Asymptotic aalysis of the Bell polyomials by the ray method arxi:0709.0252 [math.ca] Sep 2007 Diego Domiici Techische Uiersität Berli Sekretariat MA 4-5 Straße des 7. Jui 6 D-062 Berli Germay Permaet address: Departmet of Mathematics State Uiersity of New York at New Paltz Hawk Dr. New Paltz, NY 256-244 USA May 29, 208 Abstract We aalyze the Bell polyomials B x) asymptotically as. We obtai asymptotic approximatios from the differetial-differece equatio which they satisfy, usig a discrete ersio of the ray method. We gie some examples showig the accuracy of our formulas. Keywords Bell polyomials, asymptotic expasios, Stirlig umbers MSC-class: 4E05, B7, 4E20 Itroductio The Bell polyomials B x) are defied by [] B x) = Sk xk, = 0,,..., where S k k=0 is a Stirlig umber of the secod kid [2, 24,,4]. They hae the geeratig fuctio B x) t! = exp[ x e t )], ) domiici@math.tu-berli.de =0
from which it follows that ad B 0 x) = 2) B + x) = x[b x)+b x)], = 0,,... ) The asymptotic behaior of B x) was studied by Elbert [], [4] ad Zhao [5], usig the saddle poit method ad ). I this paper we will use a differet approach ad aalyze ) istead of ). The adatage of our method is that o kowledge of a geeratig fuctio is required ad therefore it ca be applied to other sequeces of polyomials satisfyig differetial-differece equatios [6], [7]. 2 Asymptotic aalysis To aalyze ) asymptotically as, we use a discrete ersio of the ray method [8]. Replacig the aszat B x) = ε F εx,ε) 4) i ), we get with Fu,+ε) = u ε F ) x +F, 5) u = εx, = ε 6) ad ε is a small parameter. We cosider asymptotic solutios for 5) of the form Fu,) exp [ ε ψu,) ] Ku,), 7) as ε 0. Usig 7) i 5) we obtai, to leadig order, the eikoal equatio e q up+) = 0 8) ad the trasport equatio K + 2 ψ 2 2K uexp ψ ) K u = 0, 9) where The iitial coditio 2), implies p = ψ x, q = ψ. 0) ψu,0) = 0, Ku,0) =. ) To sole 8) we use the method of characteristics, which we briefly reiew. Gie the first order partial differetial equatio Fu,,ψ,p,q) = 0, 2
with p,q defied i 0), we search for a solutio ψu,) by solig the system of characteristic equatios u = du dt = F p, = d dt = F q, ṗ = dp dt = F u p F dq, q = ψ dt = F q F ψ, ψ = dψ dt = p F p +q F q, where we ow cosider {u,,ψ,p,q} to all be fuctios of the ew ariables t ad s. For 8), we hae Fu,,ψ,p,q) = e q +p 2u ad therefore the characteristic equatios are Solig 2), subject to the iitial coditios we obtai where we hae used From ) ad ) we hae which implies Thus, The characteristic equatio for ψ is which together with 4) gies u+u = 0, = e q, ṗ p =, q = 0 2) u0,s) = s, 0,s) = 0, p0,s) = Bs), ) u = se t, = Bst, p = Be t, q = lbs) 0 = F t=0 = e q0,s) sb. ψ0,s) = 0, K0,s) =, 4) 0 = d ds ψ0,s) = p0,s) d ds u0,s)+q0,s) d ds 0,s) = B ) +lbs) 0 = B. u = se t, = st, p = e t, q = ls). 5) ψ = p u+q = e t ) se t) +ls)s, ψt,s) = s t e t) +ls)st. 6) We shall ow sole the trasport equatio 9). From 5), we get t u = tet st+), t = st+), s u = et t+, s = t+ 7)
ad therefore, 2 ψ = q 2 = q t t + q s s = st+). 8) Usig 7)-8) to rewrite 9) i terms of t ad s, we hae K + 2t+) K = 0 with solutio Kt,s) = t+, 9) where we hae used 4). Solig for t,s i 5), we obtai t = LW u), s = LW u where LW ) deotes the Lambert-W fuctio [9], defied by LWz)exp[LWz)] = z. Replacig 20) i 6) ad 9), we get [ ] ψu,) = LW ) +l LW ) u+), u u Ku,) = LW u) + ad from 7) we fid that { [ ] /ε Fu,) exp LW ) + ε l LW ) u u ), 20) } u+ ) ε LW ), 2) u + as ε 0. Usig 6) ad 2) i 4), we coclude that { [ ] } B x) exp LW ) +l LW ) x+) x x LW ), 22) x + as. Remark The fuctio LWz) has two real-alued braches for e z < 0, deoted by LW 0 z) the pricipal brach of LW) ad LW z), satisfyig LW 0 : [ e,0 ) [,0), LW : [ e,0 ), ], 4
with LW 0 e ) = = LW e ). For z 0, LWz) has oly oe real-alued brach LW 0 : [0, ) [0, ) ad for z < e, LW 0 z) ad LW z) are complex cojugates. Therefore, for 22) to be well defied, we eed to cosider three separate regios:. A expoetial regio for x > 0 or x < e. Here we hae B x) Φ x;0),, 2) where { [ ] } Φ x;k) = exp LW ) +l k LW ) x+) x k x LW ). k x + 2. A oscillatory regio for e < x < 0. I this iteral, B x) Φ x;0)+φ x; ),. 24) I Figure a) we plot B 5 x) ad the asymptotic approximatios 2) +++) ad 24) ooo), all multiplied by e x for scalig purposes, i the iteral 0,0). We see that our formulas are quite accurate ee for small alues of ad that the trasitio betwee 2) ad 24) is smooth.. A trasitio regio for x e. We will aalyze this regio i the ext sectio. I Figureb) we plot B 5 x) ad 2) +++)ad 24) ooo), all multiplied by e x, itheiteral 20,0). We obsere that the approximatios 2) ad 24) break dow i the eighborhood of e5,59. 2. The trasitio regio Whe x = e, the quatity LW x) + aishes ad 2) is o loger alid. To fid a asymptotic approximatio i a eighborhood of e, we itroduce the stretched ariable β defied by x = e β, β = O). 25) For alues of z close to z 0 = e, the Lambert-W fuctio ca be approximated by [9, 4.22)] LWz) + 2ez z 0 ) 2 ez z 0)+ 2e 6 z z 0 ), z e. 26) Usig25) i 26), we hae, ) LW + 2 2e β e β 5 e β 2 7 6 2e β, β 0. 27)
00 80 2 60 y 40 20 20 8 6 4 2 0 8 6 4 2 x 0 0 8 6 4 2 0 2 4 6 8 0 x 2 a) b) Figure : A compariso of the exact solid cure) ad asymptotic ooo), +++) alues of B 5 x). Hece, { [ ] } exp LW ) +l k LW ) x+) ϕβ,), β 0, x k x for k = 0, with x = e β ad ϕβ,) = ) exp We ow cosider solutios for ) of the form [ B x) = ϕβ,)λβ) = ϕ { [l)+e 2] e ) } β. 28) e+ x ] [ ) 2, Λ e+ x ) ] 2, 29) for some fuctio Λβ). Replacig 29) i ) ad usig 25) we obtai, to leadig order Λ 2e βλ = 0, with solutio ) ) Λβ) = C Ai 2 e β +C 2 Bi 2 e β, 0) where Ai ), Bi ) are the Airy fuctios. To determie the costats C,C 2 i 0), we shall match 2) with 29). Usig 25) ad 27) i 2), we hae B x) ϕβ,)exp 2 ) 2e 2e 2 β 2 β ) 4 6, β 0 +. ) 6
O the other had, the Airy fuctios hae the well kow asymptotic approximatios [2, 0.4.59, 0.4.6)] Aiz) 2 π exp 2 ) z 2 z 4, z, ad therefore we coclude that Biz) π exp 2 z 2 ) z 4, z C = π2 5 6 6, C2 = 0. 2) Replacig 0) ad 2) i 29), we fid that for x e, we hae B x) ) π2 5 6 6 ϕβ,)ai 2 e β,. This cocludes the asymptotic aalysis of B x) for large. Ackowledgemet 2 This work was completed while isitig Techische Uiersität Berli ad supported i part by a Sofja Koaleskaja Award from the Humboldt Foudatio, proided by Professor Olga Holtz. We wish to thak Olga for her geerous sposorship ad our colleagues at TU Berli for their cotiuous help. Refereces [] E. T. Bell, Expoetial polyomials, A. of Math. 2) 5 2) 94) 258 277. [2] M. Abramowitz, I. A. Stegu Eds.), Hadbook of mathematical fuctios with formulas, graphs, ad mathematical tables, Doer Publicatios Ic., New York, 992. [] C. Elbert, Weak asymptotics for the geeratig polyomials of the Stirlig umbers of the secod kid, J. Approx. Theory 09 2) 200) 28 228. [4] C. Elbert, Strog asymptotics of the geeratig polyomials of the Stirlig umbers of the secod kid, J. Approx. Theory 09 2) 200) 98 27. [5] Y.-Q. Zhao, A uiform asymptotic expasio of the sigle ariable Bell polyomials, J. Comput. Appl. Math. 50 2) 200) 29 55. [6] D. E. Domiici, Asymptotic aalysis of the Hermite polyomials from their differetial-differece equatio 2007) To appear i the Joural of Differece Equatios ad Applicatios. [7] D. E. Domiici, Asymptotic aalysis of the asymptotic aalysis of geeralized Hermite polyomials 2007) Submitted. [8] E. Giladi, J. B. Keller, Euleria umber asymptotics, Proc. Roy. Soc. Lodo Ser. A 445 924) 994) 29 0. [9] R. M. Corless, G. H. Goet, D. E. G. Hare, D. J. Jeffrey, D. E. Kuth, O the Lambert W fuctio, Ad. Comput. Math. 5 4) 996) 29 59. 7