Abstact Vaious applicatios of the (epoetial coplete Bell polyoials Doal F. Coo 6 Jauay I a athe staightfowa ae, we evelop the well-ow foula fo the Stilig ubes of the fist i i tes of the (epoetial coplete Bell polyoials whee the aguets iclue the geealise haoic ubes. We also show how the (epoetial coplete Bell polyoials featue i a ube of othe aeas of atheatical iteest. EXPLICIT FORMULA FOR TE STIRLING NUMBERS OF TE FIRST KIND The Stilig ubes s (, of the fist i [3, p.56] ae efie by the followig geeatig fuctio (the bacet sybol is also eploye (. ( ( + s(, which we ay also epess as the ifiite seies (. ( ( s(, + whee we efie s (, fo +. Lettig i (. esults i (.3 + + ( +...( + ( s(, ( s(, We also cosie the asceig factoial sybol (, also ow as the Pochhae sybol, efie by [3, p.6] as (.4 ( + ( + ( + if > (
The gaa fuctio satisfies the well ow ecuece elatio Γ Γ ( + a it is easily pove by iuctio that this ay be etee to (.4. ( + ( + ( + Γ Γ ( + ece we have (.5 Γ ( + Γ ( + Γ Γ ( + Fo the easos iscusse below, we shall fi it oe coveiet to eal with the fuctio ( ivie by. We cosie the fist eivative Γ ( + Γ ( + ψ ψ (.6 [ ( + (+ ] whee ψ is the igaa fuctio efie by (.6. ψ log Γ By efiitio we have ψ ( + ψ ( + log Γ ( + log Γ (+ Γ ( + log Γ ( + log log Refeig to the efiitio (.4 of the Pochhae sybol, we have a iffeetiatio esults i log log + log( + + + log( +
+ + + + + ( whee (.7 ( is the geealise haoic ube fuctio efie by ( + a we ote that ( Theefoe we have the well ow foula [3, p.4] ψ ( + ψ ( + ( a we eote g. + + This gives us the elatioship (which i tu leas us to cosie the (epoetial coplete Bell polyoials + [ ψ( ψ(+ ] (.8 We see that g g( (! g ( + + g ( ( (! + As ote by Kölbig [4] a Coffey [4] we have 3
e e Y f f f f f ( ( (.9 (,,..., whee the (epoetial coplete Bell polyoials ay be efie by! (. Y(,...,... π!!...!!!! Y whee the su is tae ove all patitios π of, i.e. ove all sets of iteges that + + 33+ + a fo j such The efiitio (. ieiately iplies the followig elatio (.. Y ( a, a,..., a a Y (,..., a with a we have (.. Y (,,...,( ( Y (,..., The coplete Bell polyoials have itege coefficiets a the fist si ae set out below (Cotet [6, p.37] (. Y Y (, + Y (,, + 3 + 3 3 3 3 Y (,,, + 6 + 4 + 3 + 4 4 3 4 3 4 Y (,,,, + + + 5 + 5 + + 5 3 5 3 4 5 3 4 3 5 Y (,,,,, + 6 + 5 + + 5 + 5 + 6 6 3 3 6 3 4 5 6 5 4 4 3 + + 45 + 5 + 3 4 3 6 The coplete Bell polyoials ae also give by the epoetial geeatig fuctio (Cotet [6, p.34] 4
j t t (. ep j Y(,..., j j!! Usig (.9 we see that t j t ep j Y(,..., j j! t a hece we ote that (. is siply the coespoig Maclaui seies. We ote that j j j t t t t Y ( a,..., a ep a ep a ep j j j! j j! j j! j j! a thus we have a a t t Y(,..., Y( a,..., a!! Let us ow cosie a fuctio f ( t which has a Taylo seies epasio aou : we have j j f ( + t ( j t f ( j t e ep f e ep f j j! j j! We see that f (, (,..., ( e + Y f f f t! t f f ( + t f ( + t e e e t t a we theefoe obtai a eivatio of (.9 above e f f ( ( e Y f f f (,,..., Suppose that h h g a let f log h. We see that 5
h f g( h a the usig (.9 above we have log h ( ( (.3 h e hy ( g, g,..., g I paticula we have ( (, (,..., ( h h Y g g g I ou case, fo (.8 we have h g + ( ( a it is easily see that ( ( (.5 (! Y(,!,...,( (! Diffeetiatio of (.3 also esults i (.6 ( ( ( s(, + a i paticula we have (.7 + + s! (, + Theefoe, equatig (.5 a (.7, we obtai the ow elatioship fo Stilig ubes of the fist i fo (!! + + ( ( (.8 s (, + ( Y(,!,...,( (! 6
The above elatioship was peviously eive by Kölbig [4] but this appeas to be a oe iect poof of this ipotat foula. The fist few Stilig ubes s (, of the fist i ae easily eteie fo (.8; these ae also epote i [] a i the boo by Sivastava a Choi [3, p.57] (.9 s (, δ, s + (,! Usig (.8 gives us s (, ( (! ( ( {( } (! + ( s (,3 3 { } ( ( ( (3 (! s (,4 ( 3 + 6 (! s + Y (! ( + + ( ( (,,!,...,! a sice [5, p.45] Y (,..., Y (,..., (. + + + we obtai the ecuece foula give by She [] ( + s(, + s(, + + ( Let us ow see why we iitially iecte ou attetio to ( ivie by. Diffeetiatio of (.5 esults i (. Γ ( + ( s (, + Γ + a i paticula we have 7
(. + ( s! (, Alteatively we have the fist eivative Γ + Γ Γ + Γ Γ + Γ Γ Γ ( + [ ψ ( + ψ ( ] Γ Theefoe we have (.3 ( ( a it is easily see fo (.7 that ( li ece we obtai ( (! a usig (.9 we see agai that s + (,! It shoul be ote that ay also be eteie iectly; howeve, the atheatics is soewhat oe cubesoe. Fo eaple, let us cosie Diffeetiatig (.3 gives us We have by efiitio ( ( ( (. 8
( ( + + + + + + + a theefoe ( ( + + + + ( + + + + ( ( ( + + + This gives us ( ( ( + ( + + + + ( + Siilaly we have ( ( + + ( + + + ( + ( ( + a we easily see that ( ( ( ( ( ( + ( + + ( + ( + ( + This gives us a we the obtai ( li ( ( ( ( ( ( li ( ( ( li 9
(! ( ece we have usig (. s (, ( (! ( Fo (.3 we fi that ( ( Y,!,...,( (! a usig (. we see that ( ( +! (, li,!,..., s Y (! CAUCY S GENERATING FUNCTION FOR TE STIRLING NUMBERS OF TE FIRST KIND Usig the bioial theoe we have fo z < (. ( z! z a, usig (., iffeetiatio with espect to esults i (. ( log( z ( z! z Moe geeally we have log ( z z ( ( z! a lettig we see that ( log ( z z! The usig (.
+ ( s! (, we the easily obtai the well-ow Maclaui epasio ue to Cauchy [3, p.56] (.3 z log ( z! ( s(,! Sice s (, fo + this ay be epesse as z log ( z! ( s(,! A iffeet poof of (.3 was give by Póyla a Szegö i [8, p.7]. Lettig i (. we obtai the well-ow geeatig fuctio fo the haoic ubes (.3. log( z z ( z Diffeetiatig (. gives us ( ( log ( z (.4 a with we have ( z! log ( z (.5 ( ( ( z z Such seies ae cosiee i oe etail i, fo eaple, [7] a the efeeces cotaie theei. Itegatig (.5 esults i ( + log 3 ( ( z z 3 + z + ( ( z + + + ( + + +
z + + ( + + ( + ( + ( + + ( ( ( + z We theefoe have + ( z log z z Li3 ( z 3 (.6 3 ( ( whee (.7 Li ( z 3 is the polylogaith efie by z Lis ( z s Fo (.3 we have with 3 3 log ( z 6 ( s(,3 z! ( ( ( 3 z ( ( ( 3 + z i ageeet with (.6 above. COPPO S FORMULA We have the well-ow patial factio ecopositio [3, p.88] (3.! ( f ( +...( + + + { },,,...,
a we ote fo (.4. that! Γ ( + Γ ( +...( + Γ ( + + o equivaletly! Γ ( + ( +...( + ( + Diffeetiatio esults i Γ ( + ( f + ( + a efeig to (.3 we theefoe have the th eivative Γ ( + f Y,!,...,( (! We also have fo (3. ( + + + ( ( + ( (! ( + + f a we theefoe obtai ( Γ ( + ( ( +! ( ( (3. Y (!,!,..., (! + + + + + It ay be ote that Coppo [] has epesse this i a slightly iffeet fo ( Γ ( + ( +! ( ( (3.3 Y (!,!,...,(! + + + + + a efeece to (.. shows that these ae equivalet stateets. It ay be ote that efeece to the ight-ha sie of (3.3 shows that ( ( + + is positive fo > (which othewise oes ot appea to be ieiately obvious. Paticula cases of Coppo s foula ae set out below. 3
(3.4 ( Γ ( + Γ Γ ( + Γ Y ( + Γ ( + + Γ ( + + (3.5 ( ( Γ ( + Γ ( Γ ( + Γ ( Y + + ( + Γ ( + + Γ ( + + ( Γ ( + Γ Y 3 ( + Γ ( + + ( ( ( +, + Γ ( + Γ (3.6 ( ( ( + + + Γ ( + + ( Γ ( + Γ Y 4 3 ( + Γ ( + + 6 ( ( ( (3 +, +, + Γ ( + Γ 3 (3.7 ( ( ( ( (3 + + 3 + + + + Γ ( + + 6 We ow wish to cosie epesetatios fo (. We see that ( ( Γ ( + Γ + + Γ ( + + a we have the liit as Γ ( + Γ Γ ( + + Γ ( + + Γ ( + Γ ( + Γ ( + + Γ ( + + ( Γ ( + Γ Γ ( + + li Γ ( + + Applyig L ôpital s ule we obtai 4
Γ ( + Γ ( + Γ ( + + li Γ ( + + +Γ ( + + Γ ( + Γ ( Γ ( + Γ ( + ψ ( ψ ( + ( ece we obtai Eule s well-ow ietity (3.8 ( ( Siilaly we also have ( Γ ( + Γ ( + Γ ( + + ( + ( Γ ( + Γ + Γ ( + + Γ ( + + ( Γ ( + Γ ( + + Γ ( + + Γ ( + + a sice ( li we see that we ay apply L ôpital s ule to obtai ( ( ( ( Γ ( + Γ ( + [ + + ] +Γ ( + Γ ( + + Γ ( + + li Γ ( + + + Γ ( + + Sice li[ ] ( ( ( + + we have Γ + +Γ + Γ Γ + ( ( ( ( ( 5
Γ + + ψ ψ + ( ( ( ( ece we ay apply L ôpital s ule agai to obtai Γ ( + Γ ( + [ + ] Γ ( + + + Γ ( + + + Γ ( + + + Γ ( + + ( (3 + + li + li Γ ( + Γ ( + [ ] Γ ( + + + Γ ( + + + Γ ( + + + Γ ( + + ( ( + + + Γ ( + Γ ( + Γ ( + + + Γ ( + + + Γ ( + + + Γ ( + + ( + li li Γ ( + + Γ ( + + + Γ ( + + + Γ ( + + + Γ ( + + ( ( Γ ( + [ + Γ ( ] ( Γ Γ ( + L Diffeetiatio of (.6. gives us so that Γ Γ Γ ψ ς (, Γ Γ Γ ψ ς(, + whee ς ( s, is the uwitz zeta fuctio. Theefoe we have Γ ( + Γ ( + ψ ( ς(, + + + Γ ψ + ς ( ( ( a so 6
Γ ( + Γ ς ς + + ψ ψ + Γ ( + ( ( (, ( ( Fo the efiitio of the uwitz zeta fuctio it eaily follows that ς ς (, s+ (, s s ( a i paticula we have ς( ς(, + ( ( Sice ψ ( + ψ ( we have ( ψ ψ ψ ψ ( + + ( ( ( + Theefoe we obtai Γ ( + ( ( Γ ( + ψ ( ψ( ψ( + Γ ( + a thus ( ( ( ( ( ( ψ ψ ψ ψ L + + + which siplifies to so that we obtai ( ( ( L + ( ( ( (3.9 ( + The above esult is well ow; see fo eaple the pape by Flajolet a Segewic [] whee they also epote the followig ietity S (3 ( 3 3 + + 6 3 ( ( ( (3 which ay also be eive by the above liitig pocess (albeit with uch oe teious labou. 7
Defiig (3. S by ( S( the Flajolet a Segewic [] showe that S ( ca be epesse i tes of the geealise haoic ubes as (3. ( ( (3 3 ( S( + + 3 3...!! 3!...! 3 Refeig to (. we the ote that this ay be witte as a hece we have S Y!,!,...,(!! ( ( (! ( ( (3. S Y(!,!,...,(! which ay be cotaste with (3.. Rathe belately, I ote that lettig + i (3. gives us ( Γ ( + ( Y + + + ( + + ( + +! + ( ( ( (! +,! +,...,! + wheeupo lettig esults i ( ( Y + ( + +! ( ( ( (! +,! +,...,! + With this becoes ( ( Y + ( +! Reieig gives us ( ( (!,!,...,! 8
+ ( ( Y +! ( ( (!,!,...,! Sice this becoes + ( ( (3.3 Y(!,!,...,( (! ( (! a havig ega to (.. we see that ( ( ( ( (3.4 (!,!,...,! (!,!,..., Y Y (! ece we obtai as befoe + ( ( (3.5 Y(!,!,...,(! (! We ow ultiply (3. by a ae the suatio to obtai + (! (3.6 ( ( ( Y(!,!,...,(! a copaig this with (. we have (3.7 ( + ep Let The we have f log Γ ( + log Γ ( + f ψ ψ ( ( + ( ( ( (! a we theefoe have the Maclaui epasio (3.7 + ( f log Γ ( + log Γ ( + log Γ + 9
This ay be witte as (3.8 + Γ ( + ( ep Γ ( + Γ as peviously ote by Wilf [4]. We also ote that Γ ( + Γ ( ep ( Γ + With a + (3.8 becoes Γ ( + ep Γ( Γ ( + a theefoe we have Γ( Γ ( + ep Γ ( + ece we obtai (3.9 Γ( Γ ( + ( Γ ( + + Fo (3.6 we have With + ( log Γ ( + log Γ ( + log Γ + a + this becoes Γ( Γ ( + log Γ ( + Theefoe usig (6.3 we have Γ( Γ ( + ( (,! (,...,! ( Y Γ ( + Eployig (. we see that!
Y! ( ( (,!,...,! ep a hece we have (3. Γ( Γ ( + ( Γ ( + + Usig (. a (3. gives us a oe geeal ietity ( ( + ( ( t Y (!,!,..., (! + + + + Γ ( + ( +! t ep ( t + I equatio (3.67a of [7] we showe that (3. t t + ( t Li t a usig (3. we see that t Li Y t t t! ( ( (3. (!,!,...,(! whee Li is the polylogaith fuctio efie by Li We ote that ( Li( ς s whee ς is the alteatig Riea zeta fuctio a s ς s ς ( ( ( Theefoe with t / i (3. we obtai
a thus we obtai Li Y!,!,...,(! ( (! (! ( ( (3.3 ς Y (!,!,...,(! + SOME CONNECTIONS WIT TE BETA FUNCTION Eule s beta fuctio is efie fo Re ( > a Re ( y > by the itegal (, y B y t ( t t a it is well ow that Γ Γ( y By (, Γ ( + y Diffeetiatig with espect to gives us ( y y (, [ ψ ψ( + ] ( log Γ Γ B y y t t tt Γ ( + y a with y we have (4. Γ Γ t t tt log [ ] Γ ( + ψ ψ + o equivaletly (4. Γ Γ ( t ( t log tt ( Γ ( + With we obtai the well ow itegal epesetatio fo the haoic ubes (4.3 ( t log tt
Moe geeally we have ( ( (4.4 t ( t log tt Y(!,!,...,( (! Γ Γ Γ ( + a with we have ( ( (4.5 ( t log tt Y(!,!,...,( (! which ieiately gives us the specific value with ( ( (4.6 t log tt + ( The followig foula was also epote by Devoto a Due [, p.3] ( ( ( t log tt + + a the equivalece is eaily see by efeece to Aachi s foula [] ( (4.7 ( + ( ( As a atte of iteest, I also fou foula (4.7 epote by Leveso i a 938 volue of The Aeica Matheatical Mothly [5] i a poble coceig the evaluatio of (4.8 ( e log γ ς ( Γ + We also see fo (4.5 that (4.9 3 ( 3 ( ( (3 log 6 6 3 t tt + + The followig foula was also epote by Devoto a Due [, p.3] ( ( ( 3 (3 j ( t log tt 6 + + + ( + + + j j+ 3
a its equivalece to (4.9 is eaily see by efeece to Aachi s foula [] ( + ( (3 + ( ( Copaig (4.5 with (3.3 we ieiately see that (4. + ( ( ( t log tt! which was also eive i equatio (4.4.55zi of [8]. AN APPLICATION OF TE DIGAMMA FUNCTION We have the classical foula fo the igaa fuctio (5. ψ ( + a ψ ( a + ( (...( + a( a+...( a+ which coveges fo Re ( + a >. Accoig to Raia a Laa [9], this suatio foula is ue to Nölu (see [6], [7] a also Rube s ote []. The usig Γ ( + ( + ( + Γ we see that we ay wite (5. ψ ( a ψ ( a ( a Diffeetiatig this with espect to a usig (. we get ( ( ( ψ ( a Y,!,...,( (! ( a Diffeetiatig (5. with espect to a a usig a ( a ( a ( ( a 4
we obtai ( ( (5.3 ψ ( a ψ ( a Y( ( a,! ( a,...,( (! ( a ( a As show i [7] we ay obtai ueous Eule sus by utilisig these foulae. Fo (4.4 we have t ( t log tt Y! ( a,! ( a,...,( (! ( a Γ a ( ( a a substitutig this i (5.3 gives us ψ ( a ψ ( a t ( t log tt! a Diffeetiatio with espect to esults i ψ ( a t ( t log tt! ( ( + a a with we have ψ ( + ( a a t t log tt a t log t ( ( t t t Fo (.3. we see that ( log t ( t t a + ( + t log t ψ ( a t t We ay epess this as 5
a t log t ψ ( a t t We ote that [3, p.] ψ ( a (! ς ( +, a + which esults i the well-ow itegal a t log t (! ς ( +, a t t SOME OTER APPLICATIONS OF TE COMPLETE BELL POLYNOMIALS Let us cosie the fuctio h with the followig Maclaui epasio b (6. log h b + a we wish to eteie the coefficiets a such that (6. h a By iffeetiatig (6. we obtai h h b hg Fo (.3 we have h e hy g g g log h (,,..., a i paticula we have ( (, (,..., ( h h Y g g g Usig (6. the Maclaui seies gives us ( ( 6
a h! We have a thus a h ( Y g (, g (,..., g (! ( ( ( j ( ( ( j g b j ( j g ( j! b j+ Theefoe we obtai a e Y g g g! ( (, (,..., ( b (! (,!,...,! b e Y b b b Sice log h( log a b we have b (6.3 h e Y b,! b,...,(! b The efeig to (.! j t t ep j Y(,..., j j!! we see that h e b t b j j ep j j a this is whee we state fo i (6.. b log h b + Multiplyig (6. by α it is easily see that 7
α αb (6.4 h e Y αb,! αb,...,(! αb a, i paticula, with α we obtai b (6.5 e Y ( b,! b,..., (! b! h! α Diffeetiatig (6.4 with espect to α woul give us a epessio fo h log h. Usig (. we fi that ( +! e a (! e a! b b b + + givig us the ecuece elatio o equivaletly ( a a b + + + (6.6 a a b Suppose that h h g a let f log h. We see that h f g( h a the usig (.9 above we have h e hy g g g log h (,,..., ece we have the Maclaui epasio Refeece to (. gives us ( ( h h( Y g(, g (,..., g (! 8
j ( j t h h(ep g ( j j! a hece we have ( j log log ( ( + h h g j j t j! Theefoe, as epecte, we obtai ( j ( ( + f f f j j t j! Gaa fuctio We ote the well ow seies epasio ς log Γ ( + γ + (, < a hece we have (6.7 Γ ( + Y (! ς(,! ς(,...,( (! ς a + (6.8 Y (! ς(,! ς(,...,( (! ς! Γ ( +! whee ς ( is efie as equal to γ. Diffeetiatig (6.7 we get ( ς ς ς Γ + Y! (,! (,..., (! (...( +! a hece, lettig, we have the th eivative of the gaa fuctio ( (6.9 Γ ( Y (! ς(,! ς(,...,( (! ς 9
whee we agai esigate ς ( γ. It is epote as a eecise i Apostol s boo [, p.33] that ( a a eivatio is cotaie i [9]. ( Γ ( has the sae sig as A alteative poof is show below. Sice Γ Γ ψ we have a usig (3.7 ( ψ ψ ψ Γ Γ ( Y,,..., ( ψ ( p! ς ( p+, ( p p+ we ay epess ( Γ i tes of ψ a the uwitz zeta fuctios. ( ψ ς ς ( Γ Γ( Y,! (,,...,( (! (, Fo the efiitio of the (epoetial coplete Bell polyoials we have Y ( a, a,..., a a Y (,..., a thus with a we have Y (,,...,( ( Y (,..., ece we see that ( ψ ς ς ( Γ ( Γ( Y,! (,,...,(! (, It is well ow that ψ is egative i the iteval [, α whee α > is the uique solutio of ψ ( α a hece we see that ψ is positive i that iteval. Sice (,..., ( Y > whe all of the aguets ae positive, we euce that Γ has the sae sig as ( whe [, α. We also ote that Y (,,...,( ( Y (,..., + but o isceable sig patte eeges hee. We ay ote that 3
+ (,!,...,( (! ( (,!,..., (! Y b b b Y b b b ( e b h We also have the epasio [3, p.59] ( ς ( a, log Γ ( a+ log Γ ( a + ψ ( a + a hece we euce that ( ψ ς ς Γ ( a+ Γ( a Y! ( a,! (, a,...,( (! (, a Baes ouble gaa fuctio! We ote the Baes ouble gaa fuctio G efie by [3, p.5] G ( + ( π ep ( γ + + ep + The followig ietity was oigially eive by Sivastava [3, p.] i 988 + ς ( log G( + [ log( π ] ( + γ + ( a fo this we ay eteies a seies epasio fo / G+ i tes ivolvig the epoetial Bell polyoials. Usig ( we have 3 Y( c,! c,...,( (! c G( +! whee c [ log( π ] c ( + γ c ς ( fo 3 a we ote that c > fo all. We the see that 3
Y c c c G( + a otig (.. we eteie that G( + (,!,...,! ( Y c,! c,...,(! c so that the eivatives have the sae sig as (. I fact sig as ( fo all. G( + has the sae Fo eaple we have G ( + G ( G( + G ( + a hece we have G ( log [ π ] Beoulli ubes The Beoulli ubes ae efie by the geeatig fuctio t B t t e! a thus B t t e! Itegatio gives us t a t e t a log( e log( e t t e e a B log log a+ [ a ]! a hece we obtai 3
a B log log log log e e a+ [ a ]! We ay epess this as a ( e ( e B log log + [ a ] a! Usig L ôpital s ule we see that a with a ( e li log a a we obtai + ( B log e! which was pove by Raauja fo < π [3, p.9]. ece we have B B + B,,..., Y e! a we theefoe see that B B + Y,,...,(!! Equatig coefficiets gives us the ecuece elatio B B (6. B B B Y,,..., + B Eployig (. we obtai B B+ (6. B + ( + Values of the Riea zeta fuctio ς 33
This sectio is base o a pape by Sowe []. We have the well-ow ifiite pouct siπ π a we have fo < si π ς log log π b whee b ς, b +. Theefoe we have siπ Y b b π b We also have the Maclaui seies (,!,...,!! siπ ( π π (+! a equatig coefficiets gives us (! π ( +! (,!,...,! Y b b b Y b,! b,...,! b + + Fo eaple, with we obtai Eule s foula π π Y (, ς ( o ς ( 3 6 a with we have 4 π 6 ς ( ς (4 34
We obtai fo (. Y a, because Y + Y + Y + b +, we have i ou specific case Y This esults i Y π π (+! ( +!! (!! which siplifies to ς +! ( π ς (!(+! ( +! The substitutig the well ow elatio [3, p.98] ( ς! + π B we obtai! B (!(+! ( +!! o equivaletly [!] B (!(+! ( + REFERENCES [] V.S.Aachi, O Stilig Nubes a Eule Sus. 35
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