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Generalized Bell Polynomials and the Combinatorics of Poisson Central Moments

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Generalized Bell Polynomials and the Combinatorics of Poisson Central Moments Generalized Bell polynomials and the combinatorics of Poisson central moments Nicolas Privault∗ Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University SPMS-MAS-05-43, 21 Nanyang Link Singapore 637371 February 25, 2011 Abstract We introduce a family of polynomials that generalizes the Bell polynomials, in connection with the combinatorics of the central moments of the Poisson dis- tribution. We show that these polynomials are dual of the Charlier polynomials by the Stirling transform, and we study the resulting combinatorial identities for the number of partitions of a set into subsets of size at least 2. Key words: Bell polynomials, Poisson distribution, central moments, Stirling num- bers. Mathematics Subject Classification: 11B73, 60E07. 1 Introduction The moments of the Poisson distribution are well-known to be connected to the com- binatorics of the Stirling and Bell numbers. In particular the Bell polynomials Bn(λ) satisfy the relation n Bn(λ) = Eλ[Z ]; n 2 IN; (1.1) where Z is a Poisson random variable with parameter λ > 0, and n X Bn(1) = S(n; c) (1.2) c=0 ∗[email protected] 1 is the Bell number of order n, i.e. the number of partitions of a set of n elements. In this paper we study the central moments of the Poisson distribution, and we show that they can be expressed using the number of partitions of a set into subsets of size at least 2, in connection with an extension of the Bell polynomials. Consider the above mentioned Bell (or Touchard) polynomials Bn(λ) defined by the exponential generating function 1 n t X t eλ(e −1) = B (λ); (1.3) n! n n=0 λ, t 2 IR, cf. e.g. x11.7 of [4], and given by the Stirling transform n X c Bn(λ) = λ S(n; c); (1.4) c=0 where c 1 X c S(n; c) = (−1)c−l ln (1.5) c! l l=0 denotes the Stirling number of the second kind, i.e. the number of ways to partition a set of n objects into c non-empty subsets, cf. x 1.8 of [7], Proposition 3.1 of [3] or x 3.1 of [6], and Relation (1.2) above. In this note we define a two-parameter generalization of the Bell polynomials, which is dual to the Charlier polynomials by the Stirling transform. We study the links of these polynomials with the combinatorics of Poisson central moments, cf. Lemma 3.1, and as a byproduct we obtain the binomial identity n X m S (m; n) = (−1)k S(m − k; n − k); (1.6) 2 k k=0 where S2(n; a) denotes the number of partitions of a set of size n into a subsets of size at least 2, cf. Corollary 3.2 below, which is the binomial dual of the relation n X m S(m; n) = S (m − k; n − k); k 2 k=0 2 cf. Proposition 3.3 below. We proceed as follows. Section 2 contains the definition of our extension of the Bell polynomials. In Section 3 we study the properties of the polynomials using the Poisson central moments, and we derive Relation (1.6) as a corollary. Finally in Section 4 we state the connection between these polynomials and the Charlier polynomials via the Stirling transform. 2 An extension of the Bell polynomials We let (Bn(x; λ))n2IN denote the family of polynomials defined by the exponential generating function 1 n t X t ety−λ(e −t−1) = B (y; λ); λ, y; t 2 IR: (2.1) n! n n=0 Clearly from (1.3) and (2.1), the definition of Bn(x; λ) generalizes that of the Bell polynomials Bn(λ), in that Bn(λ) = Bn(λ, −λ); λ 2 IR: (2.2) When λ > 0, Relation (2.1) can be written as 1 X tn ety IE [et(Z−λ)] = B (y; −λ); y; t 2 IR; λ n! n n=0 which yields the relation n Bn(y; −λ) = Eλ[(Z + y − λ) ]; λ, y 2 IR; n 2 IN; (2.3) which is analog to (1.1), and shows the following proposition. Proposition 2.1 For all n 2 IN we have n k X n X B (y; −λ) = (y − λ)n−k λiS(k; i); y; λ 2 IR; n 2 IN: (2.4) n k k=0 i=0 3 Proof. Indeed, by (2.3) we have n Bn(y; −λ) = Eλ[(Z + y − λ) ]; n X n = (y − λ)n−kE [Zk] k λ k=0 n X n = (y − λ)n−kB (λ) k k k=0 n k X n X = (y − λ)n−k λiS(k; i); y; λ 2 IR: k k=0 i=0 3 Combinatorics of the Poisson central moments As noted in (1.1) above, the connection between Poisson moments and polynomials is well understood, however the Poisson central moments seem to have received less attention. In the sequel we will need the following lemma, which expresses the central moments of a Poisson random variable using the number S2(n; b) of partitions of a set of size n into b subsets with no singletons. Lemma 3.1 Let Z be a Poisson random variable with intensity λ > 0. We have n n X a Bn(0; −λ) = Eλ[(Z − λ) ] = λ S2(n; a); n 2 IN: (3.1) a=0 Proof. We start by showing the recurrence relation n−1 X n E [(Z − λ)n+1] = λ E (Z − λ)i ; n 2 IN; (3.2) λ i λ i=0 for Z a Poisson random variable with intensity λ. We have 1 X λk E [(Z − λ)n+1] = e−λ (k − λ)n+1 λ k! k=0 4 1 1 X λk X λk = e−λ (k − λ)n − λe−λ (k − λ)n (k − 1)! k! k=1 k=0 1 X λk = λe−λ ((k + 1 − λ)n − (k − λ)n) k! k=0 1 n−1 X λk X n = λe−λ (k − λ)i k! i k=0 i=0 n−1 1 X n X λk = λe−λ (k − λ)i i k! i=0 k=0 n−1 X n = λ E [(Z − λ)i]: i λ i=0 Next, we show that the identity n−1 a n X a X Y kl+1 − 1 Eλ[(Z − λ) ] = λ (3.3) kl a=1 0=k1···ka+1=n l=1 holds for all n ≥ 1, where a b means a < b − 1. Note that the degree of (3.3) in λ is the largest integer d such that 2d ≤ n, hence it equals n=2 or (n − 1)=2 according to the parity of n. Clearly, the identity (3.3) is valid when n = 1 and when n = 2. Assuming that it holds up to the rank n ≥ 2, from (3.2) we have n−1 X n E [(Z − λ)n+1] = λ E (Z − λ)k λ k λ k=0 n−1 X n = λ + λ E (Z − λ)k k λ k=1 n−1 k−1 b X n X X Y kl+1 − 1 = λ + λ λb k kl k=1 b=1 0=k1···kb+1=k l=1 n−1 k b−1 X n X X Y kl+1 − 1 = λ + λ λb−1 k kl k=1 b=2 0=k1···kb=k l=1 n−1 kb b−1 X n X X Y kl+1 − 1 = λ + λ λb−1 kb kl kb=1 b=2 0=k1···kb l=1 5 n−1 kb b X X X Y kl+1 − 1 = λ + λ λb−1 kl kb=1 b=2 0=k1···kbkb+1=n+1 l=1 n kb b X X X Y kl+1 − 1 = λ + λ λb−1 kl kb=1 b=2 0=k1···kbkb+1=n+1 l=1 n b X X Y kl+1 − 1 = λ + λb kl b=2 0=k1···kb+1=n+1 l=1 n b X X Y kl+1 − 1 = λb ; kl b=1 0=k1···kb+1=n+1 l=1 and it remains to note that b X Y kl+1 − 1 = S2(n; b) (3.4) kl 0=k1···kb+1=n l=1 equals the number S2(n; b) of partitions of a set of size n into b subsets of size at least 2. Indeed, any contiguous such partition is determined by a sequence of b − 1 o integers k2; : : : ; kb with 2b ≤ n and 0 k2 · · · kb n so that subset n i has size ki+1 − ki ≥ 2, i = 1; : : : ; b, with kb+1 = n, and the number of not necessarily contiguous partitions of that size can be computed inductively on i = 1; : : : ; b as b n − 1 kb − 1 k2 − 1 Y kl+1 − 1 ··· = : n − 1 − kb kb − 1 − kb−1 k2 − 1 − k1 kl l=1 For this, at each step we pick an element which acts as a boundary point in the subset o n i, and we do not count it in the possible arrangements of the remaining ki+1 −1−ki elements among ki+1 − 1 places. Lemma 3.1 and (3.4) can also be recovered by use of the cumulants (κn)n≥1 of Z − λ, defined from the cumulant generating function 1 X tn log E [et(Z−λ)] = λ(et − 1) = κ ; λ n n! n=1 i.e.
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