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 . Concise derivations of Gauss’s multiplication theorem for the 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 [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 ⎛⎞rx+ qs sqsx,(,)s ∑ςς⎜⎟=−s r=1 ⎝⎠qx which may be written as

q−1 ⎛⎞⎛⎞⎛⎞rx+ x x qs sssqsx,,,1(,)s ∑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.

6. Some connections with the Stieltjes constants

The generalised Euler-Mascheroni constants γ n (or Stieltjes constants) are the coefficients of the Laurent expansion of the ς ()s about s =1

∞ n 1(1)− n (6.1) ςγ()ss=+∑ n ( − 1) sn−1!n=0

The Stieltjes constants γ n ()x are the coefficients in the Laurent expansion of the Hurwitz zeta function ς (,su ) about s =1

∞∞11(1)− n (6.2) (,sx )==+()(x s 1)n ςγ∑∑s n − nn==00()nx+− s 1 n !

and γ 0 ()x =−ψ ()x , where ψ ()x is the which is the logarithmic d derivative of the gamma function ψ ()x =Γ log()x . It is easily seen from the definition dx of the Hurwitz zeta function that ς (s ,1)= ς (s ) and accordingly that γ nn(1) = γ .

⎡1 ⎤ Since lim⎢ς (s ) −=⎥γ it is clear that γ 0 = γ . It may be shown, as in [12, p.4], that s→1 ⎣⎦s −1

9 ⎡⎤NNlognnkN log +1 ⎡ logn ktN log n⎤ (6.3) γ n =−=−lim lim dt NN→∞ ⎢⎥∑∑→∞ ⎢ ∫ ⎥ ⎣⎦kk==11kn+1 ⎣ k1 t⎦ where, throughout this paper, we define log0 1= 1.

It was previously shown in [10] that

∞ i 11⎛⎞i jn+1 (6.4) γ n ()x =− ∑∑⎜⎟(1)log(−xj + ) ni++11ij==00⎝⎠j

We see from (6.2) that for n ≥ 0

n+1 d n (6.5) n+1 [(ssx−=−+ 1)ςγ ( , )] ( 1) ( n 1)n (x ) ds s=1

We multiply (2.3) by (s −1)

q−1 s ⎛⎞rx+ qs(1)(,)−=−ςς sx∑ (1), s⎜⎟ s r=0 ⎝⎠q and, using the Leibniz rule to differentiate this n +1 times, we obtain

n+1 ⎛⎞n +1 dddnk+−11k q−1 n+ ⎡ ⎛r+ x ⎞⎤ [(ssxq 1) ( , )] s ( s 1) s , ∑∑⎜⎟nk+−11−=−ςςk n+ ⎢ ⎜ ⎟⎥ kr==00⎝⎠k ds ds ds ⎣ ⎝q ⎠⎦

Evaluating this at s =1 results in

n q−1 nn+−1 ⎛⎞n +1 k⎛⎞rx++ kn⎛⎞rx qqqlog +−−+=−+∑∑⎜⎟( 1) ( nk 1)γγnk− ⎜⎟ logq ( 1) ( n 1) n⎜⎟ kr==00⎝⎠k ⎝⎠qq⎝⎠

(where we have isolated the (n + 1)th term using lim[(ssx− 1)ς ( , )]= 1) s→1

nk−+1⎛⎞⎛nn+1 ⎞ Using the binomial identity ⎜⎟⎜= ⎟ this may be expressed as n +1 ⎝⎠⎝kk⎠

q−1 n+1 n ⎛⎞rx+ nklog q ⎛⎞n k (6.6) ∑∑γγnn⎜⎟=−qq(1) +⎜⎟(1) − −k ()log xq rk==00⎝⎠qn+1 ⎝⎠k and noting that

10 qq−−11⎛⎞rx+− ⎛ mx11+− ⎞ q ⎛ mx+− ⎞⎛ qx1+⎞ ∑∑ff⎜⎟==+ ⎜ ⎟ ∑ f ⎜ ⎟⎜ f⎟ rm==01⎝⎠qq ⎝ ⎠ m = 1 ⎝ q ⎠⎝q⎠ we see that for integers q ≥ 2 and x =1

q−1 n+1 n ⎛⎞rqnjlog ⎛⎞n k (6.7) ∑∑γγnn⎜⎟=− +qq(1) − +⎜⎟(1) − γn−k logq rk==10⎝⎠qn+1 ⎝⎠k which was previously derived by Coffey [8] using the relation (2.3).

With n = 0 in (6.7) we have

q−1 ⎛⎞r (6.8) ∑γ 0 ⎜⎟=−γγ +qqqlog + r=1 ⎝⎠q

Since ψ ()x =−γ 0 ()x we see from (2.7) that

1 q−1 ⎛rx+ ⎞ (6.9) γγ00()xq=− log + ∑ ⎜⎟ qqr=0 ⎝⎠ and this concurs with (6.6) when n = 0 .

With x =1 this becomes

11qq−−12⎛⎞rr++ 11⎛⎞1 γ 00=−log qq + ∑∑γγ⎜⎟=−log + 0⎜⎟+ γ0 qqrr==00⎝⎠ qq⎝⎠q and therefore we obtain (6.8) again.

Letting x →+1 x in (6.9) we obtain

11q−1 ⎛⎞rx+ + γγ00(1+=−+xq ) log ∑ ⎜⎟ qqr=0 ⎝⎠

1 q ⎛⎞mx+ =−log q + ∑γ 0 ⎜⎟ qqm=1 ⎝⎠ and we then have

11 111q−1 ⎛⎞mx+ ⎛⎞⎛ x qx+ ⎞ γγγ00(1+=−+xq ) log ∑ ⎜⎟−0 ⎜⎟⎜ +γ0⎟ qqqqqqm=0 ⎝⎠ ⎝⎠⎝⎠

Comparing this with (6.9) we obtain

11⎛⎞x ⎛x ⎞ (6.10) γγγγ0000(1+=xx ) ( ) −⎜⎟ + ⎜1 + ⎟ qqq⎝⎠ ⎝ q ⎠

For example, letting q = 2 we see that

11⎛⎞x ⎛x ⎞ γγγγ0000(1+=xx ) ( ) −⎜⎟ + ⎜1 + ⎟ 222⎝⎠ ⎝ 2 ⎠

Since ψ ()x =−γ 0 ()x we may express (6.10) as

11⎛⎞x ⎛x ⎞ ψ (1+−xx )ψψ ( ) =⎜⎟ 1 + − ψ ⎜⎟ qqq⎝⎠ ⎝q⎠ and this may be easily verified by noting that [15, p.14]

111 ψ (1+−xx )ψ ( ) == x qxq(/)

Letting x →+1 x in (6.6) we obtain

q−1 n+1 n ⎛⎞rx++1lnkog q ⎛⎞n k ∑∑γγnn⎜⎟=−qq(1) +⎜⎟(1) −−k (1 + x )logq rk==00⎝⎠qn+1 ⎝⎠k and noting that

qq−−11⎛⎞⎛⎞⎛⎞⎛rx++1 rx + x x⎞ ∑∑fff⎜⎟⎜⎟⎜⎟⎜=++−1 f⎟ rr==00⎝⎠⎝⎠⎝⎠⎝qqqq⎠ we deduce that

n n ⎛⎞nnkk⎛xx ⎞ ⎛ ⎞ ⎛⎞ k k qxqq∑∑⎜⎟(−++ 1)γγγnk− ( )logn⎜ 1 ⎟−=−+n ⎜ ⎟ ⎜⎟( 1) γnk− (1 x )log q k =0 ⎝⎠kk⎝qq ⎠ ⎝ ⎠k =0 ⎝⎠ or equivalently

12 n 1 ⎡⎤⎛⎞⎛⎞⎛⎞xx n kk ⎢⎥γγnn⎜⎟⎜⎟⎜⎟1(+− =∑ −1)[ γγn−−kn(1 +−x)k(xq)]log qqq⎣⎦⎝⎠⎝⎠⎝⎠k =0 k

With the reindexing kn=−m we have

n ⎛⎞n kk ∑⎜⎟(1)[−+−γγnk−− (1x )nk ()]logxq k =0 ⎝⎠k

0 ⎛⎞n nm−−nm =−+−∑ ⎜⎟(1)[γγmm (1x ) ()]logxq mn= ⎝⎠nm−

n nn ⎛⎞n m −m =−(1)logqx∑⎜⎟ (1)[−γγmm (1 + ) − ()]logxq m=0 ⎝⎠m

REFERENCES

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[7] J. Choi, A duplication formula for the double gamma function Γ2 . Bull. Korean Math. Soc. 33 (1996), No.2, 289-294. http://www.mathnet.or.kr/mathnet/kms_tex/1054.pdf

[8] M.W. Coffey, New results on the Stieltjes constants: Asymptotic and exact evaluation. J. Math. Anal. Appl., 317 (2006) 603-612. arXiv:math-ph/0506061 [ps, pdf, other]

[9] D.F. Connon, Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume II(a), 2007.

13 arXiv:0710.4023 [pdf]

[10] D.F. Connon, Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume II(b), 2007. arXiv:0710.4024 [pdf]

[11] E.R. Hansen and M.L. Patrick, Some Relations and Values for the Generalized cccccRiemann Zeta Function. xx x xx Math. Comput., Vol. 16, No. 79. (1962), pp. 265-274.

[12] A. Ivić, The Riemann Zeta- Function: Theory and Applications. Dover Publications Inc, 2003.

[13] S. Kanemitsu and H. Tsukada, Vistas of special functions. World Scientific Publishing Co. Pte. Ltd., 2007.

[14] J. Milnor, On , Hurwitz zeta functions, and the Kubert identities. L'Enseignement Math., 29 (1983), 281-322.

[15] H.M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht, the Netherlands, 2001.

[16] E.T. Whittaker and G.N. Watson, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions. Fourth Ed., Cambridge University Press, Cambridge, London and New York, 1963.

Donal F. Connon Elmhurst Dundle Road Matfield Kent TN12 7HD [email protected]

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