E.W.BARNES APPROACH OF THE MULTIPLE GAMMA FUNCTIONS

Transcription

1 J. Korea Math. Soc. 9 (99), No., pp E.W.BARNES APPROACH OF THE MULTIPLE GAMMA FUNCTIONS JUNESANG CHOI AND J. R. QUINE I this paper we provide a ew proof of multiplicatio formulas for the simple ad double gamma fuctios ad also give some related asymptotic expasios.. E.W.Bares defiitio of multiple gamma fuctios r, I [3] E. W. Bares itroduces the multiple Hurwitz ζ -fuctio, for Re s > ζ r (s, a w,...,w r )= m,m,...,m r =0 (a + ) s where = m w +m w + +m r w r ad also represets the r-ple Hurwitz ζ -fuctio by the cotour itegral ζ r (s, a w,w,...,w r )= iɣ( s) π L e az ( z) s r k= ( e w kz ) dz where the coditios for a ad w,...,w r are described i [3] ad the possible cotour L is give by Received March 5, 99. Revised September 5, 99.

2 8 Juesag Choi ad J. R. Quie For our purpose we restrict these whe w k =, k =,,..., ad the cotour C is the same as Fig. I i [4]. That is to say, a > 0, Re s >, ζ (s, a) = (a + k + k + +k ) s. k,k,...,k =0 The ζ (s, a) ca be cotiued to a meromorphic fuctio with poles s =,,...,,a >0, for by the cotour itegral represetatio iɣ( s) e az ( z) s ζ (s, a) = π c ( e z ) dz the itegral is valied for a > 0adalls,soζ (s,a)has possible poles oly at the poles of Ɣ( s), i.e., s =,, 3,... But by the series defiitio ζ (s, a) is holomorphic for Re s > [4]. I particular, whe =, ζ (s, a) = (a + k) s = ζ(s,a) k=0 is the well-kow Hurwitz ζ -fuctio, which ca be cotiued to a meromorphic fuctio with oly simple pole at s = havig its residue, by the cotour itegral represetatio [], [4], [9]. Now we summarize some kow propositios. PROPOSITION.. [7]. Let ζ(s,a) = k=0 (k + a) s be the Hurwitz ζ -fuctio, where a > 0 ad Re s >, thewehave Ɣ(a) = eζ (0,a) R, where R is a cotat ad ζ (s, a) = s ζ(s,a).

3 E.W.Bares approach of the multiple gamma fuctios 9 Proof. As above, by the cotour itegral represetatio of ζ(s,a), ζ(s,a) is aalytically cotiued for all s, (a > 0). ζ(s,a +) = ζ(s,a) a s, ζ (s,a +) = ζ (s +a)+a s log a, ζ (0, a + ) = ζ (0, a) + log a. Lettig G (a) = e ζ (0,a),wehaveG (a+)=ag (a), a > 0, ad d da log G (a) = d da d ds ζ(s,a) s=0. Ay by the aalytic cotiuatio of ζ(s,a)oe sees that G (a) is C o R +. So by the Bohr-Mollerup Theorem G (a) = Ɣ(a)R, R costat. Note that R = e ζ (0) sice ζ(s,) = ζ(s) ad so Now defie R = G () = e ζ (0,). G (a) = e ζ (0,a), where ζ (s, a) = s ζ (s, a). The basic properties of G (a) are ow give by the followig propositio. PROPOSITION.. [7]. (a) G + (a + ) = G +(a) G (a). (b) G (a) ca be cotiued a meromorphic fuctio o C with poles at the egative itegers ad a simple pole at zero. (c) Let R = lim a 0 ag (a),theg ()=R /R, where R 0 =. I particular, whe =,

4 30 Juesag Choi ad J. R. Quie COROLLARY.3. (a) G (a + ) = G (a)/g (a). (b) G (a) ca be cotiued to a meromorphic fuctio o C with poles at the egative itegers ad a simple pole at zero. Now we ca get the relatioship betwee multiple gamma fuctios ad multiple Hurwitz ζ -fuctios. PROPOSITION.4. [7]. ( Ɣ (a) = R ( )m ( a m= m+ m ) ) G (a). Proof. See [4] ad [7]. I particular, whe =. COROLLARY.5. Ɣ (a) = (R R a )G (a) where R = e ζ (0) e ζ (0,) = lim a ag (a).. Multiplicatio formulas for Ɣ ad Ɣ I this sectio we provide other proofs of multiplicatio formulas for the simple ad double gamma fuctios. THEOREM.. m Ɣ(ka+ kl m )=(π)/m /k ( k k m )mak+/(mk m k) k, m =,, 3,... Ɣ(ma+ m k ),

5 E.W.Bares approach of the multiple gamma fuctios 3 Proof. Note that {i = 0,,,...}={kj +,0 k, j = 0,,,...}. m ζ(s,ka + kl m m ) = (ka + kl m + i) s Now we have m m = ( m m k )s = ( m m k )s = ( m k k )s = ( m k k )s = ( m k k )s k m (ma + l + m k i)s ζ(s,ka + kl m ) = (m k k )s ζ (s, ka + kl m ) =( m k )s log( m k k ) (ma + l + m (kj +))s k (ma + m k (ma + m + j) k s ζ(s,ma + m k ). ζ(s,ma + m k ) + (m k k )s + mj +l) s ζ(s,ma + m k ). ζ (s,ma + m k ), where the accet deotes the differetiatio with respect to s. Wehave m ζ (0,ka + kl ( k m ) = log(m k ) k + ζ (0,ma + m k ). ζ(0,ma + m k ) )

6 3 Juesag Choi ad J. R. Quie Sice ζ(0,a) = a,wehave k ζ(0,ma + m k k ) = (/ ma m k ) = (/ ma)k m (k ). Thus m e ζ (0,ka+ kl m ) = ( m k k )(/ ma)k m (k ) By Propositio., e ζ (0,a) = e ζ (0) Ɣ(a), sowehave e ζ (0,ma+ m k ). m e mζ (0) m Note that Ɣ(ma+ kl Ɣ(ka+ kl m )=(m k )(/ ma)k m m ) = e(k m)ζ (0) ( m k )(/ ma)k m ζ (0) = /log(π), k (k ) e kζ (0) e (k m)ζ (0) = (π) /(m k), Ɣ(ma+ m k ). (k ) k ( m k )(/ ma)k m (k ) = ( k m )mak+/(mk m k). This completes the proof of Theorem.. Ɣ(ma+ ma k ). Now we get the classical Gauss multiplicatio formula as the special case of Theorem.. COROLLARY.. m Ɣ(a + m ) = (π)/m / m / ma Ɣ(ma), m =, 3, 4,...

7 E.W.Bares approach of the multiple gamma fuctios 33 Proof. Plug k =, m =, 3, 4,... i the formula of Theorem.. Fially we provide aother proof of the multiplicatio formula for Ɣ. THEOREM.3. [],[4]. Ɣ (x + i + j ) = K (π) ( ) x x x Ɣ (x) where K = A e (π) ( ) 5. LEMMA. ζ (s, x) = ζ(s,x)+( x)ζ(s, x). I particular, ζ (s, ) = ζ(s ) ad so ζ (0, ) = ζ ( ). Note that the classical result is (See Chapter i [4]) ζ( m,x) = B m+(x) m+, m=0,,, B 0 (x)=, B (x)= x, B (x)= x x + 6,. Proof of Lemma. Note that ζ (s, x) = (x + k + k + +k ) s = k,...,k =0 k=0 ( k+ ) (x +k) s. This is because the umber of solutios of k + k + +k = k, k = 0,,,..., (k,k,...,k ) N is equal to the coefficiet of x k i the expasio of the Maclauri series of ( x), ( ) k + i.e.,.

8 34 Juesag Choi ad J. R. Quie I particular, ζ (s, x) = = = k,k =0 k=0 k=0 (x + k + k ) s k + (x + k) s (x + k) (x + k) + x s (x + k) s k=0 = ζ(s,x)+( x)ζ(s, x). Thus we have ζ (s, x) = ζ(s,x)+( x)ζ(s, x). Proof of Theorem.3. Cosider ζ (s, x + i + j ) = = s k,k =0 (x + i + j + k + k ) s (x + i + j + k + k ) s k,k =0 = s k,k =0 = s ζ (s,x). (x + k + k ) s ζ (s, x + i + j ) = (log )s ζ (s, x) + s ζ (s,x), where the accet deotes the differetiatio with respect to s. Therefore we

9 E.W.Bares approach of the multiple gamma fuctios 35 have ζ (0, x + i + j ) = (log )ζ (0, x) + ζ (0,x). e ζ i+ j (0,x+ ) = ζ(0,x) e ζ (0,x). G (x + i + j ) = ζ (0,x) G (x). Note that G (x) = R R a Ɣ (x), by Corollary.5. R Ɣ (x + i + j i+ j )R (x+ ) = ζ(0,x) R R x Ɣ (x). Thus we have i+ j (x+ R ) = R Ɣ (x + i + j i+ j (x+ ) = R ( x+ ). ) = F(x, )Ɣ (x) where F(x, ) = ζ(0,x) R R x x+. R = e ζ (0) = e /log(π) = (π) /. R = e ζ (0) e ζ (0,) = (π) / e ζ ( ) (Lemma) = (π) / A e (see Chapter i [4]). ζ (0, x) = ζ(,x) + ( x)ζ(0, x) = / x +/x + ( x)(/ x) = / x x + 5.

10 36 Juesag Choi ad J. R. Quie The we have F(x, ) = / x x+ 5 (π) /( ) A e ( ) (π) /( x x) /( ) = K(π) ( )x x x where K = A e (π) 5. Therefore we have Ɣ (x + i + j ) = K (π) ( ) x x x Ɣ (x). We make x = 0 i the formula just obtaied. COROLLARY.4. Ɣ ( i + j ) = K where the accet deotes that we remove the case i = 0, j = 0. Proof. Note that Ɣ(x) lim x 0 Ɣ(x) =. From the first expressio o Ɣ (x + ), Ɣ(x) Ɣ (x) = Ɣ (x + ) = A(x) where A(x) = (π) x e x(x+) γ x k= ((+ x ) k )k x x+ e k.

11 E.W.Bares approach of the multiple gamma fuctios 37 The we have Ɣ(x) = A(x)Ɣ (x), lim A(x) = = lim A(x). x 0 x 0 Ɣ (x) lim x 0 Ɣ (x) = lim Ɣ(x) A(x) x 0 A(x) Ɣ(x) = lim Ɣ(x) x 0 Ɣ(x). lim x 0 lim x 0 xe γx k= xe γx k= Ɣ (x + i + j Ɣ (x + i + j ( ) + x e x k k ( + x k ) = Ɣ (x) ) e x k =. Ɣ (x + i + j ). ) = K lim Ɣ (x) x 0 Ɣ (x) = K. 3. Some related asymptotic expasios PROPOSITION 3.. (a) s ζ(s,a) s=0 = log Ɣ(a) /log(π) (b) For real a > 0, where 0 <θ(a)<. s ζ(s,a) s=0 = (a /)log a a + θ(a) a, Proof. (a) We kow that Ɣ(a) = e ζ (0,a) /R, from Propositio., where R = e ζ (0). Thus e ζ (0,a) = e ζ (0) Ɣ(a) = (π) / Ɣ(a) [4],[5]. ζ (0, a) = s ζ(s,a) s=0 = log Ɣ(a) /log(π). (b) It is kow [] that for real x > 0, Ɣ(x) = π x x / e x e θ(x) x with 0 <θ(x)<.

12 38 Juesag Choi ad J. R. Quie The log Ɣ(a) = log(π)+(a θ(a) )log a a + a. Therefore, as i (a), for real a > 0, where 0 <θ(a)<. ζ (0, a) = (a /) log a a + θ(a) a, PROPOSITION 3.. (a) s ζ (s, x) s=0 = log Ɣ (x) + /(x ) log(π)+ log A where A is the Kikeli s costat. (b) For real x > 0, s ζ (s, x) s=0 = 3x 4 x ( x x + 5 ) log x + O( ), x. x Proof. (a) We kow that Ɣ (x) = (R R x )eζ (0,x), where R = e ζ (0), R = e ζ (0) e ζ (0,), Corollary.5. The we have ζ (0, x) = s ζ (s, x) s=0 = log Ɣ (x) + (log R x log R ) = log Ɣ (x) + ( x)ζ (0) + ζ (0, ) = log Ɣ (x) + /(x ) log(π)+ζ ( ), sice ζ (0, ) = ζ ( ), ζ (0) = /log(π). ζ (0,x) = log Ɣ (x) + /(x ) log(π)+/ log A sice A = e / ζ ( ) ad so ζ ( ) = / log A.

13 E.W.Bares approach of the multiple gamma fuctios 39 (b) From Stirlig s formula, for real x > 0, log Ɣ (x + ) = x log π log A + 3x 4 + ( x ) log x + O( ), x. x We kow that Ɣ (x + ) = Ɣ (x)ɣ(x). Thus log Ɣ (x) = log Ɣ (x + ) log Ɣ(x). I the course of proof of (b), Propositio 3.. log Ɣ(x) = /log(π)+(x /)log x x + O( ), x, x>0. x Therefore we have From (a), log Ɣ (x) = ( x /) log(π)+(x x + 5 ) log x 3 4 x + x log A + + O( ), x. x / sζ (s, x) s=0 = 3 4 x x ( x x + 5 ) log x + O( ), x. x Refereces. Lars V. Ahlfors, Complex aalysis (third editio), McGraw-Hill Book Compoy, E. W. Bares, The theory of G-fuctio, Quarterly Joural of Mathematics 3 (899), , O the theory of the multiple gamma fuctio, Philosophical Trasactios of the Royal Society (A), XIX (904), Juesag Choi, Determiats of laplacias ad multiple gamma fuctio, Ph.D. Dissertatio, Florida State Uiversity W. Magus, F. Oberhettiger ad P. P. Soi, Formulas ad theorems for the special fuctios of mathematical physics (third editio), Spriger-Varlag, J. R. Quie, S. H. Heydari ad R. Y. Sog, Zeta regularized products, Trasactios of Amer. Math. Soc., to appear.

14 40 Juesag Choi ad J. R. Quie 7. Ila Vardi, Determiats of Laplacias ad multiple gamma fuctios, SIAM Joural o Mathematical Aalysis, 9(988), A. Voros, Spectral fuctios, special fuctios ad the selberg zeta fuctio, Commu. Math. Phys. 0(987), E. T. Whittaker ad G. N. Watso, A course of moder aalysis (fourth editio), Cambridge Uiversity Press, 963. Departmet of Mathematics Dogguk Uiversity Kyoug Ju , Korea Departmet of Mathematics Florida State Uiversity Tallahassee, Florida 3306 U.S.A.