JACKSON S INTEGRAL OF MULTIPLE HURWITZ-LERCH ZETA FUNCTIONS AND MULTIPLE GAMMA FUNCTIONS

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1 JACKSON S INTEGRAL OF MULTIPLE HURWITZ-LERCH ZETA FUNCTIONS AND MULTIPLE GAMMA FUNCTIONS SU HU, DAEYEOUL KIM, AND MIN-SOO KIM axv: v3 [math.nt 9 Aug 28 Abstact. Usng the Jackson ntegal, we obtan the q-ntegal analogue of Raabe s 843 fomula fo Banes multple Huwtz-Lech zeta functons and Banes and Vad s multple gamma functons. Ou esults genealze q-ntegal analogue of the Raabe type fomulas fo the Huwtz zeta functons and log gamma functons n[n. Kuokawa, K. Mmach, and M. Wakayama, Jackson s ntegal of the Huwtz zeta functon, Rend. Cc. Mat. Palemo , no., Dung the poof we also obtan a new fomula on the elatonshp between the hghe and lowe odes Huwtz zeta functons.. Intoducton Raabe s 843 fomula fo Gamma functon s as follows: Γat. log dt = aloga a, a 2π see [, p. 89; see also [3, p. 29, Eq. 4 and Eq. 4. The Huwtz zeta functon ζs,a s defned by.2 ζs, a = na s n= fo Res > and a C\Z. Hee Z = {,, 2,...} and C s the set of complex numbes. It also satsfes the Raabe type fomula:.3 ζs,atdt = a s s. The gamma functon appeas n the devatve of the Huwtz zeta functon as ndcated by the followng Lech s fomula [ see also [3, p. 59, Eq. 7: s ζs,a s= = log Γa. 2π The Jackson ntegal s, when q >, defned as.4 fad q a = q fq n q n n= 2 Mathematcs Subject Classfcaton. M35, 33B5. Key wods and phases. Jackson ntegal, Raabe type fomula, Multple Huwtz-Lech zeta functon, Banes gamma functon.

2 2 SU HU, DAEYEOUL KIM, AND MIN-SOO KIM see [8. Usng.4, Kuokawa, Mmach and Wakayama poved q-ntegal analogue of the Raabe type fomulas fo the Huwtz zeta functons ζs, a and log Gamma functons log Γa, espectvely. See [9, Theoems 2 and 3. Themultpleo,smply, k-tuplehuwtz-lechzetafunctonφ k z,s,a s defned by.5 Φ k z,s,a = m,...,m k = z m m k m m k a s fo a C\Z,s C when z < and Res > k when z = see [6, 6. It s analytcally contnued to s C as a meomophc functon by the usual method see [4 and holomophc at s C\{,...,k} when z =. In ts specal case when z =, the defnton.5 yelds the famla multple o, smply, k-tuple Huwtz zeta functon ζ k s,a = Φ k,s,a:.6 ζ k s,a = m,...,m k = m m k a s, whee Res > k and a C\Z. Lettng k = n.5, we obtan the Huwtz-Lech zeta functon Φ z,s,a = Φz,s,a. Futhemoe, when k = and z =,.5 also yelds the Huwtz zeta functon Φ,s,a = ζs,a. In addton, lettng k = and a = n.6, the Remann zeta functon ζ s, = ζs s obtaned fo detals, see [2, 7, 3, 5. We emak hee that the geneal fom of the multple Huwtz zeta functon ζ k s,a was studed systematcally by Banes [3 dung hs eseach on the theoy of the multple Gamma functon Γ k a. In 977, Shntan [2 evaluated the specal values of cetan L-functons attached to eal quadatc numbe felds n tems of Banes s multple Huwtz zeta functons, and the wok of Banes has also been appled n the study of the detemnants of the Laplacans aound the mddle of 98s see [4. In ths pape, by usng the Jackson ntegal, we extend Kuokawa, Mmach and Wakayama s q-ntegal analogue of the Raabe type fomula fo the Huwtz zeta functons and log gamma functons to the multple case. See Theoems 3. and 4. below. Ou pape s oganzed as follows. In Sec. 2, we shall ecall Banes and Vad s defntons of multple gamma functons. In Sec. 3 and Sec. 4, we shall state and pove Theoems 3. and Theoem 4., espectvely.

3 2. Multple gamma functons ThemultplegammafunctonΓ k abythefollowngecuence-functonal equaton fo efeences and a shot hstocal suvey see [3, 3, 4: 2. Γ k a = Γ ka, a C, k N, Γ k a Γ a = Γa, Γ k =. The multple gamma functons wee fst defned n a slght dffeent fom by Banes [3. The defnton 2. s due to Vad [4 and seems moe natual than that of Banes. Defne 2.2 G k a = expζ k,a, whee ζ k s,a = s ζ ks,a. Poposton 2. Vad [4, Poposton 2.3. The multple gamma functon Γ k a may be expessed by [ k Γ k a = R a G k a, whee = k m R m = exp ζ, = wth R =. In patcula, the specal cases of Poposton 2. when k = and k = 2 gve othe foms of the smple and double gamma functons Γ a = Γa and Γ 2 a, espectvely: Γa = R G a = exp ζ ζ,a = 2πexpζ,a, whee ζs = ζs, s the Remann zeta functon and Γ 2 a = R 2 Ra G 2a = exp ζ ζ 2,aζ expζ 2,a = exp 2 ζ exp aζ exp = A2π 2 a 2 exp 2 ζ 2,a, 2 ζ 2,a whee we have used ζ 2 s,a = aζs,aζs,a,ζ = log 2π and the known dentty loga = 2 ζ, n whch As the Knkeln s constant see [4, p. 496,p. 499 and Poposton

4 4 SU HU, DAEYEOUL KIM, AND MIN-SOO KIM The Genealzed Steltjes constants ae denoted as γ n a and ase when consdeng the Lauent expanson of the Huwtz zeta functon ζs, a at smple pole s =. Fo < a and fo all complex s, we have 2.3 ζs,a = s n= n γ n as n. n! See[4,Theoem.TheEuleconstantγ sapatculacaseofthesteltjes constants, that s, γ = γ, and the Eule constant s expessed as n γ = lm n logn = 3. Jackson s ntegal of multple Huwtz-Lech zeta functons 3.. Results. The multple Lpschtz-Lech zeta functons L k z,s wll be nvolved n ou fomula, whch s defned as z m m k 3. L k z,s = m m k s, m,...,m k = whee s C when z < and Res > k when z =. When k = n 3., one knows that z m 3.2 L z,s = L s z = m s, whch s the polylogathm. Hee s s usually called the ode. Fo each fxed s C, the sees 3.2 defnes an analytc functon of z fo z < ; n patcula, L z = z/ z and L z = log z. Lettng z = n 3.2, the Remann zeta functon L s = ζs s obtaned. It s easy to see that L k,s s expessed as the multple Huwtz zeta functon m= 3.3 L k,s = ζ k s,k. Denote by [n q = qn q. Ou man esult n ths secton s as follows: Theoem 3.. Let k be an postve ntege. Then k k s Lk z,sl 3.4 Φ k z,s,ad q a = [ s q l [l q s establshed n the egon z and Res <. And s 3.5 Φ k z,s,k ad q a = z k l Lk z,sl, l [l q = l= l=

5 3.6 s Φ k z,s,k ad q a = z k Lk z,sl l [l q ae establshed n the egon s C when z < and s C\{,...,k} when z =. Lettng k = and z = n Theoem 3., we mmedately obtan the followng esult. Coollay 3.2 Kuokawa, Mmach and Wakayama [9, Theoem 2 and 3.. We have s ζsl 3.7 ζs,ad q a = [ s q l [l q fo Res <, and fo all s C\{}. ζs, ad q a = l= l= s ζsl l, l [l q l= ζs,ad q a = s ζsl l [l q Dung the poof Theoem 3., we also obtan the followng esult on the elatonshp between the hghe and lowe odes Huwtz zeta functons. Lemma 3.3. Let Φ k z,s,a be the multple Huwtz-Lech zeta functon. Then 3. Φ k z,s,a = a k k z k Φ s k z,s,ak = s establshed n the egon s C when z < and s C\{,...,k} when z = Poof of Lemma 3.3. Fst of all, lettng k = 2 n.5, we have z m m 2 Φ 2 z,s,a = m m,m 2 = m 2 a s 3. z m 2m 3 = m 2 m 3 a z m 3 s m 3 a s. Thus 3.2 Φ 2 z,s,a = m 2 =m 3 = m 2,m 3 = m 3 = l= m 3 = z m 2m 3 m 2 m 3 a s z m 3 m 3 a s a s. m 2 = z m 2 m 2 a s 5

6 6 SU HU, DAEYEOUL KIM, AND MIN-SOO KIM Lettng k = 3 n.5, we have the followng decomposton: 3.3 Φ 3 z,s,a = = m,m 2,m 3 = m = m 2,m 3 = z m m 2 m 3 m m 2 m 3 a s z m m 2 m 3 m m 2 m 3 a s Φ 2z,s,a. We also have 3.4 z m m 2 m 3 m m =m 2,m 3 = m 2 m 3 a = s m,m 2 =m 3 = 3.5 m = m 3 = z m m 3 m m 3 a s, z m m 2 m 3 m m,m 2 = m 3 = m 2 m 3 a = s m,m 2,m 3 = and 3.6 m =m 3 = m,m 2 = z m m 2 m m 2 a s z m m 3 m m 3 a = s m,m 3 = z m m 2 m 3 m m 2 m 3 a s z m m 2 m 3 m m 2 m 3 a s z m m 3 m m 3 a s In vew of 3.2, 3.3, 3.4, 3.5 and 3.6, we get m = z m m a s. 3.7 Φ 3 z,s,a = 3 a s m 2 = m 2,m 3 = m,m 2,m 3 = z m 2 m 2 a s z m 2m 3 m 2 m 3 a s z m m 2 m 3 m m 2 m 3 a s. In what follows, we use the above easonng the symmety of the multple sum m,...,m k = n the m j s vaables to pove an analogue of 3.7 fo geneal k. Denote by S = {m,...,m k }.

7 Then 3.8 Φ k z,s,a = Snce m,...,m k = m,...,m k = = a k s fom 3.8, we have z m m k m m k a s = {m,...,m k } S m,...,m k = z m m k m m k a = s 3.9 Φ k z,s,a = a k s Fom 3.9, we get 3.2 Φ k z,s,a = a k s = m,...,m k = = a k s = = k z k k m,...,m k = k m,...,m k = z m m k 7. m m k a s z m m k m m k a s, z m m k m m k a s. z m m k m m k ak s z k Φ k z,s,ak fo s C when z < and Res > k when z =, and analytcally contnued to s C\{,...,k} f z =. Ths completes the poof Poof of Theoem 3.. To pove Theoem 3., we need the followng lemma. Lemma 3.4. Let k be an postve ntege and Φ k z,s,a be the multple Huwtz-Lech zeta functon. Then s Φ k z,s,ak d q a = z k Lk z,sl l [l q s establshed n the egon s C when z < and s C\{,...,k} when z =. l=

8 8 SU HU, DAEYEOUL KIM, AND MIN-SOO KIM Poof. Usng.4 and 3., we obtan the followng equalty Φ k z,s,ak d q a = q = q z k = q z k = z k Φ k z,s,q n k q n n= n= n= q n q n m,...,m k = m,...,m k = q n m m k n= m,...,m k = z m m k m m k q n s z m m k m m k s s z m m k m m k s l s q nl q l m m k s Lk z,sl l= = z k l= l [l q fo s C when z < and Res > k when z =, and analytcally contnued to s C\{,...,k} f z =. Ths completes the poof. Poof of Theoem 3.. Nowwepove 3.4.Suppose z andres <. Fom Lemma 3.3, we have 3.2 Φ k z,s,ad q a = [ a k s = k z k Φ k z,s,ak d q a. Thus fomlemma 3.4andthefollowng fomulawhch svaldfores < e.g., see [9, p. 5: a sd qa = q q n s q n = n= [ s q, we have shown 3.4 n the egon Res <.

9 Now we pove 3.5. Suppose s C when z < and Res > k when z =. As n the pove of Lemma 3.4, we have Φ k z,s,k ad q a = q = q = q z k Φ k z,s,k q n q n n= n= n= = q z k n= q n q n q n m,...,m k = m,...,m k = m,...,m k = q n s z m m k m m k k q n s z m m k m m k q n s z m m k m m k s m m k = z m m k z k m n= m,...,m k = m k s l s l q nl q l m m k l= = s l Lk z,sl, z k l [l q l= so t s vald fo s C when z <, and fom the analytc contnuaton t s also vald fo all s C\{,...,k} when z =. Ths completes the poof of 3.5. The same pocedue poves Jackson s ntegal of multple gamma functons 4.. Results. Let [ be the absolute Stlng numbes of the fst knd see, e.g., [3, Secton.6, defned ecusvely by 4. [ [ = [, [ = { f =, f, see [, 2. Let H l := l j= l N be the hamonc numbes see, e.g., [3, p. j 77. By vtue of Theoem 3. wth z = and Poposton 2. above, we obtan the followng esult.

10 SU HU, DAEYEOUL KIM, AND MIN-SOO KIM Theoem 4.. Let k be an postve ntege. Then we have k logr k [ logγ k ad q a =! [ = = q qlogq k q k ζ k,k whee k = k = k = k l l[l q l= l=k [ k j= j l P j,k k ζl j,k k! P l,k k H l γ k k! k l ζ k l,k, l[l q P j,k a = k [ k a j j =j and [ k s the absolute Stlng numbes of the fst knd. Hee, the sum H s undestood to be. The case k = of Theoem 4. educes to Coollay 4.2 Kuokawa, Mmach and Wakayama [9, Theoem 3. logγad q a = qlogq q γ l ζl. [2 q l[l q 4.2. Poof of Theoem 4.. The poof of Theoem 4. s based on the followng two lemmas. Lemma 4.3. We have logg k ad q a = k = logr k! logγ k ad q a = l=2 [ [ q. Poof. Fom 2.2 and Poposton 2., the multple gamma functon Γ k a may be expessed by k a 4.2 logγ k a = logg k a logr k, whch s equvalent to 4.3 logg k ad q a = = logγ k ad q a k = logr k! a d q a,

11 whee a denotes the fallng factoal powe. Note that 4.4 a d q a = [ q fo see [9, p. 47 and 4.5 a = aa a = [ = a, whee [ k s the absolute Stlng numbes of the fst knd. Usng 4.3, 4.4 and 4.5, the desed fomula follows. Lemma 4.4 Cho[5,Eq.2.5. The multple zeta functon ζ k s,a defned by.6 may be expessed by means of the Huwtz functon k ζ k s,a = P j,k aζs j,a. k! j= Now we ae at the poston to pove ou man esult n ths secton. Poof of Theoem 4.. Fom Theoem 3. wth z =, we obtan 4.6 ζ k s,ad q a = k k [ s q = l= s l ζk sl,k [l q, whee we have used L k,s = ζ k s,k. The followng denttes wll be used to pove the theoem: lm ζk s,a s = logg s k a. 2 lm s s [ s q = qlogq. q 3 lm s s s l = s s s l lms = l. s l! l 4 Fom 2.3, we have lm s sζs,a = lm s s 5 Fo k, we have lm s s = γ a. s n γ n as n n! n= s ζ s, = ζ,.

12 2 SU HU, DAEYEOUL KIM, AND MIN-SOO KIM 6 Let k. By Lemma 4.4 and 4 above, we have lm s s s ζ s, =! j= =! P j, lm s s sζs j, P, γ P j, ζ j,. j= 7 Let k. By Lemma 4.4 and the above 4, we have lm s s s ζ s, = =! j=! P j, lm s s j= j s ζs j, P j, ζ j,! P, H γ, lm s s whee we have used s = =! ζs, [ lm s s H γ. s s n γ n s n n! n= Dffeentatng both sdes of 4.6 and then settng s =, we fnd that 4.7 lm s s ζ ks,ad q a = lm s s [ s q k k [ s lm [l q s s l = l= ζ k sl,k,

13 then afte usng -7 and eplacng by k, we obtan logg k ad q a = qlogq k q k ζ k,k k k k = l= k = l l[l q = [ k j= j l P j,k k ζl j,k k! P l,k k H l γ k k! k l ζ k l,k, l[l q l=k whee the sum H s undestood to be. Theefoe, by Lemma 4.3, we obtan the desed esult. Acknowledgment The authos ae gateful to the anonymous efeee fo hs/he helpful comments. Refeences [ V.S. Adamchk, The multple gamma functon and ts applcaton to computaton of sees, Ramanujan J. 9 25, no. 3, [2 T.M. Apostol, On the Lech zeta functon, Pac. J. Math. 95, [3 E.W. Banes, On the theoy of the multple gamma functons, Tans. Cambdge Phlos. Soc. 9 94, [4 B.C. Bendt, On the Huwtz zeta-functon, Rocky Mountan J. Math , no., [5 J. Cho, Explct fomulas fo Benoull polynomals of ode n, Indan J. Pue Appl. Math , no. 7, [6 J. Cho, D.S. Jang and H.M. Svastava, A genealzaton of the Huwtz-Lech zeta functon, Integal Tansfoms Spec. Funct. 9 28, no. -2, [7 K. Dlche, Sums of poducts of Benoull numbes, J. Numbe Theoy 6 996, [8 F.H. Jackson, On q-defnte ntegals, Quat. J. Pue Appl. Math. 4 9, no. 5, [9 N. Kuokawa, K. Mmach and M. Wakayama, Jackson s ntegal of the Huwtz zeta functon, Rend. Cc. Mat. Palemo , no., [ M. Lech, Dalsš stude v obou Malmsténovských řad, Rozpavy České Akad., 3 894, no. 28, 6. [ N. Nelsen, Handbuch de Theoe de Gammafunkton, Chelsea, New Yok, 965, epnt of 96 edton. [2 T. Shntan, On a Konecke lmt fomula fo eal quadatc felds, J. Fac. Sc. Unv. Tokyo Sect. IA Math , no., [3 H.M. Svastava and J. Cho, Zeta and q-zeta Functons and Assocated Sees and Integals, Elseve Scence Publshes, Amstedam, London and New Yok, 22. [4 I. Vad, Detemnants of Laplacans and multple Gamma functons, SIAM J. Math. Anal , [5 K.S. Wllams and N.Y. Zhang, Specal values of the Lech zeta functon and the evaluaton of cetan ntegals, Poc. Ame. Math. Soc ,

14 4 SU HU, DAEYEOUL KIM, AND MIN-SOO KIM Depatment of Mathematcs, South Chna Unvesty of Technology, Guangzhou, Guangdong 564, Chna E-mal addess: Depatment of Mathematcs and Insttute of Pue and Appled Mathematcs, Chon-buk Natonal Unvesty, 567 Baekje-daeo, deokjn-gu, Jeonjus, Jeollabuk-do 54896, Republc of Koea E-mal addess: Dvson of Mathematcs, Scence, and Computes, Kyungnam Unvesty, 7Woyeong-dong kyungnamdaehak-o, Masanhappo-gu, Changwons, Gyeongsangnam-do 5767, Republc of Koea E-mal addess: