of the Barnes function.

Transcription

1 O the Bares fuctio Victor Adamchik Caregie Mello Uiversity Abstract. The multiple Bares fuctio, defied as a geeraliatio of the Euler gamma fuctio, is used i may applicatios of pure ad applied mathematics ad theoretical physics. This paper presets ew itegral represetatios as well as special values of the Bares fuctio. Moreover, the Bares fuctio is expressed i a closed form by meas of the Hurwit eta fuctio. These results ca be used for umeric ad symbolic computatios of the Bares fuctio.

2 Preamble I 899, Bares [,,3 itroduced ad studied the geeraliatio of the Euler gamma fuctio defied by the followig fuctioal equatio: G+ / *+/ G+/, ± C G+/ where * is the gamma fuctio. For the iteger positive values of, the Bares G fuctio is simply a product of factorials: G+ / Ç k k Ç *+k/ k From the above fuctioal equatio ad the Weierstrass caoical product for the gamma fuctio, *+/ J/È m/ ˆ exp+ exp+ m m Bares derived G+ / + S/s exp L M M + J/ c È ˆ k - L exp M kk k where J is the Euler-Mascheroi costat. The above product is a etire fuctio i a whole complex plae ad ca serve as a explicit defiitio of the Bares fuctio. Origially, the G fuctio was itroduced (i a differet form) by Kikeli [5 (see also Glaisher [6) i his research o the asymptotics behavior of this product at ˆ:... G+ / Kikeli also applied the theory of the G fuctio to the class of trigoometric itegrals à log trig+x/ dx () where trig(x) is ay trigoometric or hyperbolic fuctio. All such itegrals ca be expressed i fiite terms of the G fuctio.

3 3 Alexeiewsky [7 geeralied the Kikeli product () to which is related to the multiple Bares fuctio. p p... p exp+ + p, / + p/ / () These days the G fuctio remais the active topic of research (see [8--5.) The theory of the Bares fuctio has bee related to certai spectral fuctios i mathematical physics, to the study of determiats of Laplacias, ad to the Hecke L-fuctios. I [ ad [5 the G fuctio is expressed i terms of the Hurwit eta fuctio by log G+ / log *+/ +/ +, /, R+/! where +t, / = c d +t, / +/ ad is related to Glaisher's costat A: d t log A ccc +/ J log+ S/ c +/ S I [5 Adamchik obtaied the closed form represetatio for the class of itegrals ivolvig cyclotomic polyomials ad ested logarithms i terms of the Hurwit eta fuctio. I the view of (3) the results ca be traslated ito the G fuctio otatio, for example +x à 4 6 x / ccc log +x / 3 logx dx 4 logl M 3 G+ 4 / c G+ 4 / 3 log *- 4 log 3 *- Choi et al. [3, 4 ad Adamchik [5-7 cosidered a class of sums ivolvig the Riema eta fuctio which ca be evaluated by meas of the G fuctio. Here is a series represetatio for the G fuctio for ƒ, ˆ log G+ / log+ S/ J + / Æ k +/ k +k/ k k 4 (3) ad for Glaisher's costat log A ˆ log Å k + k / +4 k 7/ +7 k 8/ L M c cc 36 +k / +k /

4 4 Bares [4 (followed by Vigeras [8 ad Vardi [) geeralied the G fuctio to the multiple gamma fuctio G by the recurrece formula G + / G +/ G +/ G +/ c G +/ G+/ *+/, ± C, ± Here G+/ is the Bares fuctio ad *+/ is the Euler gamma fuctio. Vigeras ad Vardi cosidered a slightly differet form of the multiple gamma fuctio defied as a reciprocal to the Bares fuctio *+/ c G. Vardi derived a implicit represetatio for * +/ i terms +/ of the multiple eta fuctio. I this paper we derive a closed form of G i fiite terms of the Hurwit fuctio ad special polyomials. Here are two particular cases rewritte i terms of the derivatives of the Hurwit fuctio whe 3 (4) ad 4 log G 3 + / log *+/ + / log G + / +/ +, / (5) 6 log G 4 + / 3 log *+/ 6 + / log G 3 + / +3 3 / log G + / +3/ +3, / (6) We also preset various itegral ad series represetatios as well as some special values of the Bares fuctio. These results ca be used i umeric ad symbolic computatios of the G fuctio. I believe the multiple gamma fuctio is of direct iterest for computer algebra researchers ad users, because of its sigificat applicatios i physics, umber theory, combiatorics ad applied mathematics. The Bares fuctio deserves to be implemeted i computer algebra systems.. G fuctio of the ratioal argumet There are a few kow special cases whe the G fuctio is expressible i a closed form. The first oe is due to Bares [: log log log G- S 4 4 the other two are due to Choi ad Srivastava [4: 3 log A 8

5 ccc 5 log G G S 3 4 log *- log G 3 5 G - log G- 4 4 S log A 3 ccc 3 log + S / c 8 where G is Catala's costat. I a view of closed form represetatio for the derivatives of the Hurwit fuctios, obtaied i [8, ad the formula (3), ad it is easy to derive the additioal special cases of log G+ p/ for p Propositio. 3, 6, 3, 5 6. log log 3 G- 3 7 S 8 r 3 3 log * log A +/ + 3 / r 3 S 9 (7) log G- log S 8 r 3 3 log * log A +/ + 3 / r 3 S 9 (8) log G- log 6 cc 44 S r log * log A +/ + 6 / 4 r 3 S 5 cc 7 (9) log G- 5 log 6 cc 44 S r 3 6 log * log A 5 +/ + 6 / 4 r 3 S 5 cc 7 () where +/ +/ log *+/ is the polygamma fuctio. Similar represetatios ca be derived for the multiple G fuctio.. Coectio to the polylogarithm From the Lerch fuctioal equatio for the Hurwit eta fuctio ad formula (3), we ca easily derive log L G+ / M ccc G+ / S i B S +/ log - si+s / i Li +e i S /, S where B +/ is the secod Beroulli polyomial, ad Li +/ is the polylogarithm.the idetity ca be rewritte i a alterative form by meas of the Clause fuctio Cl +/: () where log L G+ / S M ccc log - Cl + S /, G+ / si+s / S Cl +x/ Im+Li +e ix // ()

6 c c cc 6 Obviously eough, the G fuctio is related to the Dirichlet L-series. Here is oe of the formulas: log L M G+ 7 M 6 / G+ 5 6 / 6 log+ S/ 3 r 3 8 S c L 3 +/ (3) 3. Biet-like represetatio The Biet formula for the gamma fuctio is log *+/ - log log,r S à ˆ ex ccc - x e x x dx I this sectio we derive a similar represetatio for the Bares fuctio. Recall the wellkow itegral for +s, /: +s, / *+s/ à x ˆ s e x Åx, R+s/ e x!, R+/! cc The itegral ca be aalytically cotiued to the domai R+s/!. To do so, we perform the stadard procedure of removig the itegrad sigularity by subtractig the trucated Taylor series at x : +s, / s cc x s e x x - ccc s s s s cc *+s/ à ˆ e x x dx Differetiatig the above formula with respect to s ad computig the limit at s, we get +, / log ccc cc 4 ccc B +/ à ˆ ex t cc x - e x ccc t dx where B +/ is the secod Beroulli polyomial. From here it immediately follows (4) Propositio. The Bares G fuctio admits the Biet itegral represetatio: log log *+/ log A 4 ccc B +/ à ˆ ex t ccc x - e x t cc I a similar way we derive the represetatio for Glaisher's costat: log+ S/ log A c c cc à x log x S ˆ cc e x dx log G+ / cc dx, R+/! (5) (6)

7 c cc c c 7 4. The multiple G fuctio For the multiple G fuctio defied by the fuctioal equatio (4), Vardi [ obtaied the followig formula L log G +/ lim +s, / M (7) s Å +/ k L M R s k k where ad +s, / is the multiple eta fuctio: +s, / ˆ Å k ˆ...Å k R Å k lim s k L ccc s M k+s, / ˆ +k k... k / Å s k L +k s M / k The aim of this sectio is to fid a closed form represetatio for G +/ i terms of the Hurwit eta fuctio. For clarity of expositio, we first cosider polyomials P k, +/ Å i k which ca be rewritte i the alterative form P k, +/ Å i L M +/ L ik M i i k + / $ +i / $ i ( i k ( where % i ) are usiged Stirlig umbers of the first kid. The polyomials P k, +/ satisfy the fuctioal equatio P k, +/ P k, +/ P k, + / P k, +/, k (8) (9) () ()

8 cc 8 Here are the mai properties of these polyomials: P, +/ P k, +/ $ P, +/ + / P, - + 3/ ccc + / cc P, +/ L M ( k + / () Lemma. The multiple eta fuctio +s, / defied by (9) may be expressed by meas of the Hurwit fuctio / +s, cc + / Å L M j P j, +/ +s j, / Proof. Recall the defiitio of usiged Stirlig umbers of the first kid k cc + / Å k i $ i ( Expadig k i ++k / / i by the biomial theorem, implies that L k L i M cc Æ M +k / j +/ i j $ + / j Iterchagig the order of summatio ad makig use of (), we obtai L k M cc Å P j, +/+k / j + / i i Å j We complete the proof by substitutig this ito (9). Lemma. The multiple eta fuctio R defied by (8) may be expressed by meas of the derivatives of the eta fuctio j i i ( (3) R + / Å k +k/ $ k ( (4) Proof. First we make use of the Lemma ad the idetity () for P j, +/ L lim M s k +s, / cc c s cc + / Å j P j, +/ + j/ c + / Å j + j/ $ ( j

9 cc 9 Further, i a view of (8), we have R Å Å k cc +k / k j + j/ $ k j ( +/ Å j j/æ + k j k +k / $ j Takig ito accout that the ier sum o the right had side is (it follows from () with ) we complete the proof. Å k j + / cc +k / $ k j ( $ j ( Propositio 3. The multiple Bares fuctio G +/ may be expressed by meas of the derivatives of the eta fuctios log G +/ cc + / Å j where the polyomials P j, +/ are defied by (). P j, +/+ + j/ + j, // ( (5) 5. Itegral represetatios via polygammas I [5 Adamchik derived a closed form solutio to the itegral à x +x/ dx, R+/!, ± i fiite terms of the Hurwit eta fuctio: à x +x/ dx +/ +/ +/ Å k +/ k L M cc B H k - +k, / k B k +/ H k k c where B ad H are Beroulli ad Harmoic umbers respectively. If the itegral (6) leads to the followig represetatio for the Bares G fuctio: log G+ / c log, r (7) Sà x +x/ dx, R+/! This represetatio demostrates that the complexity of computig G+/ depeds at most o the complexity of computig the polygamma fuctio. The restrictio R+/! ca be easily (6)

10 removed by aalyticity of the polygamma. For example, by resolvig the sigularity of the itegrad at the pole x, we cotiue log G+ / to the wider area R+/!: log G+ / c log, r Slog+ / à x +x/ Aother way to cotiue (7) ito the left half-plae is to use the idetity +x/ which upo substitutig it ito (7) yields +x/ Scot+S x/ x x dx (8) log G+ / log G+ / log+ S/ à S x cot+s x/ dx (9) The idetity (9) holds everywhere i a complex plae of, except the real axes, where the itegrad has simple poles. Therefore, i a view of (9) we ca cotiue (7) to /. ote, if is egative we ca still use the represetatio (7), but with a cotour of itegratio deformed i such a way that it does ot cross poles. For, the formula (6) yields à ad for 3 : x +x/ dx log +/ G 3 + / +/ log G + / cc +6 3 / (3) à x 3 +x/ dx 6 log G 4 + / 6 log G 3 + / log G + / 3 +/ 3 +/ 3 +/ ccc / The geeral formula ca also be derived. Skippig the techical details, we have (3) Propositio 4. Let be a positive iteger ad ad R+/!, the à x +x/ dx Æ +/ k L M k Å k k k +/ k k! log G k + / B k +/ H k k - +k/ ccc k +/ B B +/ H ccc (3) where! are Stirlig umbers of the secod kid. k

11 We ca resolve the equatio (3) with respect to G k + /. I particular, combiig (7) with (3) leads us to the followig itegral represetatio for the triple Bares fuctio: log G 3 + / à x+ x/ +x/ dx + / +/ +/ ccc / ad combiig (7), (3) ad (3) yields: (33) 6 log G 4 + / à x+x /+x / +x/ dx + /+ / +/ 3 + / +/ 3 +/ ccc / Formulas (33) ad (34) are suitable for umeric computatio of G 3 +/ ad G 4 +/ for ay ± s. (34) 6. Implemetatio remarks We have preseted here may available results o the Bares fuctio as well as ew results o umeric ad symbolic computatios. I this sectio we discuss a umeric computatioal scheme for the Bares G fuctio. The itegral represetatio via polygamma fuctios cosidered i the previous sectio seems to be a efficiet umeric procedure for evaluatig G fuctio. Based o (7), we defie the double Bares fuctio G+/ G +/ as G+/ + S/ exp L M M + /+ / c ccc à x +x/ dx, arg+/ œs This represetatio is valid for ± s. If is a egative real, we have (35) G+/, ± + /+/ M c cc G+/ + S/ exp L x +x/ dx, arg +/ S ÃJ where the cotour of itegratio J is a lie betwee ad that does ot cross the egative real axis; for example, J could be the followig path:, i, i,. (36)

12 c Aother method of defiig the Bares G fuctio for egative reals is to use the formula (). This gives G+/ +/ GsW ƒsi+s G+ / L / M S exp- cc cc S Cl + S + GW// where GW is the floor fuctio, ad Cl +/ is the Clause fuctio. Here is a picture of the G fuctio over the iterval (-3,3): (37) I a similar maer, takig ito cosideratio the formulas (33) ad (34), we ca defie G 3 ad G 4 respectively. There are may applicatios, particularly i umber theory, where the logarithm of the Bares fuctio ofte appears. However, because of a brach cut of logarithm, the fuctio log G+/ (where G is defied by (35)) icludes spurious discotiuities for complex argumet. The followig the picture of Im+log G+ i y//, where y ±, demostrates this:

13 c cc 3 Therefore, we defie a additoal fuctio LogG+/ (the logarithm of the Bares fuctio) which is a aalytic fuctio throughout the complex plae + / + / LogG+/ log+ S/ Ã x +x/ Åx If is a egative real, we uderstad the path of itegratio as i (36). Here is the picture of Im+LogG+ iy//, where y ±, : (38) Refereces. E. W. Bares, The theory of the G-fictio, Quart. J. Math., 3(899), E. W. Bares, Geesis of the double gamma fuctio, Proc. Lodo Math. Soc., 3(9), E. W. Bares, The theory of the double gamma fuctio, Philos. Tras. Roy. Soc. Lodo, Ser. A, 96(9), E. W. Bares, O he theory of the multiple gamma fuctio, Tras. Cambridge Philos. Soc., 9(94), Kikeli, Ueber eie mit der Gammafuctio verwadte Trascedete ud dere Awedug auf die Itegralrechug, J.Reie Agew.Math., 57(86), J. W. L. Glaisher, O a umerical cotiued product, Messeger of Math., 6(877), 7-76.

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