S
ome
I
nequalities of the
B
ell
P
olynomials
Feng Qi
Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China; Department of Mathematics, College of Science, Tianjin
Polytechnic University, Tianjin City, 300387, China
Abstract. In the paper, the author
(1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Fa`a di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds;
(2) recovers an explicit formula and its inversion formula for the Bell poly-nomialsintermsoftheStirlingnumbersofthefirstandsecondkinds,with theaidoftheaboveexplicitformulaanditsinversionformulaforhigher orderderivativesofgeneratingfunctionsoftheBellpolynomials;
(3) constructs some determinantal and product inequalities and deduces the logarithmic convexity of the Bell polynomials, with the assistance of the complete monotonicity of generating functions of the Bell polynomials. These inequalities are main results of the paper.
Contents
1. Introduction 1
2. Higher order derivatives for generating functions 2
3. An explicit formula and its inversion formula 4
4. Inequalities for the Bell polynomials 5
References 8
1. Introduction
In combinatorics, the Bell numbers, usually denoted byBkfork∈ {0}∪N, where N denotes the set of all positive integers, count the number of ways a set with k
elements can be partitioned into disjoint and nonempty subsets. These numbers
E-mailaddress: [email protected], [email protected], [email protected]
2010 Mathematics Subject Classification. Primary 11B83; Secondary 11A25, 11B73, 11C08, 11C20, 15A15, 26A24, 26A48, 26C05, 26D05, 33B10, 34A05.
Key words and phrases. Bell polynomial; Bell number; Bell polynomial of the second kind; higher order derivative; generating function; Fa`a di Bruno formula; inversion theorem; Stirling number of the first kind; Stirling number of the second kind; explicit formula; inversion formula; absolute monotonicity; complete monotonicity; determinantal inequality; product inequality.
This paper was typeset usingAMS-LATEX.
1
have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan, but they are named after Eric Temple Bell, who wrote about them in the 1930s. The Bell numbersBk fork≥0 can be generated by
eet−1=
∞ X
k=0 Bk
tk
k! = 1 +t+t 2+5
6t 3+5
8t 4+13
30t 5+203
720t
6+ 877 5040t
7+· · ·
and the first ten Bell numbersBk for 0≤k≤9 are
B0= 1, B1= 1, B2= 2, B3= 5, B4= 15,
B5= 52, B6= 203, B7= 877, B8= 4140, B9= 21147.
In [1, Section 3], Asai, Kubo, and Kuo mentioned applications of the Bell numbers Bn and its generalizations to white noise distribution theory. For more information on the Bell numbers Bk, please refer to [1, 2, 3, 4, 5, 15, 16, 20] and plenty of references therein.
As well-known generalizations of the Bell numbers Bk fork≥0, the Bell poly-nomialsBk(x) fork≥0 can be generated by
ex(et−1)=
∞ X
k=0 Bk(x)
tk
k! = 1 +xt+ 1
2x(x+ 1)t 2+1
6x x
2+ 3x+ 1
t3
+1 24x x
3+ 6x2+ 7x+ 1
t4+ 1 120x x
4+ 10x3+ 25x2+ 15x+ 1
t5+· · ·
and the first seven Bell polynomialsBk(x) for 0≤k≤6 are
1, x, x(x+ 1), x x2+ 3x+ 1
, x x3+ 6x2+ 7x+ 1
,
x x4+ 10x3+ 25x2+ 15x+ 1
, x x5+ 15x4+ 65x3+ 90x2+ 31x+ 1
.
In [18] it was pointed out that there have been studies on interesting applications of the Bell polynomials Bk(x) in soliton theory, including links with bilinear and trilinear forms of nonlinear differential equations which possess soliton solutions. See, for example, [8, 9, 10]. Therefore, applications of the Bell polynomialsBk(x) to integrable nonlinear equations are greatly expected and any amendment on multi-linear forms of soliton equations, even on exact solutions, would be beneficial to interested audiences in the research community. For more information about the Bell polynomialsBk(x), please refer to [6, 7, 18, 19] and closely related references therein.
In this paper, continuing the article [16], we present an explicit formula and its inversion formula for higher order derivatives with respect tot of generating func-tionsexe±tfor the Bell polynomialsB
k(x) with the help of the Fa`a di Bruno formula, properties of the Bell polynomials of the second kind Bn,k(x1, x2, . . . , xn−k+1), and the inversion theorem for the Stirling numbers s(n, k) and S(n, k), recover an ex-plicit formula and its inversion formula for the Bell polynomialsBk(x) in terms of the Stirling numberss(n, k) andS(n, k), construct some determinantal and product inequalities for the Bell polynomials Bk(x), and deduce the logarithmic convexity of the Bell polynomialsBk(x).
2. Higher order derivatives for generating functions
and S(n, k), we present an explicit formula and its inversion formula for higher order derivatives with respect totof the generating function exe±t.
Theorem 2.1. Forn≥0, thenth derivative of the generating functionexe±t with respect to tcan be computed by
∂nexe±t
∂tn = (±1) nexe±t
n X
k=0
S(n, k) xe±tk
(2.1)
and the generating functionexe±t satisfies a family of nonlinear ordinary differential equations
n X
k=0
(±1)ks(n, k)f(k)(t) =f(t)[lnf(t)]n, (2.2)
wherex∈C,S(n, k) forn≥k≥0, which can be generated by
(ex−1)k k! =
∞ X
n=k
S(n, k)x n
n!,
represent the Stirling numbers of the second kind, ands(n, k)forn≥k≥0, which can be generated by
[ln(1 +x)]k
k! =
∞ X
n=k
s(n, k)x n
n!, |x|<1,
stand for the Stirling numbers of the first kind.
Proof. In combinatorics, the Bell polynomials of the second kind Bn,k are defined by
Bn,k(x1, x2, . . . , xn−k+1) =
X
`1,...,`n∈{0}∪N Pn
i=1i`i=n
Pn i=1`i=k
n!
Qn−k+1 i=1 `i!
n−k+1
Y
i=1
xi
i!
`i
forn≥k≥0, see [3, p. 134, Theorem A], and satisfy identities
Bn,k abx1, ab2x2, . . . , abn−k+1xn−k+1
=akbnBn,k(x1, x2, . . . , xn−k+1) (2.3)
and
Bn,k(1,1, . . . ,1) =S(n, k), (2.4) see [3, p. 135], whereaandbare any complex numbers. The Fa`a di Bruno formula for computing higher order derivatives of composite functions can be described in terms of the Bell polynomials of the second kind Bn,k by
dn
dtnf ◦g(x) = n X
k=0
f(k)(g(x))Bn,k g0(x), g00(x), . . . , g(n−k+1)(x)
, (2.5)
see [3, p. 139, Theorem C]. Applying f(u) =exu andu =g(t) =et to (2.5) and making use of identities (2.3) and (2.4) yield
∂nexet ∂tn =
n X
k=0 ∂kexu
∂uk Bn,k e
t, et, . . . , et
= n X
k=0
xkexuektBn,k(1,1, . . . ,1) =exe t
n X
k=0
The explicit formula (2.1) for the plus sign case is thus proved.
In [21, p. 171, Theorem 12.1], it is stated that, ifbα andak are a collection of constants independent ofn, then
an= n X
α=0
S(n, α)bα if and only if bn= n X
k=0
s(n, k)ak. (2.6)
Combining this inversion theorem for the Stirling numbers with (2.1) arrives at equations
n X
k=0
s(n, k)∂ kexet
∂tk =e
xet xetn
which is equivalent to that the generating functionexet
satisfies the family of non-linear ordinary differential equations in (2.2) for the plus sign case.
The equation (2.12) in the second proof of [16, Theorem 2.2] reads that
Bn,n(α) =αn and Bn+k+1,k(α,0, . . . ,0) = 0, (2.7)
whereα∈Candk, n∈ {0} ∪N. Applyingf(u) =exe
u
andu=g(t) =−tin (2.5), takingα=−1 in (2.7), and utilizing (2.1) lead to
∂nexe−t ∂tn =
n X
k=0 ∂kexeu
∂uk Bn,k(−1,0, . . . ,0) =
∂nexeu
∂un Bn,n(−1)
= (−1)nexeu
n X
k=0
S(n, k) xeuk
= (−1)nexe−t
n X
k=0
S(n, k) xe−tk
.
The formula (2.1) for the minus sign case follows immediately.
Employing the inversion theorem for Stirling numbers, demonstrated in (2.6), to consider the formula (2.1) for the minus sign case give
n X
k=0
s(n, k)(−1)k∂ kexe−t
∂tk =e xe−t
xe−tn
which implies that the generating function exe−t satisfies the family of nonlinear ordinary differential equations in (2.2) for the minus sign case. The proof of
Theo-rem 2.1 is thus complete.
3. An explicit formula and its inversion formula
In this section, with the aid of Theorem 2.1, we recover an explicit formula and its inversion formula for the Bell polynomialsBn(x) in terms of the Stirling numbers
s(n, k) andS(n, k).
Theorem 3.1. Forn≥0, the Bell polynomials Bn(x)can be computed by
Bn(x) = n X
k=0
S(n, k)xk (3.1)
and satisfy
n X
k=0
Proof. In light of the theory of series, it is easy to see that Bn(x) = (±1)nlim
t→0
∂nex(e±t−1)
∂tn = (±1)
ne−xlim t→0
∂nexe±t ∂tn . Combining this with (2.1) gives
Bn(x) = (±1)ne−xlim t→0(±1)
nexe±t n X
k=0
S(n, k) xe±tk
= n X
k=0
S(n, k)xk.
The formula (3.1) follows.
Substitutingf(t) =exe±t into (2.2) and takingt→0 result in n
X
k=0
(±1)ks(n, k)(±1)kexBk(x) =exxn
which can be simplified as (3.2). The proof of Theorem 3.1 is complete.
4. Inequalities for the Bell polynomials
In light of complete monotonicity of generating functions exe−t and with the assistance of properties of completely monotonic functions, we now start out to construct some determinantal and product inequalities for the Bell polynomials Bn(x). From these inequalities, we can derive the logarithmic convexity of the sequence of the Bell polynomialsBn(x). These inequalities are our main results in this paper.
Theorem 4.1. Letm≥1be a positive integer and let|aij|mdenote a determinant of ordermwith elementsaij.
(1) If ai for1≤i≤m are non-negative integers, then
Bai+aj(x)
m≥0, x >0 (4.1)
and
(−1)ai+ajBai+aj(x)
m≥0, x >0. (4.2)
(2) If a = (a1, a2, . . . , an) and b = (b1, b2, . . . , bn) are non-increasing n-tuples of non-negative integers such thatPk
i=1ai≥P k
i=1bifor1≤k≤n−1and Pn
i=1ai=P n
i=1bi, then n Y
i=1
Bai(x)≥ n Y
i=1
Bbi(x), x >0. (4.3)
Proof. Recall from [13, 25] that a function f is said to be absolutely monotonic on an interval I if it has derivatives of all orders and f(k−1)(t)≥0 for t∈ I and k∈N. Recall from [13, Chapter XIII], [22, Chapter 1], and [25, Chapter IV] that an infinitely differentiable functionf is said to be completely monotonic on an interval Iif it satisfies (−1)kf(k)(x)≥0 onIfor allk≥0. Theorem 2b in [25, p. 145] reads that, iff1(x) is absolutely monotonic andf2(x) is completely monotonic on their defined intervals, then their composite functionf1(f2(x)) is completely monotonic on its defined interval. Consequently, the function exe−t for x > 0 is completely monotonic with respect tot∈(0,∞).
In [12] and [13, p. 367], it was obtained that if f is completely monotonic on [0,∞), then
f(ai+aj)(t)
and
(−1)ai+ajf(ai+aj)(t)
m≥0. (4.5)
Applyingf(t) to the functionexe−t in (4.4) and (4.5) and taking the limitt→0+ give
lim t→0+
e
xe−t(ai+aj)
m=
(−1)ai+ajexBai+aj(x)
m≥0
and
lim t→0+
(−1)
ai+aj exe−t(ai+aj)
m=
(−1)ai+aj(−1)ai+ajexBai+aj(x)
m≥0.
The determinant inequalities (4.1) and (4.2) follow.
In [13, p. 367, Theorem 2], it was stated that if f is a completely monotonic function on [0,∞), then
n Y
i=1
(−1)aif(ai)(t)≥ n Y
i=1
(−1)bif(bi)(t). (4.6)
Applyingf(t) to the functionexe−t in (4.6) and taking the limitt→0+ give
lim t→0+
n Y
i=1
h
(−1)ai exe−t(ai)i =
n Y
i=1
[exBai(x)]
≥ lim t→0+
n Y
i=1
h
(−1)bi exe−t(bi)i= n Y
i=1
[exBbi(x)].
The product inequality (4.3) follows. The proof of Theorem 4.1 is complete.
Corollary 4.1. The sequence{Bn(x)}n≥0 forx >0 is logarithmically convex.
Proof. In [13, p. 369] and [14, p. 429, Remark], it was stated that if f(t) is a completely monotonic function such thatf(k)(t)6= 0 fork≥0, then the sequence
ln(−1)k−1f(k−1)(t), k≥1 (4.7)
is convex. Applying this result to the functionexe−t forx >0 figures out that the sequence
lnh(−1)k−1 exe−t(k−1)i
→x+ lnBk−1(x), t→0+
fork≥1 is convex. Hence, the sequence{Bn(x)}n≥0 is logarithmically convex. Alternatively, letting
`≥1, n= 2, a1=`+ 2, a2=`, and b1=b2=`+ 1
in the inequality (4.3) leads to B`(x)B`+2(x) ≥ B2`+1(x) which means that the sequence{Bk(x)}k∈Nis logarithmically convex. The proof of Corollary 4.1 is
com-plete.
Corollary 4.2. If `≥0 andn≥k≥0, then
[Bn+`(x)]k[B`(x)]n−k ≥[Bk+`(x)]n, x >0.
Proof. This follows from taking
a= (
k z }| {
n+`, . . . , n+`,
n−k z }| {
`, . . . , `) and b= (k+`, k+`, . . . , k+`)
Theorem 4.2. If `≥0,n≥k≥m,2k≥n, and2m≥n, then
Bk+`(x)Bn−k+`(x)≥Bm+`(x)Bn−m+`(x), x >0. (4.8)
Proof. In [23, p. 397, Theorem D], it was recovered that iff(t) is completely mono-tonic on (0,∞) and ifn≥k≥m,k≥n−k, andm≥n−m, then
(−1)nf(k)(t)f(n−k)(t)≥(−1)nf(m)(t)f(n−m)(t).
Replacingf(t) by the function (−1)` exe−t(`)
in the above inequality leads to
(−1)n exe−t(k+`) exe−t(n−k+`)≥(−1)n exe−t(m+`) exe−t(n−m+`). Further taking t → 0+ finds the inequality (4.8). The proof of Theorem 4.2 is
complete.
Theorem 4.3. Forx, `≥0 andm, n∈N, let
G`,m,n=B`+2m+n(x)[B`(x)]2−B`+m+n(x)B`+m(x)B`(x)
−B`+n(x)B`+2m(x)B`(x) +B`+n(x)[B`+m(x)]2,
H`,m,n=B`+2m+n(x)[B`(x)]2−2B`+m+n(x)B`+m(x)B`(x)
+B`+n(x)[B`+m(x)]2,
I`,m,n=B`+2m+n(x)[B`(x)]2−2B`+n(x)B`+2m(x)B`(x)
+B`+n(x)[B`+m(x)]2.
Then
G`,m,n≥0, H`,m,n≥0,
H`,m,nQG`,m,n whenm≶n, I`,m,n≥ G`,m,n≥0 when n≥m.
(4.9)
Proof. In [24, Theorem 1 and Remark 2], it was obtained that iff is completely monotonic on (0,∞) and
Gm,n= (−1)nf(n+2m)f2−f(n+m)f(m)f−f(n)f(2m)f+f(n)f(m)
2 ,
Hm,n= (−1)nf(n+2m)f2−2f(n+m)f(m)f +f(n)f(m)
2 ,
Im,n= (−1)n
f(n+2m)f2−2f(n)f(2m)f +f(n)f(m)2
forn, m∈N, then
Gm,n≥0, Hm,n≥0,
Hm,nQGm,n whenm≶n,
Im,n≥Gm,n≥0 whenn≥m.
(4.10)
Replacingf(t) by (−1)` exe−t(`)
inGm,n,Hm,n, andIm,nand simplifying produce
Gm,n= (−1)`+n n
exe−t(`+2m+n)h
exe−t(`)i2 − exe−t(`+m+n)
exe−t(`+m)
exe−t(`)
− exe−t(`+n) exe−t(`+2m) exe−t(`)+ exe−t(`+n)h exe−t(`+m)i 2o
,
Hm,n= (−1)`+n n
−2 exe−t(`+m+n)
exe−t(`+m)
exe−t(`)
+ exe−t(`+n)h
exe−t(`+m)i2o
,
Im,n= (−1)`+n n
exe−t(`+2m+n)h
exe−t(`)i2
−2 exe−t(`+n) exe−t(`+2m) exe−t(`)+ exe−t(`+n)h exe−t(`+m)i 2o
.
Further takingt→0+ reveals
lim
t→0+Gm,n=e 3G
`,m,n, lim
t→0+Hm,n=e 3H
`,m,n, and lim
t→0+Im,n=e 3I
`,m,n.
Substituting these quantities into (4.10) and simplifying bring about inequalities
in (4.9). The proof of Theorem 4.3 is complete.
Theorem 4.4. Forx, k≥0 andn∈N, we have " n
Y
`=0
Bk+2`(x)
#1/(n+1) ≥
"n−1
Y
`=0
Bk+2`+1(x)
#1/n
. (4.11)
Proof. Iff is a completely monotonic function on (0,∞), then, by the convexity of the sequence (4.7) and Nanson’s inequality listed in [11, p. 205, 3.2.27],
" n Y
`=0
(−1)k+2`+1f(k+2`+1)(t)
#1/(n+1) ≥
" n Y
`=1
(−1)k+2`f(k+2`)(t)
#1/n
fork≥0. Replacingf(t) byexe−t in the above inequality results in " n
Y
`=0
(−1)k+2`+1 exe−t(k+2`+1)
#1/(n+1) ≥
" n Y
`=1
(−1)k+2` exe−t(k+2`) #1/n
for k ≥0. Letting t → 0+ in the above inequality leads to (4.11). The proof of
Theorem 4.4 is complete.
Remark 4.1. This paper is a revised version of the preprint [17]. References
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