Some Identities for a Sequence of Unnamed Polynomials Connected with the Bell Polynomials

SOME IDENTITIES FOR A SEQUENCE OF UNNAMED POLYNOMIALS CONNECTED WITH THE BELL POLYNOMIALS

FENG QI

Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China; College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China;

Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China

DA-WEI NIU

Department of Mathematics, East China Normal University, Shanghai City, 200241, China

BAI-NI GUO

School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

Abstract. In the paper, using two inversion theorems for the Stirling num-bers and binomial coefficients, employing properties of the Bell polynomials of the second kind, and utilizing a higher order derivative formula for the ratio of two differentiable functions, the authors present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, derive two identities connecting the sequence of unnamed poly-nomials with the Bell polypoly-nomials, and recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials.

E-mail addresses:[email protected], [email protected], [email protected], [email protected], [email protected], [email protected].

2010 Mathematics Subject Classification. Primary 11B83; Secondary 05A15, 11A25, 11B65, 11B73, 11C08, 33B10.

Key words and phrases. identity; Bell polynomial; unnamed polynomial; explicit formula; inversion theorem; Stirling number; binomial coefficient.

This paper was typeset usingAMS-LA_TEX. 1

1. Motivations

In [4, pp. 257–258] and [5], the functions in

F(t, x) = √ 1

1−t exp

x

1

√

1−t −1

= ∞

X

n=0 hn(x)

tn

n! (1.1)

were considered. In [4, pp. 257–258], it was pointed out that the unnamed polyno-mialshn(x) satisfy

hn(x) =

1 n!

x ex

_d

d(x2₎

n

x2n−1ex. (1.2)

In [5], it was pointed out that the unnamed polynomials hn(x) and the Bell

poly-nomialsBn(x) are connected by the identity n

X

k=0

_n

k

Bk(x) = (−2)n n X

k=0

(−1)khk(x)S(n, k), n≥0, (1.3)

where the Bell polynomialsBk(x) can be generated by

ex(et−1)= ∞

X

k=0 Bk(x)

tk

k!

and the Stirling numbers of the second kindS(n, k) can be generated by

(ex−1)k

k! =

∞

X

n=k

S(n, k)x

n

n!, k∈ {0} ∪N.

It was pointed out in [4, pp. 257–258] that the expression (1.2) had been applied in 1937 to the theory of hyperbolic differential equations. It was also pointed out in [10] that there have been some studies on interesting applications of Bell poly-nomials Bk(x) in soliton theory, including links with bilinear and trilinear forms

of nonlinear differential equations which possess soliton solutions. See, for exam-ple, [6, 7, 8]. Therefore, applications of the Bell polynomials Bk(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 au-diences in the research community.

To simplify main results in [5], among other things, the following conclusions were established in the newly published paper [10].

Theorem 1.1 ([10, Theorem 1]). Forn≥0, the nth derivative of the generating function F(t, x)of the polynomials hn(x)in (1.1)can be computed by

dnF(t, x) dtn =

F(t, x) 2n

n X

k=0

" n X

`=k

`!(2n−2`−1)!!

_2n−`−1

2(n−`)

_`

k

# _xk

k!(1−t)n+k/2,

where the double factorial of negative odd integers−(2n+ 1) is defined by

(−2n−1)!! = (−1)

n

(2n−1)!!= (−1)

n2nn!

(2n)!, n≥0

and the conventions 0₀

= 1 and p_q

= 0 forq > p≥0 are adopted. Consequently, the polynomialshn(x)forn≥0 can be expressed as

hn(x) =

1 2n

n X

k=0

" n X

`=k

`!(2n−2`−1)!!

_2n−`−1

2(n−`)

_`

k

#_xk

Theorem 1.2 ([10, Theorem 3]). For n≥0, the Bell polynomials Bn(x)and the

polynomialshn(x) can be expressed each other by

Bn(x) = (−1)n n X

`=0 (−1)`

" n X

k=`

2k

_n

k

S(k, `)

#

h`(x) (1.4)

and

hn(x) =

1 2n

n X

`=0

" n X

k=` _n

k

(2n−2k−1)!!2k−`s(k, `)

#

B`(x), (1.5)

wheres(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.

Theorem 1.3 ([10, Theorem 4]). For n≥0, the Bell polynomials Bn(x)and the

polynomialshn(x) satisfy the identities n

X

k=0

_n

k

2k(2n−2k−3)!!hk(x) =−2n n X

k=0 s(n, k)

2k Bk(x) (1.6)

and

n X

k=0

_n

k

"n−k X

`=0

s(n−k, `)

2` B`(−x) #

hk(x) =

(2n−1)!!

2n . (1.7)

In this paper, using two inversion theorems for the Stirling numberss(n, k) and S(n, k) and binomial coefficients n_k, employing properties of the Bell polynomials of the second kind B(n, k), and utilizing a higher order derivative formula for the ratio of two differentiable functions, the authors present two explicit formulas, a determinantal expression, and a recursive relation forhn(x), derive two identities

connectinghn(x) withBn(x), and recover the identity (1.3).

Our main results can be stated as the following four theorems.

Theorem 1.4. Forn≥0, the polynomials hn(x)satisfy

hn(x) =

1 n!

n X

k=0 (−1)k

2k _n

k

n−1

2

n−k k X

`=0

(−1)`[2(k−`)−1]!!

_2k −`−1 2(k−`)

x`,

where

hxin= n−1

Y

k=0

(x−k) =

(

x(x−1)· · ·(x−n+ 1), n≥1

1, n= 0

is the falling factorial.

Theorem 1.5. Forn≥0, the Bell polynomials Bn(x)andhn(x)satisfy

Bn(x) =−2n n X

k=0 S(n, k)

2k k X

`=0

_k

`

2`(2k−2`−3)!!h`(x). (1.8)

Theorem 1.6. Forn≥0, the identity (1.3)is valid and

hn(x) = n X

k=0

(−1)n−ks(n, k) 2k

k X

`=0

_k

`

Theorem 1.7. Forn≥0, the polynomialshn(x)can be computed by the

determi-nantal expression

hn(x) = (−1)n

1 λ0(x) 0 · · · 0 0

1!!

2 λ1(x) λ0(x) · · · 0 0

3!!

22 λ2(x)

2 1

λ1(x) · · · 0 0

..

. ... ... . .. ... ...

(2n−5)!!

2n−2 λn−2(x) n−2₁ λn−3(x) · · · λ0(x) 0 (2n−3)!!

2n−1 λn−1(x) n−1₁ λn−2(x) · · · n_n−1₋₂λ1(x) λ0(x)

(2n−1)!!

2n λn(x) n₁

λn−1(x) · · · _n−2n λ2(x) _n−1n

λ1(x)

,

(1.10)

by the recursive relation

n X

r=0

n r

λn−r(x)hr(x) =−

(2n−1)!!

2n , (1.11)

and by the explicit formula

hn(x) =n! n X

k=0

[2(n−k)−1]!! [2(n−k)]!!

λk(−x)

k! (1.12)

forn≥0, where

λk(x) =

1 2k

k X

`=0

" ` X

p=0 (−1)p

_`

p

k−1 Y

q=0

(p+ 2q)

#

x`

`!, k≥0.

2. _Lemmas

In order to prove our main results, we recall several lemmas below.

Lemma 2.1 ([11, Theorem 1.4]). Forn≥0, we have

dn e√u

dun =e

√

u(−1)n

(2u)n n X

k=0

(−1)k(2n−2k−1)!!

_2n−k−1

2(n−k)

uk/2.

Lemma 2.2([14, p. 171, Theorem 12.1]). Ifbαandak 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.

Lemma 2.3 ([14, p. 83, Eq. (7.12)]). If ak and bk for k ≥ 0 are a collection of

constants independent ofnsuch that n≥k≥0, then

a(n) =

n X

k=0 (−1)k

_n

k

b(k) if and only if b(n) =

n X

k=0 (−1)k

_n

k

a(k).

Lemma 2.4 ([3, pp. 134 and 139]). For n ≥k ≥0, the Bell polynomials of the second kind, or say, partial Bell polynomials, denoted by Bn,k(x1, x2, . . . , xn−k+1),

are defined by

Bn,k(x1, x2, . . . , xn−k+1) =

X

1≤i≤n,`i∈{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

x_i

i!

The Fa`a di Bruno formula can be described in terms of the Bell polynomials of the second kindBn,k(x1, x2, . . . , xn−k+1)by

dn

dtnf◦h(t) = n X

k=0

f(k)(h(t))Bn,k h0(t), h00(t), . . . , h(n−k+1)(t)

. (2.1)

Lemma 2.5 ([3, p. 135]). Forn≥k≥0, we have

Bn,k abx1, ab2x2, . . . , abn−k+1xn−k+1

=akbnBn,k(x1, x2, . . . , xn−k+1), (2.2)

whereaandb are any complex numbers.

Lemma 2.6 ([9, Remark 1]). Forn≥k≥0, we have

Bn,k 1,1−λ,(1−λ)(1−2λ), . . . , n−k

Y

`=0

(1−`λ)

!

=(−1)

k

k!

k X

`=0 (−1)`

_k

`

n−1 Y

q=0

(`−qλ). (2.3)

Lemma 2.7 ([1, p. 40, Entry 5)]). Let p=p(x)andq=q(x)6= 0be two differen-tiable functions. Then

_p(x)

q(x)

(k)

=(−1)

k

qk+1

p q 0 · · · 0 0

p0 q0 q · · · 0 0

p00 q00 2₁q0 · · · 0 0

..

. ... ... . .. ... ...

p(k−2) _q(k−2) k−2 1

q(k−3) _{· · · q 0}

p(k−1) _q(k−1) k−1 1

q(k−2) _{· · ·} k−1

k−2

q0 q p(k) _q(k) k

1

q(k−1) _{· · ·} k k−2

q00 k k−1

q0

(2.4)

fork≥0. In other words, the formula (2.4)can be rewritten as

dk dxk

p(x) q(x)

= (−1)

k

qk+1_(x)

W(k+1)×(k+1)(x)

, (2.5)

where|W(k+1)×(k+1)(x)|denotes the determinant of the (k+ 1)×(k+ 1)matrix

W(k+1)×(k+1)(x) = U(k+1)×1(x) V(k+1)×k(x)

,

the quantityU(k+1)×1(x)is a(k+ 1)×1matrix whose elementsu`,1(x) =p(`−1)(x)

for1≤`≤k+ 1, andV(k+1)×k(x)is a(k+ 1)×k matrix whose elements

vi,j(x) = 



 _i−1

j−1

q(i−j)_{(x), i−j≥0}

0, i−j <0

Lemma 2.8 ([2, p. 222, Theorem] and [12, Remark 3]). Let M0= 1and

Mn=

m1,1 m1,2 0 · · · 0 0

m2,1 m2,2 m2,3 · · · 0 0

m3,1 m3,2 m3,3 · · · 0 0

..

. ... ... . .. ... ...

mn−2,1 mn−2,2 mn−2,3 · · · mn−2,n−1 0 mn−1,1 mn−1,2 mn−1,3 · · · mn−1,n−1 mn−1,n

mn,1 mn,2 mn,3 · · · mn,n−1 mn,n

forn∈N. Then the sequenceMn forn≥0satisfies M1=m1,1 and

Mn=mn,nMn−1+

n−1

X

r=1

(−1)n−rmn,r n−1

Y

j=r

mj,j+1

!

Mr−1, n≥2. (2.6)

3. _{Proofs of main results}

Now we are in a position to prove our main results.

Proof of Theorem 1.4. By virtue of Lemma 2.1, the formula (1.2) implies that

hn(x) =

1 n!

x ex

_d

du

n

un−1/2e √

u_{, u=x}2

= 1 n!

x ex

n X

k=0

_n

k

un−1/2(n−k)

e √

u(k)

= 1 n!

x ex

n X

k=0

_n

k

n−1

2

n−k

uk−1/2(−1)

k

(2u)k

×e √

u k X

`=0

(−1)`[2(k−`)−1]!!

_2k−`−1

2(k−`)

u`/2

= 1 n!

n X

k=0 (−1)k

2k

n k

n−1

2

n−k k X

`=0

(−1)`[2(k−`)−1]!!

2k−`−1 2(k−`)

x`.

The proof of Theorem 1.4 is complete.

Proof of Theorem 1.5. The identity (1.6) in Theorem 1.3 can be rearranged as

−1

2n n X

k=0

_n

k

2k(2n−2k−3)!!hk(x) = n X

k=0

s(n, k)Bk(x) 2k .

Further utilizing Lemma 2.2 yields

Bn(x)

2n =− n X

k=0

S(n, k) 1 2k

k X

`=0

_k

`

2`(2k−2`−3)!!h`(x).

The proof of Theorem 1.5 is complete.

Proof of Theorem 1.6. Interchanging the order of sums in the right hand side of the identity (1.4) in Theorem 1.2 arrives at

(−1)nBn(x) = n X

k=0 (−1)k

_n

k

(−2)k

k X

`=0

Further applying Lemma 2.3 leads to

(−2)n

n X

`=0

S(n, `)(−1)`h`(x) = n X

k=0 (−1)k

_n

k

(−1)kBk(x) = n X

k=0

_n

k

Bk(x)

which is equivalent to (1.3).

Employing Lemma 2.2 in (1.3) produces

(−1)nhn(x) = n X

k=0

s(n, k)(−1)

k

2k k X

`=0

_k

`

B`(x).

The proof of Theorem 1.6 is complete.

Proof of Theorem 1.7. By straightforward computation, we obtain

dk dtk

(1−t)−1/2

=

−1

2

k

(−1)k(1−t)−1/2−k

→

−1

2

k

(−1)k=

₁

2

k

=(2k−1)!!

2k , t→0

and, when denoting u=u(t) = (1−t)−1/2 _{and making use of the formulas (2.1)} and (2.2),

dkexp−x(1−t)−1/2

dtk =

k X

`=0

e−xu(`)Bk,` u0(t), u00(t), . . . , u(k−`+1)(t)

=

k X

`=0

(−x)`e−xuBk,`

−1

2

1

−1 (1−t)3/2,

−1

2

2 1

(1−t)5/2, . . . ,

−1

2

k−`+1

(−1)k−`+1 (1−t)k−`+3/2

→e−x

k X

`=0

(−x)`Bk,` _1!!

2 , 3!! 22, . . . ,

(2k−2`+ 1)!! 2k−`+1

, t→0

= e −x

2k k X

`=0

(−x)`Bk,`(1!!,3!!, . . . ,(2k−2`+ 1)!!).

Takingλ=−2 in (2.3) gives

Bn,k(1!!,3!!, . . . ,(2n−2k+ 1)!!) =

(−1)k

k!

k X

`=0 (−1)`

_k

`

n−1 Y

q=0

(`+ 2q)

forn≥k≥0. Therefore, we obtain

lim

t→0

dkexp

−x(1−t)−1/2

dtk =

e−x

2k k X

`=0 x`

`!

` X

p=0 (−1)p

_`

p

k−1 Y

q=0

(p+ 2q) =e−xλk(x).

The equation (1.1) means that

hn(x) = lim t→0

dn dtn

(1−t)−1/2 exp

x 1−(1−t)−1/2 =e

−x_lim t→0

dn dtn

(1−t)−1/2 exp

Further applying the formula (2.5) to

p(t) = (1−t)−1/2 and q(t) = exp

−x(1−t)−1/2

results in

hn(x) = (−1)ne−xlim t→0exp

(n+ 1)x(1−t)−1/2

×lim t→0

p q 0 · · · 0 0

p0 q0 q · · · 0 0

p00 q00 2₁

q0 · · · 0 0

..

. ... ... . .. ... ...

p(n−2) _q(n−2) n−2 1

q(n−3) _{· · · q 0}

p(n−1) q(n−1) n−1₁ q(n−2) · · · n_n−1₋₂

q0 q p(n) _q(n) n

1

q(n−1) _{· · ·} n n−2

q00 n n−1 q0

= (−1)nenx

1 q0(x) 0 · · · 0 0

1!!

2 q1(x) q0(x) · · · 0 0

3!!

22 q2(x)

2 1

q1(x) · · · 0 0

..

. ... ... . .. ... ...

(2n−5)!!

2n−2 qn−2(x) n−2₁ qn−3(x) · · · q0(x) 0

(2n−3)!!

2n−1 qn−1(x) n−1₁ qn−2(x) · · · n_n−1₋₂q1(x) q0(x) (2n−1)!!

2n qn(x) n₁qn−1(x) · · · _nn₋₂q2(x) _nn₋₁q1(x)

= (−1)n

1 λ0(x) 0 · · · 0 0

1!!

2 λ1(x) λ0(x) · · · 0 0

3!!

22 λ2(x) 21

λ1(x) · · · 0 0

..

. ... ... . .. ... ...

(2n−5)!!

2n−2 λn−2(x) n−2₁

λn−3(x) · · · λ0(x) 0

(2n−3)!!

2n−1 λn−1(x) n−1₁ λn−2(x) · · · n_n−1₋₂λ1(x) λ0(x)

(2n−1)!!

2n λn(x) n₁

λn−1(x) · · · _nn₋₂λ2(x) _n−1n

λ1(x)

,

whereqk(x) =e−xλk(x) fork≥0. The determinantal expression (1.10) is proved.

Applying (2.6) to the determinantal expression (1.10) and consideringλ0(x) = 1 derive

(−1)nhn(x) =nλ1(x)(−1)n−1hn−1(x) + (−1)n−1

(2n−1)!! 2n

+

n X

r=2

(−1)n−r+1

_n

r−2

λn−r+2(x)(−1)r−2hr−2(x), n≥2.

This can be reformulated as (1.11).

From the equation (1.1) and the above arguments, it follows that

hn(x) =e−xlim t→0

dn dtn

₁

√

1−t exp

_x

√

1−t

=e−xlim

t→0 n X k=0 _n k

_dn−k

dtn−k

1

√

1−t dk dtkexp

_x

√

1−t

=e−x

n X

k=0

_n

k

lim

t→0 dn−k dtn−k

1

√

1−t tlim→0 dk dtkexp

_x

√

1−t

=e−x

n X

k=0

_n

k

_{(2n−2k−1)!!}

2n−k

ex

2k k X

`=0 (−x)`

`!

` X

p=0 (−1)p

_`

p

k−1 Y

q=0

(p+ 2q)

= 1

2n n X

k=0

n k

(2n−2k−1)!!

k X

`=0 (−1)`

`! x

` ` X

p=0 (−1)p

` p

k−1 Y

q=0

(p+ 2q)

which can be rewritten as (1.12). The proof of Theorem 1.7 is complete.

4. _Remarks

Finally we list several remarks about our main results and other things.

Remark 4.1. Recently a new inversion theorem was discovered in [13, Theorem 4.3] which can be reformulated as that

sn= n X

k=1

_k

n−k

Sk if and only if (−1)nnSn= n X

k=1 (−1)kk

_2n−k−1

n−1

sk,

wheresk andSk are two sequences independent ofnsuch thatn≥k≥1.

Remark 4.2. The identity (1.4) is slightly different from (1.8).

Remark 4.3. The identity (1.9) is obviously simpler than (1.5) in their forms.

Remark 4.4. Comparing (1.7) with (1.11) reveals that

n−k X

`=0

s(n−k, `)

2` B`(−x) =−λn−k(x),

that is,

n X

`=0

s(n, `)B`(x)

2` =−λn(−x).

Further considering Lemma 2.2 deduces

Bn(x) =−2n n X

`=0

S(n, `)λ`(−x).

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