arXiv:1508.00138v1 [math.PR] 1 Aug 2015 sspotdb h rn E-010//T/01 fNationa of DEC-2011/02/A/ST1/00119 Grant the by supported is distribution. deg (1.1) (1.2) with hc saayi nanihoho f0: of (1.4) neighborhood a in analytic is which if namely operators, Q D (1.3) cigo h ierspace linear the on acting 15 x ( exp (1.5) namely ioiltp satisfying type binomial called sequence A e od n phrases. and words Key 2010 hr soet-n orsodnebtensqecso bino of sequences between correspondence one-to-one is There ieroperator linear A aua a fotiigasqec fbnma yei osatwit start to is type binomial of sequence a obtaining of way natural A = eoe h eiaieoperator: derivative the denotes .M sspotdb h oihNtoa cec etrgatN.2 No. grant Center Science National Polish the by supported is M. W. w g a n ( 1 ai basic ( ahmtc ujc Classification. Subject Mathematics D insmgopo nes asindsrbtos eas n probab find also We ν distributions. are Gaussian inverse polynomials of semigroup Bessel-Carlitz tion corresponding the that show Abstract. uneo ioiltp n eain ihFs ubr.I h cas the In numbers. Fuss with relations and type binomial of quence t .Te ioilsqec per nteTyo xaso fexp of expansion Taylor the in appears sequence binomial a Then 0. 6= t = ) , ,where ), > t AIYO EUNE FBNMA TYPE BINOMIAL OF SEQUENCES OF FAMILY A n ,frwhich for 0, n o every for and { w n o et operator delta For ( OCEHMLTOSI NAROMANOWICZ ANNA M LOTKOWSKI, WOJCIECH t Q ) Q } n ∞ fteform the of sadlaoeao hnteei nqesequence unique is there then operator delta a is =0 { for Qw y euneo ioiltp,Bse oyoil,ivreGaussian inverse polynomials, Bessel type, binomial of Sequence n w R f ( fplnmasi adt eof be to said is polynomials of g Q t n [ ( Q 0 ) ( n x ( x } ( x s fplnmas scle a called is polynomials, of ] = . h eslplnmasat polynomials Bessel the , t = ) ≥ = ) and 0 = ) + c 1! and 0 1 t 1. aD = ) t a 1! D c 1! D 1 1 · Introduction x x f =0and 0 := 1 − + rmr 54;Scnay6E7 44A60. 60E07, Secondary 05A40; Primary ( X k + + bD ,t s, x =0 n c 2! )= )) 2 Qw a c 2! 2! p D  2 +1 2 ∈ 1 x x 2 n k n 2 2 efidtecrepnigplnma se- corresponding the find we R +  X n ( + ∞ + =0 t w ehave we = ) c 3! Dt k c 3! 3 a 3! w ( 3 D 3 s x n n n x n ) 3 3 ( ! w 3 := t + · + t ) + n stemmn sequence. moment the is , w x − , . . . n . . . . n n , . . . k eteo Science. of Centre l − · ( et operator delta t t 1 n ) ( . ioiltype binomial − t for ) 1 1/5BS1066 .R. A. 012/05/B/ST1/00626, for lt distributions ility fteconvolu- the of s n iltp n delta and type mial n e ≥ ≥ D .These 1. .W ilwrite will We 1. − if function a h 2 { 1 D c w se[0)if [10]) (see 1 2 n .Here 0. 6= ( ( we t t ) · } w f n ∞ n =0 ( x are )), of f 2 WOJCIECH MLOTKOWSKI, ANNA ROMANOWICZ and the coefficients of wn are partial of a1, a2,.... More generally, we can merely assume that f (as well as g) is a formal . Then f is the composition inverse of g: f(g(x)) = g(f(x)) = x. The aim of the paper is to describe the sequences of polynomials of binomial type, which correspond to delta operators of the form Q = aD bDp+1. Then we discuss the 1 2 − special case, D 2 D , studied by Carlitz [3]. We find the corresponding semigroup of probability distributions− an also the family of distributions corresponding to the Bessel polynomials.

2. The result Theorem 2.1. For a = 0, b R and for p 1 let Q := aD bDp+1. Then the basic polynomials are as follows:6 w∈(t)=1 and for≥n 1 − 0 ≥ n−1 [ p ] j (n + j 1)!b − (2.1) w (t)= − tn jp. n j!(n jp 1)!an+j j=0 X − − In particular w1(t)= t/a.

Proof. We have to show that if n 1 then Qwn(t)= n wn−1(t). It is obvious for n = 1, so assume that n 2. Then we have≥ · ≥ n−1 [ p ] j (n + j 1)!(n jp)b − − aDw (t)= − − tn 1 jp n j!(n jp 1)!an−1+j j=0 X − − n−1 n−1 [ p ] j t (n + j 1)!(n jp)b − − = n + − − tn 1 jp a j!(n jp 1)!an−1+j j=1   X − − and n−p−1 [ p ] j+1 (n + j 1)!(n jp)b − − − bDp+1w (t)= − − tn 1 jp p. n j!(n jp p 1)!an+j j=0 X − − − Now we substitute j′ := j + 1:

n−1 [ p ] ′ ′ j′ p+1 (n + j 2)!(n j p + p)b n−1−j′p bD wn(t)= ′ − ′ − n−1+j′ t . ′ (j 1)!(n j p 1)!a Xj =1 − − − If j 1 and jp +1 < n then ≥ (n + j 1)!(n jp) (n + j 2)!(n jp + p) − − − − j!(n jp 1)! − (j 1)!(n jp 1)! − − − − − (n + j 2)! (n + j 1)(n jp) j(n jp + p) = − − − − − j!(n jp 1)!  − −  n(n + j 2)!(n jp 1) n(n + j 2)! = − − − = − j!(n jp 1)! j!(n jp 2)! − − − − A FAMILY OF SEQUENCES OF BINOMIAL TYPE 3 and if jp +1= n then this difference is 0. Therefore we have

n−2 n−1 [ p ] j t n(n + j 2)!b − − aD bDp+1 w (t)= n + − tn 1 jp − n a j!(n jp 2)!an−1+j   j=1 − −  X

n−2 [ p ] j (n + j 2)!b n−1−jp = n − t = n w − (t), j!(n jp 2)!an−1+j · n 1 j=0 X − − which concludes the proof. 

np+1 1 Recall, that Fuss numbers of order p are given by n np+1 and the corresponding  ∞ np +1 xn (2.2) (x) := Bp n np +1 n=0 X   is determined by the equation

(2.3) (x)=1+ x (x)p. Bp · Bp In particular

1 √1 4x (2.4) (x)= − − . B2 2x For more details, as well as combinatorial applications, we refer to [4]. Now we can exhibit the function f corresponding to the operator aD bDp+1 and to the polynomials (2.1). −

Corollary 2.1. For the polynomials (2.1) we have

∞ w (t) exp (t f(x)) = n xn, · n! n=0 X where x bxp (2.5) f(x)= . a · Bp+1 ap+1   Proof. Since f is the inverse function for g(x) := ax bxp+1, it satisfies af(x) = x + bf(x)p+1, which is equivalent to −

af(x) bxp af(x) p+1 =1+ . x ap+1 x     p p+1 Comparing with (2.3), we see that af(x)/x = p+1 (bx /a ). B ′  Alternatively, we could apply the formula an = wn(0) for the coefficients in (1.4). 4 WOJCIECH MLOTKOWSKI, ANNA ROMANOWICZ

3. The operator D 1 D2 − 2 One important source of sequences of binomial type are moments of semigroups of probability measures on the real line. In this part we describe an example 1 2 of such semigroup, which corresponds to the D 2 D . For more details we refer to [3, 5, 7, 10] and to the entry A001497 in [11]. − In view of (2.1), the sequence of polynomials of binomial type corresponding to the delta operator D 1 D2 is − 2 − n 1 (n + j 1)! tn−j n (2n k 1)! tk (3.1) w (t)= − = − − , n j!(n j 1)!2j (n k)!(k 1)!2n−k j=0 X − − Xk=1 − − with w0(t) = 1. They are related to the Bessel polynomials n j (n + j)! t 1/t 2 (3.2) y (t)= = e K− − (1/t) , n j!(n j)! 2 πt n 1/2 j=0 r X −   where Kν(z) denotes the modified Bessel function of the second kind. Namely, for n 1 we have ≥

n n t 2t (3.3) wn(t)= t yn−1 (1/t)= t e K1/2−n(t) r π for n 0. Applying≥ Corollary 2.1 and (2.4) we get ∞ a (3.4) f(x)= x (x/2)=1 √1 2x = n xn, · B2 − − n! n=1 X where an = (2n 3)!! . The function f(xi) admits Kolmogorov representation (see formula (7.15) in− [12]) as +∞ e−u/2 (3.5) 1 √1 2xi= xi+ euxi 1 uxi du, 3 − − 0 − − √2πu Z  with the probability density function √ue−u/2/√2π on [0, + ). Therefore ∞ φ(x) = exp 1 √1 2xi − − is the characteristic function of some infinitely divisible probability measure. It turns out that the corresponding convolution semigroup µt t>0 is contained in the family of inverse Gaussian distributions, see 24.3 in [1]. { }

Theorem 3.1. For t> 0 define µt := ρt(u) du, where

− − 2 t exp (u t) · 2u (3.6) ρt(u) := √2πu3  for u > 0 and ρt(u) = 0 for u 0. Then µt t>0 is a convolution semigroup, ≤ { } ∞ exp t t√1 2xi is the characteristic function of µt, and wn(t) n=0, defined by (3.1),− is the moment− sequence of µ . { }  t A FAMILY OF SEQUENCES OF BINOMIAL TYPE 5

Proof. It is sufficient to check moments of µt. Substituting u := 2v and applying formula: 1 t p ∞ t2 dv (3.7) K (t)= exp v p 2 2 − − 4v vp+1   Z0   (see (10.32.10) in [9]), we obtain −(u−t)2 ∞ t exp t ∞ 2 2u te u t − un · du = exp − un 3/2 du 3 0 √2πu  √2π 0 2 − 2u Z Z   t n−1 ∞ 2 t n−1 n−1/2 te 2 t n−3/2 te 2 t = exp v v dv = 2 K − (t), √π − − 4v √π 2 1/2 n Z0     which, by (3.3), is equal to wn(t).  4. Probability measures corresponding to the Bessel polynomials In this part we are going to give some remarks concerning Bessel polynomials (3.2). First we compute the exponential generating function (cf. formula (6.2) in [5]).

Proposition 4.1. For the exponential generating function of the sequence yn(t) we have { } ∞ y (t) exp 1 1 √1 2tx (4.1) n xn = t − t − . n! √1 2tx n=0  X − Proof. By (3.3) we have ∞ ∞ ∞ y (t) tn+1w (1/t) tnw (1/t) n xn = n+1 xn = n xn−1 n! n! (n 1)! n=0 n=0 n=1 − X ∞ X X d tnw (1/t) d 1 1 = n xn = exp √1 2tx , dx n! dx t − t − n=0 ! X   which leads to (4.1).  Now we represent the Bessel polynomials (3.2) as a moment sequence. Theorem 4.1. For n 0 and t> 0 we have ≥ − − 2 ∞ exp (u 1) n 2tu (4.2) yn(t)= u du. √2πtu  Z0 Proof. Substituting u := 2tv, applying (3.7) and (3.2) we get: −(u−1)2 ∞ exp 1/t ∞ 2tu e − u 1 un du = un 1/2 exp − du. √2πtu  √2πt 2t − 2tu Z0 Z0   e1/t(2t)n+1/2 ∞ 1 = vn−1/2 exp v dv √2πt − − 4t2v Z0   1/t 2 = e K−n−1/2 (1/t)= yn(t), rπt which concludes the proof. Alternatively, we could apply Theorem 3.1 and relation (3.3).  6 WOJCIECH MLOTKOWSKI, ANNA ROMANOWICZ

Denote by νt the corresponding probability measure, i.e.

−(u−1)2 exp 2tu (4.3) νt := χ(0,+∞)(u) du. √2πtu 

Although νt t>0 is not a convolution semigroup, we will see that every νt is infinitely divisible. { } Theorem 4.2. For t> 0 we have (4.4) ν = γ D µ , t t ∗ t 1/t where γt denotes the gamma distribution with shape 1/2 and scale 2t: −u exp 2t (4.5) γt = χ(0,+∞)(u) du √2πtu and Dtµ1/t is the dilation of µ1/t by t:

−(u−1)2 exp 2tu (4.6) Dtµ1/t = χ(0,+∞)(u) du. √2πtu3 

In particular, νt is infinitely divisible.

Proof. From Theorem 4.1 we see that the characteristic function of νt: 1 1 √ exp t t 1 2txi (4.7) ψt(x)= − − , √1 2txi −  is the product of 1 , √1 2txi − the characteristic function of γt and 1 1 exp √1 2txi , t − t −   the characteristic function of Dtµ1/t, which proves (4.4). Since both γt and Dtµ1/t are infinitely divisible, so is their convolution νt.  Let us list some interesting integer sequences which arise from the polynomials (3.1) and (3.2), together with their numbers in the On-line Encyclopedia of Integer Sequences [11] and the corresponding probability distribution. For their combinatorial applications we refer to [11]:

(1) A144301: wn(1), moments of µ1, (2) A107104: wn(2), moments of µ2, (3) A043301: wn+1(2)/2, moments of the density function u ρ2(u)/2, (4) A080893: 2n w (1/2), moments of the density function ρ· (u/2) /2, · n 1/2 (5) A001515: yn(1), moments of ν1, (6) A001517: yn(2), moments of ν2, (7) A001518: yn(3), moments of ν3, (8) A065919: yn(4), moments of ν4. A FAMILY OF SEQUENCES OF BINOMIAL TYPE 7

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