Extended Bell and Stirling Numbers From Hypergeometric Exponentiation

Transcription

1 Joural of Iteger Sequeces, Vol , Article Exteded Bell ad Stirlig Numbers From Hypergeometric Expoetiatio J.-M. Sixdeiers K. A. Peso A. I. Solomo 1 Uiversité Pierre et Marie Curie, Laboratoire de Physique Théorique des Liquides, Tour 16, 5 ième étage, 4 place Jussieu, Paris Cedex 05, Frace addresses: sixdeiers@lptl.jussieu.fr, peso@lptl.jussieu.fr ad a.i.solomo@ope.ac.u Abstract Expoetiatig the hypergeometric series 0 F L 1, 1,..., 1; z, L = 0, 1, 2,..., furishes a recursio relatio for the members of certai iteger sequeces b L, = 0, 1, 2,.... For L > 0, the b L s are geeralizatios of the covetioal Bell umbers, b 0. The correspodig associated Stirlig umbers of the secod id are also ivestigated. For L = 1 oe ca give a combiatorial iterpretatio of the umbers b 1 ad of some Stirlig umbers associated with them. We also cosider the L 1 aalogues of Bell umbers for restricted partitios. The covetioal Bell umbers [1] b 0, = 0, 1, 2,..., have a well-ow expoetial geeratig fuctio B 0 z e ez 1 = b 0 z!, 1 which ca be derived by iterpretig b 0 as the umber of partitios of a set of distict elemets. I this ote we obtai recursio relatios for related sequeces of positive itegers, called b L, L = 0, 1, 2,..., 1 Permaet address: Quatum Processes Group, Ope Uiversity, Milto Keyes, MK7 6AA, Uited Kigdom. 1

2 obtaied by expoetiatig the hypergeometric series 0 F L 1, 1,..., 1; z defied by [2]: z 0F L 1, 1,..., 1; z =, 2 }{{}! L which we shall deote by 0 F L z ad which icludes the special cases 0 F 0 z e z ad 0 F 1 z I 0 2 z, where I 0 x is the modified Bessel fuctio of the first id. For L > 1, the fuctios 0 F L z are related to the so-called hyper-bessel fuctios [3], [4], [5], which have recetly foud applicatio i quatum mechaics [6], [7]. Thus we are iterested i b L give by e [ 0FLz 1] = z b L, 3! thereby defiig a hypergeometric geeratig fuctio for the umbers b L. From eq. 3 it follows formally that b L =! L For L = 0 the r.h.s of eq. 4 ca be evaluated i closed form: b 0 = 1 e =0 d dz e [ 0FLz 1] z=0. 4 { [ 1! = e z z d ]} e z. 5 dz z=1 The first equality i 5 is the celebrated Dobińsi formula [1], [8], [9]. The secod equality i eq. 5 follows from observig that for a power series Rz = =0 A z we have z d dz Rz = A z 6 =0 ad applyig eq. 6 to the expoetial series A =! 1. The reaso for icludig the divisors! rather tha! as i the usual expoetial geeratig fuctio arises from the fact that oly by usig eq. 3 are the umbers b L actually itegers. This ca be see from geeral formulas for expoetiatio of a power series [8], which employ the expoetial Bell polyomials, complicated ad rather uwieldy objects. It caot however be cosidered as a proof that the b L are itegers. At this stage we shall use eq. 3 with b L real ad apply to it a efficiet method, described i [9], which will yield the recursio relatio for the b L. For the proof that the b L are itegers, see below eq. 11. To this ed we first obtai a result for the multiplicatio of two power-series of the type 3. Suppose we wish to multiply fx = a x L! get fx gx = d x L!, where d L =! r+s= a L rc L s r! s! = r=0 r ad gx = c L x!. We a L r c L r. 7 Substitute eq. 2 ito eq. 3 ad tae the logarithm of both sides of eq. 3: z z = l b! L!. 8 =1 2

3 Now differetiate both sides of eq. 8 ad multiply by z: z z z b L!! = b L, 9! which with eq. 7 yields the desired recurrece relatio b L + 1 = + 1 b L, = 0, 1, =0 L + 1 = b L, 11 =0 b L 0 = Sice eq. 11 ivolves oly positive itegers, it follows that the b L are ideed positive itegers. For L = 0 oe gets the ow recurrece relatio for the Bell umbers [9]: b = b =0 We have used eq. 11 to calculate some of the b L s, listed i Table I, for L = 0, 1,..., 6. Eq.11, for fixed, gives closed form expressios for the b L directly as a fuctio of L colums i Table I: b L 2 = L, b L 3 = L + 3! L, b L 4 = L L L + 4! L, etc. The sets of b L have bee checed agaist the most complete source of iteger sequeces available [10]. Apart from the case L = 0 covetioal Bell umbers oly the first o-trivial sequece L = 1 is listed: 1 it turs out that this sequece b 1, listed uder the headig A i [10], ca be give a combiatorial iterpretatio as the umber of bloc permutatios o a set of objects which are uiform, i.e. correspodig blocs have the same size [12]. Eq.1 ca be geeralized by icludig a additioal variable x, which will result i smearig out the covetioal Bell umbers b 0 with a set of itegers S 0,, such that for >, S 0, = 0, ad S 0 0, 0 = 1, S 0, 0 = 0. I particular, [ B 0 z, x e xez 1 ] = S 0, x z!, 14 which leads to the expoetial geeratig fuctio of S 0, l, the covetioal Stirlig umbers of the secod id, see [1], [8], i the form e z 1 l ad defies the so-called expoetial or Touchard polyomials l 0 x as They satisfy 1 others have sice bee added l! l 0 x = = =l =1 S 0, l z, 15! S 0, x. 16 =1 l 0 1 = b 0, 17 3

4 justifyig the term smearig out used above. The appearace of itegers i eq. 3 suggests a atural extesio with a additioal variable x: [ ] B L z, x e x[ 0FLz 1] = S L, x z, 18! =1 where we iclude the right divisors! i the r.h.s of 18. This i tur defies hypergeometric polyomials of type L ad order through l L x = S L, x, 19 =1 which satisfy l L 1 = b L, 20 with the b L of eq. 10. Thus the polyomials of eq. 19 smear out the b L with the geeralized Stirlig umbers of the secod id, of type L, deoted by S L, with S L, = 0, if >, S L, 0 = 0 if > 0 ad S L 0, 0 = 1, which have, from eq. 18 the hypergeometric geeratig fuctio 0 F L z 1 l l! = =l S L, l! z, L = 0, 1, 2, Eq.21 ca be used to derive a recursio relatio for the umbers S L,, i the same maer as eq. 3 yielded eq. 12. Thus we tae the logarithm of both sides of eq. 21, differetiate with respect to z, multiply by z ad obtai: S L, l 1! z z =! which, with the help of eq. 7, produces the required recursio relatio S L + 1, l = =l S L, l! z, 22 L S L, l 1, 23 S L 0, 0 = 1, S L, 0 = 0, 24 which for L = 0 is the recursio relatio for the covetioal Stirlig umbers of the secod id [1], [8], ad i eq. 23 the appropriate summatio rage has bee iserted. Sice the recursios of eq. 23 ad eq. 24 ivolve oly itegers we coclude that S L, l are positive itegers. We have calculated some of the umbers S L, l usig eq. 21 ad have listed them i Tables II ad III, for L = 1 ad L = 2 respectively. Observe that S 1, 2 = 1 ad S L, =! L, L = 1, 2. Also, by fixig ad l, the idividual values of S L, l have bee calculated as a fuctio of L with the help of eq. 23, see Table IV, from which we observe + 1 S L, =! L, L = 1, 2, which is the lowest diagoal i Table IV. We ow demostrate that the repetitive use of eq. 23 permits oe to establish closed-form expressios for ay supra-diagoal of order p, i.e. the sequece S L + p,, 4

5 for p = 1, 2, 3,..., if oe ows the expressio for all S L +, with < p. We shall illustrate it here for p = 1, 2. To this ed fix l = o both sides of eq. 23. It becomes, upo usig eq. 25, ad defiig α L S L + 1,, a liear recursio relatio with the solutio α L = [ + 1!]L 2 L L α L 1, α L 0 = 0, 26 α L = S L + 1, = = [ ] L ! [ ] L + 1! S 0 + 1,, 28 2 which gives the secod lowest diagoal i Table IV. Observe that for ay L, S L + 1, is proportioal to S 0 +1, = +1/2. The sequece S 1 +1, = 1, 9, 72, 600, 5400, ,... is of particular iterest: it represets the sum of iversio umbers of all permutatios o letters [10]. For more iformatio about this ad related sequeces see the etry A i [10]. The S L + 1, for L > 1 do ot appear to have a simple combiatorial iterpretatio. A recurrece equatio for β L S L + 2, is obtaied upo substitutig eq. 25 ad eq. 27 ito eq. 23: β L = + 1 2! [ + 2! 2! ] L 1 2 L L L β L 1, β L 0 = It has the solutio S L + 2, = [ + 2! 2 ] L L 3 L 30 which is a closed form expressio for the secod lowest diagoal i Table IV. Clearly, eq. 30 for L = 0 gives the combiatorial form for the series of covetioal Stirlig umbers I a similar way we obtai S 0 + 2, = ! S L + 3, = [ ] L ! L L 1 8 L L 8 L 1 4 L 1 32 which for L = 0 reduces to Combied with the stadard defiitio [8], [9] S 0 + 3, = S 0, l = 1l l! l 1 =1 l. 34 5

6 eqs.28, 31 ad 33 give compact expressios for the summatio form of S 0 + p,. Further, from eq. 34, use of eq. 6 gives the followig geeratig formula [ S 0, l = 1l z d l 1 l! dz =1 [ = 1l z d ] [1 z l 1] l! dz l ] z z=1 35, l. 36 z=1 The formula 1 ca be geeralized by puttig restrictios o the type of resultig partitios. geeratig fuctio for the umber of partitios of a set of distict elemets without sigleto blocs b 0 1, is [8], [14], [15], B 0 1, z = e ez 1 z = or more geerally, without sigleto, doubleto..., p blocs p = 0, 1,... is [15] B 0 p, z = e ez p z =0! = The b 0 1, z!, 37 b 0 p, z!, 38 with the correspodig associated Stirlig umbers defied by aalogy with eq. 14 ad eq. 22. umbers b 0 1,, b 0 2,, b 0 3,, b 0 4, ca be read off from the sequeces A000296, A006505, A ad A i [10], respectively. For more properties of these umbers see [11]. We carry over this type of extesio to eq. 3 ad defie b L p, through B L p, z e 0 F Lz p =0 z! = The z b L p,, 39! where b L 0, = b L from eq. 3. We ow of o combiatorial meaig of b L p, for L 1, p > 0. The b L p, satisfy the followig recursio relatios: b L p, = p =0 + 1 L b L p,, 40 b L p, 0 = 1, 41 b L p, 1 = b L p, 2 = = b L p, p = 0, 42 b L p, p + 1 = That the b L p, are itegers follows from eq. 40. Through eq. 39 additioal families of iteger Stirliglie umbers S L,p, ca be readily defied ad ivestigated. The umbers b 0 p, are collected i Table V, ad Tables VI ad VII cotai the lowest values of b 1 p, ad b 2 p,, respectively. Formula 1 ca be used to express e i terms of b 0 i various ways. differetiatio forms are Two such lowest order i b 0 e = 1 + l = 44! b = l. 45! 6

7 I the very same way, eq. 3 ca be used to express the values of 0F L z ad its derivatives at z = 1 i terms of certai series of b L s. For L = 1, the aalogues of eq. 44 ad eq. 45 are b 1 I 0 2 = 1 + l! 2, 46 b I li 1 2 = 1 + l + 1! 2 47 ad for L = 2 the correspodig formulas are b 2 0F 2 1, 1; 1 = 1 + l! 3, 48 b F 2 1, 1; 1 + l 0 F 2 2, 2; 1 = 1 + l + 1 2! By fixig z 0 at values other tha z 0 = 1, oe ca li the umerical values of certai combiatios of 0F L 1, 1,... ; z 0, 0F L 2, 2,... ; z 0,... ad their logarithms, with other series cotaiig the b L s. The above cosideratios ca be exteded to the expoetiatio of the more geeral hypergeometric fuctios of type 0 F L 1, 2,..., L ; z where 1, 2,..., L are positive itegers. We cojecture that for every set of s a differet set of itegers will be geerated through a appropriate adaptatio of eq. 3. We quote oe simple example of such a series. For eq. 3 exteds to where the umbers 0F 2 1, 2; z = e [ 0F21,2;z 1] = z + 1! 3 50 z f 2 + 1! 3 51 f 2 = + 1! 2 [ d e[ 0F21,2;z 1] dz tur out to be itegers: f 2, = 0, 1,..., 8 are: 1, 1, 4, 37, 641, 18276, , , etc. A The aalogue of equatios 23 ad 44 is: 0F 2 1, 2; 1 = 1 + l ] z=0 52 f 2 + 1! Acowledgemets We tha L. Haddad for iterestig discussios. We have used Maple c to calculate most of the umbers discussed above. 7

8 Table I: Table of b L : L, = 0, 1,..., 6. The rows give sequeces A000110, A023998, A A L b L 0 b L 1 b L 2 b L 3 b L 4 b L 5 b L Table II: Table of S L, l: for L = 1 ad l, = 1, 2,..., 8. The triagle, read by colums, gives A061691, the rows ad diagoals give A017063, A061690, A000142, A001809, A l S 1 1, l S 1 2, l S 1 3, l S 1 4, l S 1 5, l S 1 6, l S 1 7, l S 1 8, l Table III: Table of S L, l: for L = 2 ad l, = 1, 2,..., 8. The triagle, read by colums, gives A061692, the rows ad diagoals give A061693, A061694, A001044, A l S 2 1, l S 2 2, l S 2 3, l S 2 4, l S 2 5, l S 2 6, l S 2 7, l S 2 8, l

9 Table IV: Table of S L, l: l, = 1, 2,..., 6. l S L 1, l S L 2, l S L 3, l S L 4, l S L 5, l S L 6, l ! L 3 3 L 4 4 L L 5 5 L L 6 6 L L L 3 3! L 6 12 L L L L L L 4 4! L L L L 5 5! L L 6 6! L Table V: Table of b 0 p, : p = 0, 1, 2, 3; = 0,..., 10. The colums give A000110, A000296, A006505, A b 0 0, b 0 1, b 0 2, b 0 3, Table VI: Table of b 1 p, : p = 0, 1, 2; = 0,..., 9. The colums give A023998, A061696, A b 1 0, b 1 1, b 1 2,

10 Table VII: Table of b 2 p, : p = 0, 1, 2; = 0,..., 8. The colums give A A b 2 0, b 2 1, b 2 2, Refereces [1] S.V. Yablosy, Itroductio to Discrete Mathematics, Mir Publishers, Moscow, [2] G.E. Adrews, R. Asey ad R. Roy, Special Fuctios, Ecyclopedia of Mathematics ad its Applicatios, vol. 71, Cambridge Uiversity Press, [3] O.I. Marichev, Hadboo of Itegral Trasforms of Higher Trascedetal Fuctios, Theory ad Algorithmic Tables, Ellis Horwood Ltd, Chichester, 1983, Chap. 6. [4] V.S. Kiryaova ad B.Al-Saqabi, Explicit solutios to hyper-bessel itegral equatios of secod id, Comput. ad Math. with Appl. 37, [5] R.B. Paris ad A.D. Wood, Results old ad ew o the hyper-bessel equatio, Proc. Roy. Soc. Edib. 106 A, [6] N.S. Witte, Exact solutio for the reflectio ad diffractio of atomic de Broglie waves by a travelig evaescet laser wave, J. Phys. A 31, [7] J.R. Klauder, K.A. Peso ad J.-M. Sixdeiers, Costructig coheret states through solutios of Stieltjes ad Hausdorff momet problems, Physical Review A, 64, [8] L. Comtet, Advaced Combiatorics, D. Reidel, Bosto, [9] H.S. Wilf, Geeratigfuctioology, 2 d ed., Academic Press, New Yor, [10] N.J.A. Sloae, O-Lie Ecyclopedia of Iteger Sequeces, published electroically at: /jas/sequeces/. [11] M. Berstei ad N.J.A. Sloae, Some caoical sequeces of itegers, Liear Algebra Appl., 226/228,

11 [12] D.G. Fitzgerald ad J. Leech, Dual symmetric iverse mooids ad represetatio theory, J. Austr. Math. Soc., Series A, 64, [13] P. Delerue, Sur le calcul symbolique à variables et foctios hyperbesséliees II, A. Soc. Sci. Brux. 67, [14] R. Ehreborg, The Hael Determiat of Expoetial Polyomials, Am. Math. Mothly, 207, [15] R. Suter, Two Aalogues of a Classical Sequece, J. Iteg. Seq. 3, Article Metios sequeces A A A A A A A A A A A A A A A A A A A A A A A A A A Received April 5, 2001; published i Joural of Iteger Sequeces, Jue 22, Retur to Joural of Iteger Sequeces home page. 11