Identities for partial Bell polynomials derived from identities for weighted integer compositions

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by  Steffen Eger

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Identities for partial Bell polynomials de-
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rived from identities for weighted integer com-
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positions
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t Steffen Eger
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Abstract. Wediscussclosed-form formulas forthe(n,k)-thpartialBell

1 polynomialsderivedinCvijovi´c[2].WeshowthatpartialBellpolynomi-
v als arespecial cases of weighted integer compositions, anddemonstrate
7 how the identities for partial Bell polynomials easily follow from more
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general identities for weighted integer compositions. We also provide
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short and elegant probabilistic proofs of the latter, in terms of sums
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of discrete integer-valued random variables. Finally, we outline further
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identities for the partial Bell polynomials.
.
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Mathematics Subject Classification (2010). Primary 11P81; Secondary
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60C05.
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1 Keywords.Integercomposition,PartialBellPolynomial,Sumofdiscrete
: random variables.
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1. Introduction
In a recent note, Cvijovi´c [2] has derived three new identities — which we
list in Equations (5), (6), and (7) below — for the (n,k)-th partial Bell
polynomials in the variables x ,x ,...,x , which are defined as1
1 2 n−k+1
B (x ,x ,...,x )=
n,k 1 2 n−k+1
n! x1 ℓ1··· xn−k+1 ℓn−k+1, (1)
Xℓ1!···ℓn−k+1!(cid:16)1!(cid:17) (cid:18)(n−k+1)!(cid:19)
The final publication is available at Springer via
http://dx.doi.org/10.1007/s00010-015-0338-2
1AnotherwaytodefinethepartialBellpolynomialsisbytheformalpowerseriesexpansion
1 xmtm k = Bn,k(x1,x2,...,xn−k+1)tn. This wouldalsoimmediatelylead
k!(cid:16)Pm≥1 m! (cid:17) Pn≥k n!
toCorollary1belowifwedefinedthesubsequentconceptofweightedintegercompositions
k
by the formal power series expansion f(m)tm = k tn, with notation
(cid:16)Pm≥0 (cid:17) Pn≥0(cid:0)n(cid:1)f
asexplicated below.
2 Eger
wherethesummationisoverallsolutionsinnonnegativeintegersℓ ,...,ℓ
1 n−k+1
of ℓ +2ℓ +···+(n−k+1)ℓ = n and ℓ +ℓ +···+ℓ = k.
1 2 n−k+1 1 2 n−k+1
Such identities may be important for the efficient evaluation of the partial
Bellpolynomials,ashintedatinCvijovi´c[2],whoalsoindicates applications
ofthepolynomials.Importantly,theygeneralizetheStirlingnumbersS(n,k)
of the second kind, since S(n,k)=B (1,...,1).
n,k
The purpose of the present note is to show that these identities are
special cases of identities for weighted integer compositions and to outline
very short proofs for the more general identities, based on sums of discrete
random variables. Our main point is to compile (identical or very similar)
resultsdevelopedwithinheterogeneouscommunitiesofresearch,andtocom-
binetheseresults.Finally,inSection4,wepresentthreeadditionalidentities
for the partial Bell polynomials. Throughout, we write B as a shorthand
n,k
for B (x ,...,x ), unless explicit reference to the indeterminates is
n,k 1 n−k+1
critical.
2. Preliminaries on weighted integer compositions
An integer composition of a nonnegative integer n is a k-tuple (π ,...,π ),
1 k
for k ≥ 1, of nonnegative integers such that π + ··· + π = n. We call
1 k
k the number of parts. Note that order of part matters, and this distin-
guishes integer compositions from integer partitions. Now, to generalize, we
may consider f-colored integer compositions, where each possible part size
s ∈ N = {0,1,2,...} may come in f(s) different colors, whereby f : N → N.
For example, the integer n = 4 has nine distinct f-colored integer composi-
tions, for f(0) = 2, f(1) = f(2) = 1, f(3) = f(4) = f(5) = ··· = 0, with
k =3 parts, namely,
(2,2,0),(2,2,0∗),(2,1,1),(1,2,1),(2,0,2),(2,0∗,2),(1,1,2),(0,2,2),(0∗,2,2),
whereweuse astarsuperscripttodistinguishthe twodifferentcolorsofpart
size 0. For weighted integer compositions we let, more generally, f(s) be an
arbitrary real number (or even a value in a commutative ring), the weight
of partsize s. Coloredinteger compositions have been discussed in [6, 9, 10],
and weighted integer compositions have been under review in Eger [4], but
have been investigated as early as Hoggatt and Lind’s [7] work.
Let k denote the number of f-weighted integer compositions of n
n f
with k pa(cid:0)rts(cid:1), when the range of f is N, and let k denote the total weight
n f
of all f-weighted integer compositions of n with(cid:0)k(cid:1)parts, when the range of
f is the set of reals R. We then have the following theorem.
Identities for partial Bell polynomials 3
Theorem 1. Let k,n ≥ 0 be integers, and let f : N → R be arbitrary. Then
the following identities hold.
k
= f(π )···f(π ), (2)
(cid:18)n(cid:19) 1 k
f X
k k
= f(0)ℓ0···f(n)ℓn, (3)
(cid:18)n(cid:19) (cid:18)ℓ ,ℓ ,...,ℓ (cid:19)
f X 0 1 n
where the sum in (2) is over all solutions in nonnegative integers π ,...,π
1 k
of π +···+π = n, and the sum in (3) is over all solutions in nonnegative
1 k
integers ℓ ,...,ℓ of ℓ +···+ℓ =k and 0ℓ +1ℓ +···+nℓ =n. Finally,
0 n 0 n 0 1 n
k = k! denote the multinomial coefficients.
ℓ0,ℓ1,...,ℓn ℓ0!ℓ1!···ℓn!
(cid:0) (cid:1)
The proofofTheorem1 is simple. Identity (2) is a direct applicationof
the definition of k and (3) follows, combinatorially, from (2) by rewriting
n f
the summationo(cid:0)ve(cid:1)rinteger compositionsinto asummationoverintegerpar-
titions andthen adjustingeachterminthe sumappropriately(inparticular,
the multinomial coefficients accountfor distributing the parts in partitions).
Alternatively, when k is defined as coefficient of a certain polynomial,
n f
then (3) can also be(cid:0)ar(cid:1)rived at via application of the multinomial theorem
(see [4]) orthe formulaofFaa` diBruno[8]forthe higherorderderivativesof
composite functions (see [7]).
Interpreting f(s), for s∈N, as indeterminates, Theorem 1 identity (3)
immediatelyimpliesthatf-weightedintegercompositions‘generalize’partial
Bell polynomials.
Corollary 1. Let k,n≥0 be integers. Then:
k! k
B (x ,x ,...,x )= , (4)
n! n,k 1 2 n−k+1 (cid:18)n(cid:19)
f
whereby
f(0)=f(n−k+2)=f(n−k+3)=···=0,
x
f(s)= s, for s∈{1,2,...,n−k+1}.
s!
Corollary1 has been established in [1] in the more particular setting of
restrictedinteger compositions,or,classical‘multinomial/extendedbinomial
coefficients’, where, in our notation, f(s)=1 for all s∈{1,...,q}, for some
positive integer q. This yielded the conclusion that partial Bell polynomials
‘generalize’restrictedintegercompositionsinsofarasB (1!,2!,...,q!,0,...)=
n,k
n! k , where f is the indicator function on {1,...,q}.
k! n f
(cid:0) (cid:1)Representation (2) in Theorem 1 is very useful on its own since it cap-
tures an equivalence between f-weighted integer compositions and distri-
butions of sums of independent and identically distributed (i.i.d.) discrete
random variables as explicated in the following lemma.
Lemma1. Letk,n≥0beintegersandletf :N→Rbearbitrary.Thenthere
existi.i.d.nonnegativeinteger-valuedrandomvariablesX ,...,X suchthat
1 k
4 Eger
k is given as the distribution of the sum of X ,...,X (times a suitable
n f 1 k
(cid:0)no(cid:1)rmalization factor).
Conversely, the distribution P [X +···+X = n] of arbitrary i.i.d.
f 1 k
nonnegative integer-valued random variables X ,...,X with common dis-
1 k
tribution function f is given by k .
n f
(cid:0) (cid:1)
Proof. Consider k . When f is zero almost everywhere, i.e., f(s) = 0 for
n f
all s > x, for som(cid:0)e(cid:1)x ∈ N, then let F¯ denote the sum f(s′) and let
s′∈N
g(s) = f(s) for all s ∈ N. Consider the i.i.d. random vPariables X ,...,X
F¯ 1 k
with common distribution function g, i.e., P[X = s] = g(s). By definition,
i
the distribution of the sum X +···+X is given as
1 k
P [X +···+X =n]= P[X =π ]···P[X =π ]
g 1 k 1 1 k k
π1+·X··+πk=n
= g(π )···g(π )
1 k
π1+·X··+πk=n
1 1 k
= f(π )···f(π )= .
F¯n 1 k F¯n(cid:18)n(cid:19)
π1+·X··+πk=n f
When f is not zero almost everywhere, then note that
k = k =F¯ˆnP [Y +···+Y =n],
(cid:18)n(cid:19) (cid:18)n(cid:19) gˆ 1 k
f fˆ
whereby fˆ(s) = f(s) for s ≤ n and fˆ(s) = 0 for s > n, and Y ,...,Y are
1 k
i.i.d.randomvariableswithcommondistributionfunctiongˆ(s)= fˆ(s),where
F¯ˆ
F¯ˆ = fˆ(s′).
s′∈N
PConversely, the distribution of the sum of i.i.d. nonnegative integer-
valued random variables X ,...,X with common distribution function f is
1 k
given by, as above,
P [X +···+X =n]= P[X =π ]···P[X =π ]
f 1 k 1 1 k k
π1+·X··+πk=n
k
= f(π )···f(π )= ,
1 k (cid:18)n(cid:19)
π1+·X··+πk=n f
applying Theorem 1 in the last equality. (cid:3)
Lemma 1 has appearedin [4] and,in the specialcase when f is the dis-
crete uniform measure, in [1]. Lemma 1 allows us to prove properties of the
weightedintegercompositions k byreferringtopropertiesofthe distribu-
n f
tion ofthe sum of discrete rand(cid:0)om(cid:1) variables,whichis oftentimes convenient,
as we shall see below.
Identities for partial Bell polynomials 5
3. Identities and proofs
The three identities for partial Bell polynomials that Cvijovi´c [2] introduces
are the following:
1 1 n−k n n+1
B = (k+1)− x B , (5)
n,k x n−k (cid:18)α(cid:19)(cid:20) α+1(cid:21) α+1 n−α,k
1 αX=1
n
k !k ! n
B = 1 2 B B , (6)
n,k1+k2 (k +k )! (cid:18)α(cid:19) α,k1 n−α,k2
1 2 αX=0
1 n−1 αk−1−1 n α
B = ··· ··· k−1 x ···x x .
n,k+1 (k+1)! (cid:18)α (cid:19) (cid:18) α (cid:19) n−α1 αk−1−αk αk
αX1=k αXk=1 1 k
(7)
As shown in Cvijovi´c [2], (7) easily follows inductively from (6). Therefore,
we concentrate on the identities (5) and (6). Throughout, we let k,n be
nonnegative integers and f be a function f : N → R. As we will see now,
identity (6) may be seen as a special case of the convolution formula for the
sum of discrete random variables.
Lemma 2. Let n, k and k be nonnegative integers. Then:
1 2
k +k k k
1 2 = 1 2 .
(cid:18) n (cid:19) (cid:18)x(cid:19) (cid:18)y(cid:19)
f x+Xy=n f f
Proof. By our previousdiscussion, it suffices to provethe lemma for sums of
i.i.d. integer-valued random variables X ,...,X ,Y ,...,Y . Now,
1 k1 1 k2
P [(X +···+X )+(Y +···+Y )=n]
f 1 k1 1 k2
= P [X +···+X =x]P [Y +···+Y =y]
f 1 k1 f 1 k2
x+Xy=n
by the discrete convolution formula for discrete random variables.
Toformallycompletetheproof,2 weapplyLemma1(or,moreprecisely,
its proof), leading to:
k1+k2 = k1+k2 =F¯ˆnP [X +···+X =n]
(cid:18) n (cid:19) (cid:18) n (cid:19) gˆ 1 k1+k2
f fˆ
=F¯ˆn P [X +···+X =x]P [Y +···+Y =y]
gˆ 1 k1 gˆ 1 k2
x+Xy=n
= F¯ˆxP [X +···+X =x]F¯ˆyP [Y +···+Y =y]
gˆ 1 k1 gˆ 1 k2
x+Xy=n
k k k k
= 1 2 = 1 2 .
(cid:18)x(cid:19) (cid:18)y(cid:19) (cid:18)x(cid:19) (cid:18)y(cid:19)
x+Xy=n fˆ fˆ x+Xy=n f f
(cid:3)
2Weomitthisstraightforwardstepinallsubsequent proofs.
6 Eger
Proof of identity (6). Using Lemma 2 and Corollary 1, we obtain that
n! k +k n! k k
B = 1 2 = 1 2
n,k1+k2 (k +k )!(cid:18) n (cid:19) (k +k )! (cid:18)x(cid:19) (cid:18)y(cid:19)
1 2 f 1 2 x+Xy=n f f
n! k ! k !
= 1 B 2 B
(k +k )! x! x,k1 y! y,k2
1 2 x+Xy=n
k !k ! n!
= 1 2 B B
(k +k )! x!y! x,k1 y,k2
1 2 x+Xy=n
k !k ! n
= 1 2 B B .
(k +k )! (cid:18)x(cid:19) x,k1 y,k2
1 2 x+Xy=n
(cid:3)
The proof of the following identity for weighted integer compositions
which directly entails identity (5) is a straightforwardvariation of the proof
outlinedinDePril[3],whoconsiderssumsofinteger-valuedrandomvariables.
Lemma 3. Let k,n ≥ 0 be integers and let f : N → R be arbitrary with
f(0)=0 and f(1)6=0. Then:
k 1 n+1 k
= k+1− (s+1)f(s+1) .
(cid:18)n(cid:19) f(1)(n−k) (cid:18) s+1(cid:19) (cid:18)n−s(cid:19)
f Xs≥1 f
Proof. Let X ,...,X be i.i.d. nonnegative integer-valued random vari-
1 k+1
ables, with distribution function f. Consider the conditional expectation
E[X |X +...+X = n+1]. Due to identical distribution of the vari-
1 1 k+1
ablesandlinearityofE[·|·],itfollowsthatE[X |X +...+X =n+1]=
1 1 k+1
E[X1+...+Xk+1|X1+...+Xk+1=n+1] = n+1.Thus,E[k+1X −1|X +...+X =
k+1 k+1 n+1 1 1 k+1
n+1]=0, which means that
n
k+1 P [X =s,X +···+X =n+1]
0= ( s−1) f 1 1 k+1
n+1 P [X +...+X =n+1]
Xs=1 f 1 k+1
n
1 k+1
= ( s−1)P[X =s]·P [X +···+X =n+1−s]
k+1 n+1 1 f 2 k+1
n+1 f Xs=1
(cid:0) (cid:1)
n
1 k+1 k
= ( s−1)f(s) ,
k+1 n+1 (cid:18)n+1−s(cid:19)
n+1 f Xs=1 f
(cid:0) (cid:1)
so that rewriting and shifting indices lead to the required expression. (cid:3)
We omit the proof of identity (5) since it is a simple application of
Lemma 3.
Identities for partial Bell polynomials 7
4. More identities for the partial Bell polynomials
We mention the following three additional identities for partial Bell polyno-
mials,
1 n
B = x B , (8)
n,k k (cid:18)α(cid:19) α n−α,k−1
αX≥0
n−1
B = x B , (9)
n,k (cid:18)α−1(cid:19) α n−α,k−1
αX≥1
n
B (x ,...,x )= xαB (0,x ,...,x ). (10)
n,k 1 n−k+1 (cid:18)α(cid:19) 1 n−α,k−α 2 n−k+1
αX≥0
(note that (8) is a special case of (6)) which straightforwardly follow from
the following corresponding identities for weighted integer compositions,
k k−1
= f(s) , (11)
(cid:18)n(cid:19) (cid:18)n−s(cid:19)
f Xs≥0 f
k k k−1
= sf(s) , (12)
(cid:18)n(cid:19) n (cid:18)n−s(cid:19)
f Xs≥1 f
k k k−i
= f(r)i . (13)
(cid:18)n(cid:19) (cid:18)i(cid:19)(cid:18)n−ri(cid:19)
f Xi≥0 f˜
In Equation (13), f˜(r) = 0 and f˜(s) = f(s) for all s 6= r (note that (10) is
a special case of (13) in which r = 1). Proofs of identities (11) to (13) can,
e.g., be found in Fahssi [5] and Eger [4], who also give further identities for
weighted integer compositions.
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mials, Fibonacci numbers, Bell polynomials and discrete uniform distribution.
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8 Eger
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Steffen Eger
Robert Mayer Str. 10
P.O. Box 154
D-60325 Frankfurt am Main
Germany
e-mail: [email protected]