2 113⎛2 ⎞ ⎛⎞⎛⎞⎛⎞ 311 −−+−−−+++Γ+⎜4xx 2 ⎟ log 2ς ′ ( 1) log Gx⎜⎟⎜⎟⎜⎟ xlog x 462⎝ ⎠ ⎝⎠⎝⎠⎝⎠ 222 Since GxGxx(1+= ) ( ) Γ ( ) this may be written as 1 −−Γ+Γ=−−Γ+ΓlogGx ( ) log ( x ) x log ( x ) log G (2 x ) log (2 x ) x log (2 x ) 2 113⎛2 ⎞ ⎛⎞ 1 ⎛⎞⎛⎞⎛⎞ 111 −⎜4xx −+ 2 ⎟ log 2 −ς ′ ( −− 1) logGx⎜⎟ + − log Γ+ ⎜⎟⎜⎟⎜⎟ x ++ xlog Γ+ x 462⎝ ⎠ ⎝⎠ 2 ⎝⎠⎝⎠⎝⎠ 222 and using (1.6) we thereby obtain the duplication formula for the Barnes double gamma function. In 1899 Barnes developed a multiplication formula for Gnx() (see [15, p.30]) and a particular case is set out below [15, p.29] 22⎛⎞1 (3.5) GxGx()⎜⎟+Γ= () x JxGx () (2) ⎝⎠2 where for convenience Jx() is defined by 11⎛⎞⎛2 11⎞ logJx ( )=− 3log A +−⎜⎟⎜ 2 x + 3 x − log 2 +x −⎟ logπ 41⎝⎠⎝22⎠ A different derivation of this duplication formula was given by Choi [7] in 1996 where he used the double Hurwitz zeta function defined by −s ς 21(,sa )=++∑ ( a k k2 ) kk12,0≥ 4. A multiplication formula for the Barnes double gamma function With s =−1 in (2.4) we have 1 q−1 ⎛⎞rx+ ′′ ςς(1,)−−xBxqq2 ()log =∑ ⎜⎟ − 1, 2 r=0 ⎝⎠q and with the Gosper/Vardi functional equation (3.3) this becomes 7 1 ς ′(−− 1) logGxx (1 + ) + log Γ ( x ) − Bx ( )log q 2 2 qq−−11 2 ⎛rx++ ⎞ ⎛⎞⎛⎞ rx rx+ =−−qqGς ′( 1)∑∑ log⎜ 1 + ⎟ + q ⎜⎟⎜⎟log Γ rr==00⎝qq ⎠ ⎝⎠⎝⎠q However, it is not immediately clear how this may be expressed in the form of the multiplication formula originally derived by Barnes [5, p.291]. Substituting x = qt we have 1 ς ′(−− 1) logGqtqtqtBqt (1 + ) + log Γ ( ) − ( )log q 2 2 qq−−11 2 ⎛rr ⎞ ⎛⎞⎛⎞r =−−qqGtqttς ′( 1)∑∑ log⎜ 1 ++++ ⎟ ⎜⎟⎜⎟log Γ+ rr==00⎝qq ⎠ ⎝⎠⎝⎠q 5. Other multiple gamma functions Adamchik [2] has shown that for Re ()x > 0 n ′′ n (5.1) ςς()()−−−=−nx,( n1)!(∑ kQkn,1 x)logΓk+ ( x) k =0 where the polynomials Qxkn, () are defined by n nj− ⎛⎞⎧⎫nj Qxkn, ()=−∑ (1 x ) ⎜⎟⎨ ⎬ jk= ⎝⎠⎩⎭j k ⎧⎫j and ⎨⎬ are the Stirling subset numbers defined by ⎩⎭k ⎧⎫jn ⎧−−11 ⎫ ⎧ n ⎫ ⎧⎫nn ⎧1, = 0 ⎨⎬=+k ⎨ ⎬ ⎨ ⎬, ⎨⎬= ⎨ ⎩⎭kkk ⎩ ⎭ ⎩−1 ⎭ ⎩⎭00, ⎩ n ≠ 0 We have [15, p.39] (1)− n−1 (5.2) Gxnnn(1)()(+= GxG−1 x) Γ=nn()xGx[] () and it is easily seen that 8 logGxnn (+= 1) log Gx ( ) + log Gn−1 ( x ) n−1 logΓ=−nn (x ) ( 1) logGx ( ) and from this we obtain Γn ()x (5.3) Γ+=n (1)x Γn−1()x Particular cases of (5.1) are (5.4) ςς′′()()−−−=Γ−1,xxxG 1 log ( ) log (x + 1) 2 (5.5) ςς′′()()−−−=Γ+−2,x 2 2log3 (xxGxx ) (3 2 )log ( ) −− (1 ) log Γ (x ) Hence, using (2.3) we may obtain multiplication formulae for the higher order multiple gamma functions.
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