A FAMILY OF SEQUENCES OF BINOMIAL TYPE WOJCIECH MLOTKOWSKI, ANNA ROMANOWICZ Abstract. For delta operator aD bDp+1 we find the corresponding polynomial se- − 1 2 quence of binomial type and relations with Fuss numbers. In the case D 2 D we show that the corresponding Bessel-Carlitz polynomials are moments of the− convolu- tion semigroup of inverse Gaussian distributions. We also find probability distributions νt, t> 0, for which yn(t) , the Bessel polynomials at t, is the moment sequence. { } 1. Introduction ∞ A sequence wn(t) n=0 of polynomials is said to be of binomial type (see [10]) if deg w (t)= n and{ for} every n 0 and s, t R we have n ≥ ∈ n n (1.1) w (s + t)= w (s)w − (t). n k k n k Xk=0 A linear operator Q of the form c c c (1.2) Q = 1 D + 2 D2 + 3 D3 + ..., 1! 2! 3! acting on the linear space R[x] of polynomials, is called a delta operator if c1 = 0. Here D denotes the derivative operator: D1 := 0 and Dtn := n tn−1 for n 1. We6 will write Q = g(D), where · ≥ c c c (1.3) g(x)= 1 x + 2 x2 + 3 x3 + .... 1! 2! 3! There is one-to-one correspondence between sequences of binomial type and delta ∞ operators, namely if Q is a delta operator then there is unique sequence w (t) of { n }n=0 binomial type satisfying Qw0(t) = 0 and Qwn(t)= n wn−1(t) for n 1. These wn are called basic polynomials for Q. · ≥ A natural way of obtaining a sequence of binomial type is to start with a function f which is analytic in a neighborhood of 0: a a a (1.4) f(x)= 1 x + 2 x2 + 3 x3 + ..., arXiv:1508.00138v1 [math.PR] 1 Aug 2015 1! 2! 3! with a1 = 0. Then a binomial sequence appears in the Taylor expansion of exp (t f(x)), namely 6 · ∞ w (t) (1.5) exp (t f(x)) = n xn · n! n=0 X 2010 Mathematics Subject Classification. Primary 05A40; Secondary 60E07, 44A60. Key words and phrases. Sequence of binomial type, Bessel polynomials, inverse Gaussian distribution. W. M. is supported by the Polish National Science Center grant No. 2012/05/B/ST1/00626, A. R. is supported by the Grant DEC-2011/02/A/ST1/00119 of National Centre of Science. 1 2 WOJCIECH MLOTKOWSKI, ANNA ROMANOWICZ and the coefficients of wn are partial Bell polynomials of a1, a2,.... More generally, we can merely assume that f (as well as g) is a formal power series. 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 generating function ∞ 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 convolution semigroups of probability measures on the real line. In this part we describe an example 1 2 of such semigroup, which corresponds to the delta operator 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 probability distribution 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).