Asymptotics of the Stirling numbers of the second kind revisited: A saddle point approach

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1 Asyptotics of the Stirlig ubers of the secod kid revisited: A saddle poit approach Guy Louchard March 2, 202 Abstract Usig the saddle poit ethod ad ultiseries { expasios, } we obtai fro the geeratig fuctio of the Stirlig ubers of the secod kid ad Cauchy s itegral forula, asyptotic results i cetral ad o-cetral regios. I the cetral regio, we revisit the celebrated Gaussia theore { with ore precisio. I the regio α, > α > /2, we aalyze the depedece of o α. This paper fits withi the fraework of Aalytic Cobiatorics. } Keywords: Stirlig ubers of the secod kid, Asyptotics, Saddle poit ethod, Multiseries expasios, Aalytic Cobiatorics. Itroductio Let be the Stirlig uber of the secod kid. Their geeratig fuctio is give by!! z f(z), f(z) : e z. I the sequel all asyptotics are eat for. Let us suarize the related litterature. The asyptotic Gaussia approxiatio i the cetral regio is proved i Harper 7]. See also Beder ], Sachkov 3] ad Hwag 0]. I the o-cetral regio, α, 0 > α <, ost of the previous papers use the solutio of ρe ρ e ρ. () As show i the ext sectio, this actually correspods to a Saddle poit. Let us etio ˆ Hsu 8]: For t o( /2 ) { } + t 2t 2 t + f (t) t! + f ] 2(t) , f (t) t(2t + ). 3 Uiversité Libre de Bruxelles, Départeet d Iforatique, CP 22, Boulevard du Triophe, B-050 Bruxelles, Belgiu, eail: louchard@ulb.ac.be

2 ˆ Moser ad Wya 2]: For t o( ), For,, C 3, C 4 are fuctios of ρ. ˆ Good 6]: t ( )q t + (t) 2 t 2 q + (t) ] q2 +..., q 2 t.!(e ρ ) 2ρ!(πρH) /2 H eρ (e ρ ρ) 2(e ρ ) 2, ( 5C 2 3 ρ 6ρ 2 H 3C ) ] 4 4ρH , ˆ Beder ]: + t t κ : t, (t + )!(e ρ ) ) t t!ρ t+ 2πt ( + κ ( + κ) 2 e ρ )] /2 g (κ) 3λ 4 5λ 2 3, 24 λ i κ i (ρ)/σ i, σ κ 2 (ρ) /2, κ κ, κ 2 (κ + )(ρ κ ). + g (κ) + g ] 2(κ) t t ,!e α!ρ ( + e α )σ 2π, ( + eα ) l( + e α ), ρ l( + e α ), ( ) 2 σ 2 e α l( + e α ) ], It is easy to see that ρ here coicides with the solutio of (). Beder s expressio is siilar to Moser ad Wya s result. ˆ Bleick ad Wag 2]: Let ρ be the solutio of The!(e ρ ) (2π( + )) /2!ρ ( G)/2 ρ e ρ e ρ +. 2

3 2 + 8G 20G2 (e ρ + ) + 3G 3 (e 2ρ + 4e ρ + ) + 2G 4 (e 2ρ e ρ ] + ) 24( + )( G) 3 + O(/ 2 ), G ρ e ρ. The series is coverget for for o( 2/3 ). ˆ Tee 6]: { } ( ) e A ( ) k f k (t 0 ) k, k0 ( f 0 (t 0 ) where A is a fuctio of ρ,,. ˆ Tsylova 7]: Let t + o( 2/3 ). t 0, t 0 ( + t 0 )(ρ t 0 ) ( )) /2, (γ) 2πδ(γ) exp ( t) 2 /(2δ) ] ( + o()), γ( e /γ ) γ, δ e /γ (t e /γ ). After soe algebra, this coicides with Moser ad Wya s result. ˆ Chelluri, Richod ad Tee 3]: They prove, with other techiques, that Moser ad Wya s expressio is valid if Ω( /3 ) ad that Hsu s forula is valid for y x o( /3 ) ˆ Erdos ad Szekeres: see Sachkov 4], p.64: Let < / l, ( )! exp e /] ( + o()). All these papers siply use ρ as the solutio of (). They do t copute the detailed depedece of ρ o α for our o-cetral rage, either the precise behaviour of fuctios of ρ they use. Moreover, ost results are related to the case α < /2. We will use ultiseries expasios: ultiseries are i effect power series (i which the powers ay be o-itegral but ust ted to ifiity) ad the variables are eleets of a scale: details ca be foud i Salvy ad Shackell 5]. The scale is a set of variables of icreasig order. The series is coputed i ters of the variable of axiu order, the coefficiets of which are give i ters of the ext-to-axiu order, etc. Actually we iplicitly used ultiseries i our aalysis of Stirlig ubers of the first kid i ]. Our work fits withi the fraework of Aalytic Cobiatorics. Let us fially etio that Hsu 9] cosider soe geeralized Stirlig ubers. I Sec.2, we revisit the asyptotic expasio i the cetral regio ad i Sec.3, we aalyse the o-cetral regio j α, α > /2. We use Cauchy s itegral forula ad the saddle poit ethod. 3

4 2 Cetral regio Cosider the rado variable J, with probability distributio PJ ] Z (), Z () :, B where B is the th Bell uber. The ea ad variace of J are give by Let ζ be the solutio of This iediately leads to M : E(J ) B + B, σ 2 : V(J ) B +2 B B + B. ζe ζ. ζ W (), where W is the Labert fuctio (we use the pricipal brach, which is aalytic at 0). We have the well-kow asyptotic l l() ζ l() l l() + + O(/ l() 2 ). (2) l() To siplify our expressios i the sequel, let F : e ζ, G : e ζ/2. The ultiseries scale is here {ζ, G}. Our result ca be suarized i the followig local liit theore Theore 2. Let x ( M)/σ. The Z () B e x2 /2 ( + ζ)/2 + x( 6ζ + 2x2 ζ + x 2 ] 3) 2πG 6G( + ζ) 3/2 + O(/G 2 ). Proof. By Salvy ad Shackell 5], we have M F + A + O(/F ), σ 2 F + ζ + A 3 + O(/F ), B exp(t )H 0, (3)! T l(ζ)ζf + F ζ/2 l(ζ) l(2π)/2, (4) 2 + 3/ζ + 2/ζ2 A 2( + /ζ) 2, A /ζ + /ζ2 + 9/ζ 3 + 2/ζ 4 2( + /ζ) 4, H 0 + A5 ( + /ζ) /2 /F + O(/F 2 ) ], 4

5 A /ζ + 6/ζ2 + 6/ζ 3 + 2/ζ 4 24( + /ζ) 3. This leads to (fro ow o, we oly provide a few ters i our expasios, but of course we use ore ters i our coputatios), usig expasios i G, G σ ( + ζ) /2 + A 3( + ζ) /2 2G σ G ζ l(). + O(/G 3 ), We ow use the Saddle poit techique (for a good itroductio to this ethod, see Flajolet ad Sedgewick 4], ch.v III). Let ρ be the saddle poit ad Ω the circle ρe iθ. By Cauchy s theore, Z ()! f(z) dz!b 2πi Ω z+! π!b ρ f(ρe iθ ) e iθ dθ 2π!!B ρ 2π π π e l(f(ρeiθ )) iθ dθ π! f(ρ) π {!B ρ exp 2π π 2 κ 2θ 2 i }] 6 κ 3θ , (5) ( ) i κ i (ρ) l(f(ρe u )) u. (6) u0 See Good 5] for a aciet but eat descriptio of this techique. Let us ow tur to the Saddle poit coputatio. ρ is the root (of sallest odule) of ρf (ρ) f(ρ) 0, i.e. ρe ρ e ρ, which is, of course idetical to (). After soe algebra, this gives I the cetral regio, we choose M + σx F + ρ + W ( e /). x ( + ζ) /2 G + A + xa 3( + ζ) /2 + O(/G 2 ). 2G This leads to l() ζ + x ( + ζ) /2 G + O(/G2 ), ζ ζx ( + ζ) /2 G + A ζ + ζx 2 /( + ζ) G 2 + O(/G 3 ), ζx ρ ζ ( + ζ) /2 G + ζ( A + x 2 /( + ζ) ) G 2 + O(/G 3 ), x l(ρ) l(ζ) ( + ζ) /2 G + O(/G2 ). Let us reark that the coefficiets of G powers are ice ratioal expressios i ζ. 5

6 Now we ote that e ρ ρe ρ, l (e ρ ) ρ + l(ρ) + l() l(), (7) l() ζ + l(ζ), so, by Stirlig s forula, with (4), the first part of (5) leads to!!b ρ f(ρ) exp T 2 ] H H 2, T 2 (ρ + l(ρ) ζ l(ζ)) ( + l(2π)/2 + l()/2) ζf l(ρ) T, H /H 0 ( + /ζ) /2 A 5( + /ζ) /2 / H O(/3 ) G 2 + O(/G 4 ), ] 2G 2 + x 2G 3 ( + ζ) /2 + O(/G4 ). Note carefully that there is a cacellatio of the ter l() i T 2. Usig all previous expasios, we obtai exp(t 2 ) e x2 /2+l(ζ) )H 3, (8) H 3 + x( 5ζ 6ζ2 6A + x 2 2A ζ 6A ζ 2 + 2x 2 ζ 9 6( + ζ) 3/2 G We ow tur to the itegral i (5). We copute κ 2 ρeρ ( e ρ + + ρ) ζx (e ρ ) 2 ζ ( + ζ) /2 G + O(/G2 ), + O(/G 2 ). ad siilar expressios for the ext κ i that we do t detail here. Note that κ 3, κ 5,... are useless for the precisio we attai here. Now we use the classical trick of settig ] κ 2 θ 2 /2! + κ l (iθ) l /l! u 2 /2. l3 Coputig θ as a series i u, this gives, by iversio, θ G a i u i, with, for istace a ζ/2 + ζ/2 2G 2 + O(/G3 ). Settig dθ dθ dudu, we itegrate o u.. ]: this extesio of the rage ca be justified as i Flajolet ad Segewick 4], ch.v III. Now, isertig the ter ζ coig i (8) as e l(ζ), this gives ( H 4 ζ/2 + ζ ) 2πG 2G 2 + O(/G3 ). Fially, cobiig all expasios, Z () e x2 /2 H H 2 H 3 H 4 R, (9) B 6

7 R e x2 /2 ( + ζ)/2 + x( 6ζ + 2x2 ζ + x 2 ] 3) 2πG 6G( + ζ) 3/2 + O(/G 2 ). Note that the coefficiet of the expoetial ter is asyptotically equivalet to the doiat ter of 2πσ, as expected. More ters i this expressio ca be obtaied if we copute M, σ 2, B /! with ore precisio. Also, usig (2), our result ca be put ito expasios depedig o, l,.... To check the quality of our asyptotic, we have chose This leads to ζ , G , M , σ , B , B as , where B as is give by (3). Figure shows Z () ad 2πσ exp ( M σ ) / ] Figure : Z () ad 2πσ exp ( M σ ) / ] 2 2 The/ fit sees quite good, but to have ore precise iforatio, we show i Figure 2 the quotiet Z () 2πσ exp ( ) / ] M 2 σ 2. The precisio is betwee 0.05 ad 0.0. Figure 3 shows the quotiet Z () /R. The precisio is ow betwee ad Large deviatio, α, α > /2 We set ε : α, 7

8 / Figure 2: Z () 2πσ exp ( M σ ) / ] Figure 3: Z () /R 8

9 ε α α, L : l(). The ultiseries scale is here { α, α, }. Our result ca be suarized i the followig local liit theore Theore 3. e T R, T α (T L + T 0 ), R 2π α/2 R 0 + R + R O(/3 ) R 0 R 00 + R 0 α + O(/2α ), R R 0 + R α + O(/2α ), R 2 R 20 + R 2 α + O(/2α ), ], where T i,j, R i,j are power series i ε. Proof. Usig agai the Labert fuctio, we derive successively (agai we oly provide a few ters here, we use a doze of ters i our expasios) ( ε), ε, ρ 2ε ε ε3 + O(ε 4 ), l() L ε 2 ε2 + O(ε 3 ), l(ρ) L( α) + l(2) ε + 3 ε2 + O(ε 3 ). For the first part of the Cauchy s itegral, we have, otig that ε α, ad usig (7),!!ρ f(ρ) exp(t )H 2, T (ρ + l(ρ) L) ( + l()/2) + ( + L + L/2) l(ρ) T + T 0, T α (T L + T 0 ), T 2 α, T 0 l(2) 4 3 ε 5 9 ε2 + O(ε 3 ), T 0 2 ε + 4 ε2 + O(ε 3 ), H exp(t 0 ) + 2 ε ε2 + O(ε 3 ), H ] / O(/3 ) ] O(/3 ) 9

10 ε + 2(ε ) + ε 2 288(ε ) O(ε3 / 3 ). Note agai that there are cacellatios, i T of the ters l() ad l(2π)/2. Now we tur to the itegral part. We obtai, for istace, usig (6), κ 2 ε ε ε3 + O(ε 4 ), θ a i u i, a ε 6 ε2 ] 72 ε4 + O(ε 6 ). Itegratig, this gives Now we copute with H 3 H 2π α/2 3 + H ] 32 α + O(/2α ), H 3 6 ε 72 ε2 + O(ε 3 ), H ε ε2 + O(ε 3 ). e T H H 2 H 3 e T R, (0) R R 2π α/2 0 + R + R ] O(/3 ), R 0 R 00 + R 0 α + O(/2α ), R R 0 + R α + O(/2α ), R 2 R 20 + R 2 α + O(/2α ), R ε + O(ε2 ), R ε + O(ε2 ), R 0 2 ε 9 ε2 + O(ε 3 ), R 44 ε + 08 ε2 + O(ε 3 ), R ε ε2 + O(ε 3 ), R ε ε2 + O(ε 3 ). Give soe desired precisio, how ay ters ust we use i our expasios? It depeds o α. For istace, i T, α ε k if k < α/( α). Also ε k i R 00 is less tha ε l / i R 0 / if k l > /( α). Ay uber of ters ca be coputed by alost autoatic coputer algebra. We use Maple i this paper. 0

11 To check the quality of our asyptotic, we have chose 00 ad a rage α /2, 9/0], i.e. { }/ a rage 37, 90]. We use 5 or 6 ters i our fial expasios. Figure 4 shows the quotiet (e T R). The precisio is at least Note that the rage M 3σ, M + 3σ], where the Gaussia approxiatio is useful, is here 2, 36] Figure 4: { }/ (e T R) We fially etio that our o-cetral rage is ot sacred: other types of rages ca be aalyzed with siilar ethods. Refereces ] E.A. Beder. Cetral ad local liit theores applied to asyptotics eueratio. Joural of Cobiatorial Theory, Series A, 5:9, ] W. E. Bleick ad P. C. C. Wag. Asyptotics of Stirlig ubers of the secod kid. Proceedigs of the Aerica Matheatical Society, 42(2): , ] R. Chelluri, L. B. Richod, ad N. M. Tee. Asyptotic estiates for geeralized Stirlig ubers. Techical report, CWI, MAS-R9923, ] P. Flajolet ad R. Sedgewick. Aalytic cobiatorics. Cabridge Uiversity press, ] I. J. Good. Saddle-poit ethods for the ultioial distributio. Aals of Matheatical Statistics, 28(4):86 88, ] I. J. Good. A asyptotic forula for the differeces of the power at zero. Aals of Matheatical Statistics, 32(): , 96. 7] L. H. Harper. Stirlig behaviour is asyptotically oral. Aals of Matheatical Statistics, 38:40 44, 967.

12 8] L. C. Hsu. Note o a asyptotic expasio of the th differeces of zero. Aals of Matheatical Statistics, 9: , ] L. C. Hsu. A uified approach to geeralized Stirlig ubers. Advaces i Applied Matheatics, 20: , ] H.K. Hwag. O covergece rates i the cetral liit theores for cobiatorial structures. Europea Joural of Cobiatorics, 9: , 998. ] G. Louchard. Asyptotics of the stirlig ubers of the first kid revisited: A saddle poit approach. Discrete Matheatics ad Theoretical Coputer Sciece, 8(2):67 84, ] L. Moser ad M. Wya. Stirlig ubers of the secod kid. Duke Matheatical Joural, 25:29 48, ] V.N. Sachkov. Probabilistic Methods i Cobiatorial Aalysis. Cabridge Uiversity Press, 997. Traslated ad adapted fro the Russia origial editio, Nauka,978. 4] V.N. Sachkov. Probabilistic Methods i Cobiatorial Aalysis. Cabridge Uiversity Press, ] B. Salvy ad J. Shackell. Sybolic asyptotics: Multiseries of iverse fuctios. Joural of Sybolic Coputatio, 20(6): , ] N.M. Tee. Asyptotic estiates of Stirlig ubers. Studies i Applied Matheatics, 89: , ] E. G. Tsylova. Probabilistic ethods for obtaiig asyptotic forulas for geeralized Stirlig ubers. Joural of Matheatical Scieces, 75(2):607 64,