by Jim Pitman Department of Statistics University of California March 31,1999 Abstract

Transcription

1 A lattice path model for the Bessel polyomials by Jim Pitma Techical Report No. 551 Departmet of Statistics Uiversity of Califoria 367 Evas Hall # 3860 Berkeley, CA March 31,1999 Abstract The (, 1)th Bessel polyomial is represeted by a expoetial geeratig fuctio derived from the umber of returs to 0 of a sequece with 2 icremets of 1 which starts ad eds at 0. AMS 1991 subject classicatio. Primary: 05A15. Secodary: 33C10, 33C45. It is well kow [21, x3.71 (12)],[6, (7.2(40)] that the McDoald fuctio or Bessel fuctio of imagiary argumet admits the evaluatio K (x) := 1 2 x 2, Z 1 0 t,1 e,t,(x=2)2 =t dt (1) K +1=2 (x) =r 2x e,x (x)x, ( =0; 1; 2;:::) (2) Research supported i part by N.S.F. Grat

2 where The Bessel polyomials (x) := m=0 ;,m x m with ;k := (x) ad y (x) := k=0 ( + k) 2 k (, k)k : (3) ;k x k = x (x,1 ) (4) have bee extesively studied ad applied: see the book of Grosswald [9] for a review. Dulucq ad Favreau [4, 5] gave a combiatorial model for the Bessel polyomials based o the remark that ;k = + k, k (2k, 1) (2k, 3) 1 is the umber of ivolutios of + k poits with, k xed poits ad k matched pairs of poits formig 2-cycles. Their model is similar to a well kow iterpretatio of the coeciets of the Hermite polyomials, whichwas exteded to q-hermite polyomials by Ismail, Stato ad Vieot [12]. Dulucq [3] treats a q-aalog of the Bessel polyomials. See also Leroux ad Strehl [13] for a model which iterprets the coeciets of Jacobi polyomials, ad Vieot [20] for other results i this vei. The purpose of this ote is to poit out a alterative combiatorial model for the Bessel polyomials, based o a expoetial geeratig fuctio derived from lattice path eumeratios. Call a sequece b =(b 0 ;b 1 ;:::;b 2 )alattice bridge of legth 2 if 2 b 0 = b 2 = 0 ad b i, b i,1 = 1 for every 1 i 2. Let B deote the set of all lattice bridges of legth 2. For b 2 B let r(b) betheumber of returs to 0 by b: The for each =1; 2;::: r(b) :=#fi :1 i 2 ad b i =0g: X b2b x r(b) r(b) = 2 x,1(x): (5) This formula ca be read from [15, Corollary 9], which gives various probabilistic expressios of the formula i terms of radom walks ad Browia motio. This approach coects formula (5) to the itegral represetatio (1) of K,1=2 (x), ad to formulae for geeralized Stirlig umbers due to Toscao [18, 19]. For 1 r let # ;r be the umber of lattice bridges of legth 2 with r returs to 0: # ;r := #fb 2 B : r(b) =rg: (6) 2

3 The (5) amouts via (3) to the formula which reduces to # ;r =2 r,1;,r (7) # ;r =2 r 2, r r 2, r : (8) This ca be read from Feller [7, III.7, Theorem 4]. Let # + ;r be the umber of o-egative lattice bridges of legth 2 with r returs to 0. The (8) is equivalet to # + ;r = 2, r r 2, r : (9) By the well kow bijectio of Harris [10] betwee betwee plae trees with vertices ad lattice excursios of legth 2, that is o-egative lattice bridges b of legth 2 with r(b) = 1, the umberi(9)istheumber of forests of r plae trees with vertices [14, (6.1)]. The particular case r = 1 of (8) is the stadard eumeratio # ;1 =2C,1 where C := 1 +1 is the th Catala umber [7], [17, Cor ]. The correspodig geeratig fuctio is well kow to be =1 # ;1 w =2 =1 2 (10) C,1 w =1, (1, 4w) 1=2 : (11) It was already oted by Carlitz [1] that a umber of results ivolvig the Bessel polyomials acquire their simplest form whe stated i terms of the polyomial x,1 (x) which features i (5). I particular, Carlitz gave the expoetial geeratig fuctio 1+ =1 x,1 (x) u 2 =exp[x(1, (1, u)1=2 )]: (12) Formula (5) may be regarded as a combiatorial expressio of the coectio betwee the Bessel polyomials ad the Catala umbers implied by (12) ad (11), exploitig (10) ad the decompositio of a lattice path with r returs to 0 ito its r excursios away from0. To express this i terms of geeratig fuctios, observe from (5) that # ;r is the coeciet of xr i 2 x r,1(x). Symbolically, usig the otatio of [17], x r 2 # ;r = r x,1(x): (13) 3

4 With similar otatio, Carlitz's idetity (12) ca be restated as x r u 2, x,1 (x) = (1, (1, u) 1=2 ) r : (14) r Accordig to a classical expasio of Lambert [8, (5.70)] [u ](1, (1, u) 1=2 ) r =2 r,2 2, r r 2, r : (15) Thus the form (14) of Carlitz's idetity (12) ca be read from (13), (8), ad (15). Alteratively, the lattice path represetatio (5) of the Bessel polyomials could be deduced via (13) from (8), the form (14) of Carlitz's idetity (12), ad (15). See also Roma [16, p. 78] ad Di Bucchiaico [2, p. 54] for closely related discussios based o the cosequece of (12) that the sequece of polyomials f (x) is of biomial type. Let (z j ) :=,1 Y i=0 (z, i) be the geeralized factorial with decremet, ad let S(; k; ; ) be the geeralized Stirlig umbers deed by (z j ) = k=0 S(; k; ; )(z j ) k : I particular, the S(; k; 1; 0) ad S(; k; 0; 1) are the classical Stirlig umbers of the rst ad secod kids respectively. See Hsu ad Shiue [11] for a recet review of the properties of these geeralized Stirlig umbers. Accordig to [11, (14)] the polyomials S ;; (x) := k=0 S(; k; ; )x k are determied for 6= 0 by the geeratig fuctio =0 " S ;; (x) t =exp x # (1 + t) =, 1 : (16) Compare (12) ad (16) for =,2; =,1 to deduce that for all 1 S ;,2;,1 (x) =x,1 (x) (17) 4

5 That is, from (3), S(; k;,2;,1) =,1;,k = (2, k, 1) 2,k (k, 1)(, k) : (18) This expressio for S(; k;,2;,1) is equivalet to a formula give without proof by Toscao [18, (122)],[19, (2.11)] alog with several other explicit evaluatios of geeralized Stirlig umbers. See also [15] for a probabilistic iterpretatio of the S(; k;,;,1) for arbitrary > 1 which yields asymptotic evaluatios of these umbers for large ad k. Refereces [1] L. Carlitz. A ote o the Bessel polyomials. Duke Math. J., 24:151{162, [2] A. Di Bucchiaico. Probabilistic ad aalytical aspects of the umbral calculus. Stichtig Mathematisch Cetrum, Cetrum voor Wiskude e Iformatica, Amsterdam, [3] S. Dulucq. U q-aalogue des poly^omes de Bessel. I Semiaire Lotharigie de Combiatoire (Sait-Nabor, 1992), pages 53{55. Uiv. Louis Pasteur, Strasbourg, [4] S. Dulucq ad L. Favreau. A combiatorial model for Bessel polyomials. I Orthogoal polyomials ad their applicatios (Erice, 1990), pages 243{249. Baltzer, Basel, [5] S. Dulucq ad L. Favreau. U modele combiatoire pour les poly^omes de bessel. I Semiaire Lotharigie de Combiatoire (Salzburg, 1990), pages 83{100. Uiv. Louis Pasteur, Strasbourg, [6] A. Erdelyi et al. Higher Trascedetal Fuctios, volume II of Batema Mauscript Project. McGraw-Hill, New York, [7] W. Feller. A Itroductio to Probability Theory ad its Applicatios, Vol 1,3rd ed. Wiley, New York, [8] R. L. Graham, D.E. Kuth, ad O. Patashik. Cocrete Mathematics: a foudatio for computer sciece. 2d ed. Addiso-Wesley, Readig, Mass., [9] E. Grosswald. Bessel polyomials. Spriger, Berli,

6 [10] T. E. Harris. First passage ad recurrece distributios. Tras. Amer. Math. Soc., 73:471{486, [11] L. C. Hsu ad P. J.-S. Shiue. A uied approach to geeralized Stirlig umbers. Adv. i Appl. Math., 20(3):366{384, [12] M. E. H. Ismail, D. Stato, ad G. Vieot. The combiatorics of q-hermite polyomials ad the Askey-Wilso itegral. Europea J. Combi., 8(4):379{392, [13] P. Leroux ad V. Strehl. Jacobi polyomials: combiatorics of the basic idetities. Discrete Math., 57(1-2):167{187, [14] J. Pitma. Eumeratios of trees ad forests related to brachig processes ad radom walks. I D. Aldous ad J. Propp, editors, Microsurveys i Discrete Probability, umber 41 i DIMACS Ser. Discrete Math. Theoret. Comp. Sci, pages 163{180, Providece RI, Amer. Math. Soc. [15] J. Pitma. Characterizatios of Browia motio, bridge, meader ad excursio by samplig at idepedet uiform times. Techical Report 545, Dept. Statistics, U.C. Berkeley, Available via [16] S. Roma. The Umbral Calculus. Academic Press, [17] R. Staley. Eumerative Combiatorics, Vol. 2. Cambridge Uiversity Press, [18] L. Toscao. Numeri di Stirlig geeralizzati operatori diereziali e poliomi ipergeometrici. Commetatioes Poticia Academica Scietarum, 3:721{757, [19] L. Toscao. Some results for geeralized Beroulli, Euler, Stirlig umbers. Fiboacci Quart., 16(2):103{112, [20] G. Vieot. Combiatorial theory for geeral orthogoal polyomials with extesios ad applicatios. I Poly^omes Orthogoaux et Applicatios, volume 1171 of Lecture Notes i Math., pages 139{157. Spriger, Berli-New York, [21] G.N. Watso. A treatise o the theory of Bessel fuctios. Cambridge: Uiversity Press,