Viewing Some Ordinary Differential Equations from the Angle of Derivative Polynomials

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VIEWING SOME ORDINARY DIFFERENTIAL EQUATIONS FROM THE ANGLE OF DERIVATIVE POLYNOMIALS

FENG QI

Instituteof Mathematics,Henan PolytechnicUniversity,Jiaozuo 454010, China; CollegeofMathematics, InnerMongolia UniversityforNationalities, Tongliao028043, China; Departmentof Mathematics, CollegeofScience, TianjinPolytechnic University,Tianjin300387,China

BAI-NIGUO

Schoolof MathematicsandInformatics,HenanPolytechnic University,Jiaozuo454010,China

Abstract: Inthepaper,theauthorsviewsomeordinarydifferentialequations

andtheirsolutionsfromtheangleof(thegeneralized)derivativepolynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials,intermsoftheStirlingnumbersofthefirstandsecondkinds.

1. _Introduction

We recall some results obtained in two different and independent directions. We also recall below a concept in combinatorics about higher derivatives of functions.

1.1. The first direction. In [13, p. 1127 and 1131], it was obtained that 1

(1−e−t₎2 = 1 +

1 et₋₁ −

1 et₋₁

0

and

1

(1−e−t₎3 = 1 +

1 et₋₁−

3 2

1 et₋₁

0 +1

2

1 et₋₁

00 .

In the preprint [17] and its formally published version [6], the above two identities were generalized inductively and recursively by the following eight identities:

₁

et₋₁

(k) =

k+1

X

m=1

λk,m

₁

et₋₁

m ,

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

2010 Mathematics Subject Classification. Primary 34A34, Secondary 11B68, 11B83, 26A24, 33B10.

Key words and phrases. viewpoint; ordinary differential equation; solution; derivative poly-nomial; identity; Stirling numbers; Bernoulli number; Bernoulli polypoly-nomial; Frobenius-Euler polynomial.

This paper was typeset usingAMS-LA_TEX.

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1 1−e−t

(k) =

k+1

X

m=1

µk,m

1 1−e−t

m ,

₁

1−e−t

(k) =

k+1

X

m=1

λk,m

₁

et₋₁

m ,

₁

et₋₁

(k) =

k+1

X

m=1

µk,m

₁

1−e−t

m ,

1 1−e−t

k =

k

X

m=1

ak,m−1

1 1−e−t

(m−1) ,

₁

et₋₁

k =

k

X

m=1

bk,m−1

₁

et₋₁

(m−1) ,

₁

1−e−t

k = 1 +

k

X

m=1

ak,m−1

₁

et₋₁

(m−1) ,

₁

et₋₁

k = 1 +

k

X

m=1

bk,m−1

₁

1−e−t

(m−1) ,

where 1≤m≤k, the quantities

S(k, m) = 1 m!

m

X

`=1

(−1)m−` _m

`

`k

are the Stirling numbers of the second kind,

λk,m= (−1)k(m−1)!S(k+ 1, m), µk,m= (−1)m−1(m−1)!S(k+ 1, m),

ak,m−1= (−1)m 2₊₁

Mk−m+1(k, m), bk,m−1= (−1)k−mak,m−1,

and

Mj(k, i) =

1 (i−1)!

k i

S(i+ 1, i) S(i+ 2, i) · · · S(i+j−1, i)

1 i!

k i+1

S(i+ 1, i+ 1) S(i+ 2, i+ 1) · · · S(i+j−1, i+ 1)

1 (i+1)!

k i+2

0 S(i+ 2, i+ 2) · · · S(i+j−1, i+ 2) ..

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

1 (i+j−2)!

k i+j−1

0 0 · · · S(i+j−1, i+j−1)

forj∈N. In [6, Remark 5.1], it was pointed out that the four functions

1 et₋₁,

1 1−e−t,

1

1−e−t, and 1 et₋₁

are respectively the unique solutions to the linear ordinary equations

k

X

i=1

ak,i−1y(i−1)=Fn(t) and k

X

i=1

(−1)i−1ak,i−1y(i−1)=Gn(t)

for allk∈Nandn= 1,2, where

Fn(t) =



 

 

1

(1−e−t₎k −1, n= 1; 1

(1−e−t₎k, n= 2

and Gn(t) =



 

 

1

(et₋₁₎k −1, n= 1; 1

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In [6, Remark 5.2], it was concluded that            1 et₋₁

1 et₋₁

0

1 et₋₁

00

.. .

1 et₋₁

(i−1)

1 et₋₁

(i)            =         

λ0,1 0 · · · 0 0

λ1,1 λ1,2 · · · 0 0

λ2,1 λ2,2 · · · 0 0

..

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

λk−2,1 λk−2,2 · · · λk−2,k−1 0

λk−1,1 λk−1,2 · · · λk−1,k−1 λk−1,k

                   1 et₋₁

1 (et₋₁₎2

1 (et₋₁₎3

.. .

1 (et₋₁₎i

1 (et₋₁₎i+1

          and         1 et₋₁

1 (et₋₁₎2

1 (et₋₁₎3

.. .

1 (et₋₁₎k

       

= (−1)(k−1)k/2       

a1,0 0 0 · · · 0

a2,0 a2,1 0 · · · 0

a3,0 a3,1 a3,2 · · · 0

..

. ... ... . .. ... ak,0 ak,1 ak,2 · · · ak,k−1

               1 et₋₁

1 et₋₁

0

1 et₋₁

00

.. .

1 et₋₁

(k−1)         .

All of the above results in [6, 17] were established by induction.

In [25, Theorems 3.1 and 3.2], by the Fa`a di Bruno formula and combinatorial techniques, the above identities in [6, 17] were generalized and unified as follows. For real numbers α, λ∈R,

(1) ifn∈_N, then

₁

1−λeαt

(n)

= (−1)nαn n+1

X

k=1

(−1)k−1(k−1)!S(n+ 1, k)

₁

1−λeαt

k ;

(2) ifn∈N, then

₁

1−λeαt

n

=(−1) n−1

(n−1)! n

X

k=1

(−1)k−1

αk−1 s(n, k)

₁

1−λeαt

(k−1) ;

where s(n, k) forn≥k≥1 denote the Stirling numbers of the first kind.

In [4, Theorem 2.1], the above identities in [6, 17, 25] were inductively proved once again and were rewritten as follows. For real constantsλ6= 0 andα6= 0 and fork∈N, whenλ >0 andt6=−lnαλ or whenλ <0 andt∈R, we have

dk dtk

₁

λeαt₋₁

= (−1)kαk k+1

X

m=1

(m−1)!S(k+ 1, m)

₁

λeαt₋₁

m

(1.1)

and

₁

λeαt₋₁

k

= 1

(k−1)! k

X

m=1

(−1)m−1

αm−1 s(k, m)

dm−1 dtm−1

₁

λeαt₋₁

. (1.2)

1.2. The second direction. In [9, 10], it was obtained inductively and recursively that

(1) the functionF(t) = _u₋1_et is a solution of the nonlinear differential equation

Fn(t) = n un−1

n−1

X

k=0

1 (k+ 1)!

X

`1+`2+···+`k+1=n

1 `1`2· · ·`k+1

F(k)(t), (1.3)

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(2) form∈_Nand n≥0, we have

H_n(m)(u) =m _u₋₁

u

m−1m−1 X

k=0

1 (k+ 1)!

X

`1+`2+···+`k+1=m

Hn+k(u) `1`2· · ·`k+1

,

where Hn(u) are called the nth Frobenius-Euler numbers which can be generated by

1−u et₋_u =

∞

X

n=0

Hn(u) tn

n!, u6= 1;

(3) form∈Nand n≥0, we have

X

`1+`2+···+`m=n

_n

`1, `2, . . . , `m

H`1(u)H`2(u)· · ·H`m(u)

=m

_u₋₁

u

m−1m−1 X

k=0

1 (k+ 1)!

X

`1+`2+···+`k+1=m

Hn+k(u) `1`2· · ·`k+1

;

(4) form∈Nand n≥0, we have

H_n(m)(x|u) =m _u₋₁

u

m−1m−1 X

k=0

1 (k+ 1)!

× X

`1+`2+···+`k+1=m

1 `1`2· · ·`k+1

n

X

q=0

_n

q

Hq+k(u)xn−q,

where Hn(x|u) are called the Frobenius-Euler polynomials which can be generated by

1−u et₋_ue

xt₌ ∞

X

n=0

Hn(x|u) tn

n!, u6= 1.

Influenced by the papers [9, 10], the authors in [7] derived inductively and re-cursively several formulas for the Bernoulli polynomials of therth order

_t

et₋₁

r ext=

∞

X

n=0

B_n(r)(x)t n

n!

in terms of the Bernoulli numbersBn=B (1) n (0).

In [22, Theorem 2.1], it was procured inductively and recursively that the func-tionF(t) = _et1₊₁ is a solution of the nonlinear differential equation

Fn(t) =n n−1

X

k=0

1 (k+ 1)!

X

`1+`2+···+`k+1=n

1 `1`2· · ·`k+1

F(k)(t), n∈_N. (1.4)

By virtue of this, as did in [7], the authors of [22] presented several formulas for the Euler numbers and polynomials of the rth orderEn(r)(0) andEn(r)(x) defined by

₂

et_{+ 1}

r ext=

∞

X

n=0

E_n(r)(x)t n

n!

in terms of the so-called Euler numbers En=E (1)

n (0) which are different from the classical and traditional Euler numbersEn = 2nE 1₂

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In [8], it was acquired inductively and recursively that the functionF(t) = _ln(1+1 _t₎ is a solution of the nonlinear differential equation

F(n)(t) =(−1)

n_(n₋_1)!

(1 +t)n n+1

X

j=2

(j−1)!Hn−1,j−2Fj(t) (1.5)

forn∈_N, where

Hn,0= 1, Hn,1= n

X

k=1

1

k, H0,j−1= 0, Hn,j= n

X

k=1

Hk−1,j−1

k

for 2≤j ≤n. Thereafter, the authors in [8] established an identity involving the higher order Bernoulli numbers of the second kind b(nk) defined by

_t

ln(1 +t) k

=

∞

X

n=0

b(_nk)t n

n!.

In [12], it was procured inductively and recursively that

(1) the function F(t) = ₍₁₊_λt1₎1/λ₊₁ is a solution of the nonlinear differential

equation

F(n)(t) = (−1) n

(1 +λt)n n+1

X

i=1

(−1)i−1ai(n, λ)Fi(t) (1.6)

forn∈N, where

ai(n, λ) = (i−1)!λn−i+1 n−i+1

X

mi−1=0

n−mi−1−i+1

X

mi−2=0

· · ·

n−mi−1−···−m2−i+1

X

m1=0

n−mi−1

+ i λ

mi−1

n−mi−1−mi−2−1 +

i−1 λ

mi−2

· · ·

n−mi−1

− · · · −m1−i+ 2 +

2 λ

m1

₁

λ

n−mi−1−mi−2−···−m1−i+1

, (1.7)

(2) the function F(t) = ₍₁₊_λt1₎1/λ₋₁ is a solution of the nonlinear differential

equation

F(n)(t) = (−1) n

(1 +λt)n n+1

X

i=1

ai(n, λ)Fi(t) (1.8)

forn∈N.

Hereafter, the authors in [12] gave some new identities involving degenerate Euler numbers and polynomials.

Forx∈R, let

hxin=

(

x(x−1)· · ·(x−n+ 1), n≥1

1, n= 0

denotes the falling factorial ofx. In [3, Theorem 2.1], it was presented inductively and recursively among other things that the function F(t) =(1+₂₊t_t)x is a solution of the linear differential equations

F(n)(t) = " n

X

i=0

ai(n, x)

(2 +t)i−n (1 +t)i

#

F(t),

forn≥0, wherea0(n, x) = (−1)nn! and

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×

n−j

X

ij−1=0

n−j−ij−1

X

ij−2=0

· · ·

n−j−ij−1−···−i2

X

i1=0

(n−ij−1− · · · −i1−j+ 1) (1.9)

for 1≤j≤n.

In [11], it was demonstrated inductively and recursively that

(1) the functionF(t) = _t₊1_c, t6=−c, is a solution of the nonlinear differential equationF(n)_{(t) = (}₋₁₎n_n!Fn+1_{(t) for}_n_∈

N,

(2) the functionG(t) =_et1₊₁ is a solution of the nonlinear differential equation

G(n)(t) = (−1)n n+1

X

k=1

(−1)k−1an,kGk(t), (1.10)

where

an,k= (k−1)! n−k+2

X

m1=0

n−k+2−m1

X

m2=0

· · ·

n−k+2−m1−···mk−3

X

mk−2=0

km1

×(k−1)m2_{· · ·}₃mk−2 ₂n−m1−m2−···−mk−2−k+3₋₁_.

(1.11)

Hereafter, the authors gave two identities for the Changhee polynomials of therth order Ch(_nr)(x) defined by

₂

t+ 2 r

(1 +t)x=

∞

X

n=0

Ch(nr)(x) tn n!.

1.3. Derivative polynomials. Supposef is a function whose derivative is a poly-nomial inf, that is, f0(x) =P(f(x)) for some polynomialP. Then all the higher order derivatives of f are also polynomials in f, so we have a sequence of polyno-mials Pn defined by f(n)(x) = Pn(f(x)) for n ≥ 0. As usual, we call Pn(u) the derivative polynomials off. See [2, 24] and closely related references therein.

Now we more accurately introduce a new notion below. If there exists a sequence of polynomialsPn(u) =P

n

k=0akuk forak∈Cand a sequence of functionshn,k(x) for n, k ≥ 0 such thatf(n)(x) = Pn+q

k=0akhn,k(x)fk(x) for n, q ≥ 0, then we call Pn+q(u) the generalized derivative polynomials off(x) with respect to hn,k(x).

2. _{Alternative viewpoints and derivative polynomials}

Now we are in a position to discuss the above conclusions from alternative view-points and (the generalized) derivative polynomials.

2.1. Almost all the above linear or nonlinear ordinary differential equations and their solutions can be alternatively regarded as problems of (the generalized) de-rivative polynomials.

2.2. Takingλ=−1 andα= 1 in (1.1) and simplifying yield

dk dtk

₁

et_{+ 1}

= (−1)k k+1

X

m=1

(−1)m−1(m−1)!S(k+ 1, m)

₁

et_{+ 1}

m .

Comparing this with (1.10) reveals that the coefficientsan,k in (1.11) is just equal to (k−1)!S(n+ 1, k). In other words,

S(n+ 1, k) = n−k+2

X

m1=0

n−k+2−m1

X

m2=0

· · ·

n−k+2−m1−···mk−3

X

mk−2=0

km1_(k₋₁₎m2_{· · ·}₃mk−2

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We can also say that the generalized derivative polynomials of the function _et1+1 is

Pk+1(x) = (−1)k k+1

X

m=1

(−1)m−1(m−1)!S(k+ 1, m)xm

with respect tohk,m(t)≡1 fork≥1 and 1≤m≤k+ 1. It is much straightforward and simple to see that the derivative polynomials of the function 1

t+c is Pn(x) = (−1)nn!xn+1. Consequently, we find an alternative viewpoint to examine the results in [11].

2.3. By Leibnlz’s theorem for differentiation of a product, we obtain

dn dtn

_{(1 +}_t)x

2 +t

= n

X

k=0

_n

k

_dk_{[(1 +}_t)x_]

dtk

dn−k dtn−k

₁

2 +t

= n

X

k=0

_n

k

hxik(1 +t)x−k

(−1)n−k_(n₋_k)! (2 +t)n−k+1

= n

X

k=0

_n

k

hxik

(−1)n−k_(n₋_k)! (1 +t)k_{(2 +}_t)n−k

(1 +t)x 2 +t

This implies that the coefficientsaj(n, x) in (1.9) can be simplified explicitly as

ak(n, x) = (−1)n−k(n−k)!

_n

k

hxik = (−1)n−k n!

k!hxik, 0≤k≤n.

Moreover, it follows that

_n

k

= n−k

X

ik−1=0

n−k−ik−1

X

ik−2=0

· · ·

n−k−ik−1−···−i2

X

i1=0

(n−ik−1− · · · −i1−k+ 1).

In other words, the generalized derivative polynomials of the function (1+₂₊t_t)x is P1+0(u) =uwith respect to

hn,1(t) = n

X

k=0

n k

hxik

(−1)n−k(n−k)!

(1 +t)k_{(2 +}_t)n−k, n≥0.

2.4. In combinatorial analysis, the Fa`a di Bruno formula plays an important role and can be described in terms of the Bell polynomials of the second kind

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! `i

forn≥k≥0, see [1, p. 134, Theorem A], by

dn

dtn[f◦h(t)] = n

X

k=0

f(k)(h(t))Bn,k h0(t), h00(t), . . . , h(n−k+1)(t) (2.1)

for n ≥ 0, see [1, p. 139, Theorem C]. Replacing f(u) and u = h(t) in (2.1) respectively by _u_±11 and (1 +λt)1/λ _yields

f(k)(u) = (−1) k_k!

(u±1)k+1, h (k)_{(t) =}

"k−1

Y

`=0

(1−`λ) #

(1 +λt)1/λ−k,

and

₁

(1 +λt)1/λ_±₁

(n) =

n

X

k=0

(−1)kk!

(1 +λt)1/λ_±₁k+1Bn,k (1 +λt)

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(1−λ)(1 +λt)1/λ−k, . . . , "n−k

Y

`=0

(1−`λ) #

(1 +λt)1/λ−n+k−1 !

= 1

(1 +λt)n n

X

k=0

(−1)k_{k!(1 +}_λt)k/λ

(1 +λt)1/λ_±₁k+1Bn,k 1,1−λ, . . . ,

n−k

Y

`=0

(1−`λ) !

,

where we used in the above line the formula

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

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

listed in [1, p. 135].

In [1, p. 133], it was listed that

1 k!

∞

X

m=1

xm tm m!

!k =

∞

X

n=k

Bn,k(x1, x2, . . . , xn−k+1)

tn n!

fork≥0. Therefore, we obtain that

∞

X

n=k

Bn,k 1,1−λ, . . . , n−k

Y

`=0

(1−`λ) !

tn n! =

1 k!

∞

X

m=1 m−1

Y

`=0

(1−`λ)t m

m! !k

= 1 k!

(λt+ 1)1/λ−1k = 1

k! k

X

`=0

(−1)k−` _k

`

(λt+ 1)`/λ.

Accordingly, it follows that

Bn,k 1,1−λ,(1−λ)(1−2λ), . . . , n−k

Y

`=0

(1−`λ) !

= 1 k!tlim→0

k

X

`=0

(−1)k−` _k

`

(λt+ 1)`/λ(n)

= 1 k!tlim→0

k

X

`=0

(−1)k−` _k

` n−1

Y

q=0

(`−qλ)(1 +λt)`/λ−n

=(−1) k

k! k

X

`=0

(−1)` _k

` n−1

Y

q=0

(`−qλ).

In a word, we derive that

₁

(1 +λt)1/λ_±₁

(n) =

n

X

k=0

" _k X

`=0

(−1)` _k

` n−1

Y

q=0

(`−qλ) #

(1 +λt)k/λ−n

(1 +λt)1/λ_±₁k+1 (2.3)

for n ≥ 0. This means that the functions ₍₁₊_λt1₎1/λ_±1 have the same generalized

derivative polynomials

Pn+1(u) = n+1

X

k=1

"_k₋₁ X

`=0

(−1)` _k₋₁

` n−1

Y

q=0

(`−qλ) #

uk

with respect tohn,k(t) = (1 +λt)(k−1)/λ−n forn≥0 and 1≤k≤n+ 1.

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2.5. In [14, 18], it was obtained that

₁

ln(1 +t) (m)

= 1

(1 +t)m m

X

i=0

(−1)ii! s(m, i)

[ln(1 +t)]i+1, m≥0

and

1 lnx

(n)

= (−1) n

xn n+1

X

i=2

an,i

(lnx)i, n∈N,

where an,2= (n−1)! and

an,i= (i−1)!(n−1)! n−1

X

`1=1

1 `1

`1−1

X

`2=1

1 `2

· · ·

`i−4−1

X

`i−3=1

1 `i−3

`i−3−1

X

`i−2=1

1 `i−2

forn+ 1≥i≥3. Comparing this with (1.5) and rearranging lead to

Hn−1,i−1= n−1

X

`1=1

1 `1

`1−1

X

`2=1

1 `2

· · ·

`i−4−1

X

`i−3=1

1 `i−3

`i−3−1

X

`i−2=1

1 `i−2

= (−1)n+i s(n, i) (n−1)!

for 3≤i≤n. This connects (1.5) with the Stirling numbers of the first kinds(n, i) and the generalized derivative polynomials: the generalized derivative polynomials of _ln(1+1 _t₎ is

Pm+1(u) = m+1

X

i=1

(−1)i−1(i−1)!s(m, i−1)ui

with respect tohm,i≡ ₍₁₊1_t₎m form≥0. Hence, we are viewing the paper [8] from

a different angle.

2.6. Lettingλ=−1 andα= 1 in (1.2) results in

₁

et_{+ 1}

k

= 1

(k−1)! k

X

m=1

(−1)m+ks(k, m) d m−1

dtm−1

₁

et_{+ 1}

. (2.4)

Comparing this with (1.4) gives

s(n, k+ 1) = (−1)n+k+1 n! (k+ 1)!

X

Pk+1

q=1`q=n

k+1

Y

q=1

1 `q

, n∈N. (2.5)

As a result, Theorem 2.1 in [22] can be simplified as (2.4) and has something to do with the Stirling numbers of the first kinds(n, k).

2.7. Settingλ= 1_u andα= 1 in (1.2) produces

₁

u−et

k

=(−1) k−1

(k−1)! 1 uk−1

k−1

X

m=0

(−1)ms(k, m+ 1)d m

dtm

₁

u−et

fork∈N. Comparing this with (1.3) recovers the formula (2.5) which is an alterna-tive expression for the Stirling numbers of the first kinds(n, k). Therefore, all the above-mentioned results in [9, 10] can be restated simply in terms of the Stirling numbers of the first kinds(n, k).

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2.9. In [5, 15, 20, 21, 23, 24], there are more information and new conclusions about (the generalized) derivative polynomials, explicit formulas for the Bernoulli numbers and polynomials, for the Euler numbers and polynomials, for higher derivatives of some elementary functions, properties of the functions _e±±1t₋₁, and the like. Due to

limitation on the length of the paper, we will not elaborate in further detail yet.

2.10. It should be common knowledge that mathematicians should try to repre-sent, explain, or interpret a new mathematical quantities in terms of some known and popular quantities in mathematics. Once a new mathematical quantity were established connections or relations with some famous or important quantities, it would be more meaningful in mathematics.

3. _Remarks Finally we would like to give several remarks.

Remark 1. From the derivation of the equation (2.3), we can conclude that

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) (3.1)

for n≥k ≥0. This formula can be used to derive many known and new special values of the Bell polynomials of the second kind Bn,k such as, when taking λ= 0,−1,2,1₂,−2,−1

2 in (3.1) respectively,

Bn,k(1,1,1, . . . ,1) = (−1)k

k! k

X

`=0

(−1)` _k

`

`n =S(n, k), (3.2)

Bn,k(1!,2!,3!, . . . ,(n−k+ 1)!) = (−1)k

k! k

X

`=0

(−1)` _k

` n−1

Y

q=0

(`+q)

= _n

k

_n₋₁

k−1

(n−k)!

=L(n, k),

(3.3)

Bn,k((−1)!!,1!!,3!!, . . . ,(2(n−k)−1)!!) = (−1)k

k! k

X

`=0

(−1)` _k

` n−1

Y

q=0

(`−2q), (3.4)

Bn,k

1,1

2,0, . . . ,0

=(−1) k

k! k

X

`=0

(−1)` _k

` n−1

Y

q=0

`−q

2

=(n−k)! 4n−k

_n

k

_k

n−k

,

(3.5)

Bn,k(1!!,3!!,5!!, . . . ,(2(n−k) + 1)!!) = (−1)k

k! k

X

`=0

(−1)`

k `

n−1 Y

q=0

(`+ 2q), (3.6)

and

Bn,k

1,3

2,3, 15

2 . . . ,

(n−k+ 2)! 2n−k+1

= (−1) k

k! k

X

`=0

(−1)` _k

` n−1

Y

q=0

`+q

2

(3.7)

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be found in [16, Theorem 1]. The identity (3.4) recovers the main results [20, Theorems 1 and 2]. By virtue of the formula (2.2), the identity (3.5) is a special case of

Bn,k(x,1,0, . . . ,0) =

(n−k)! 2n−k

_n

k

_k

n−k

x2k−n

which was established alternatively in [5, Theorem 4.1], [20, Eq. (2.8)], and [21, Section 3]. The identities (3.6) and (3.7) are, to the best of our knowledge, new.

Remark 2. By the way, the formulas (3.4) and (3.5) in [18, p. 325, Corollary 3.1] should be slightly corrected as

x ln(1 +x)

(i)

= (−1) i

(1 +x)i i+1

X

k=2

xai,k−i(1 +x)ai−1,k

[ln(1 +x)]k , i≥2,

_x

ln(1 +x) (i)

= 1

(1 +x)i i

X

k=1

(−1)k_{k![xs(i, k) +}_{i(1 +}_x)s(i₋_{1, k)]}

[ln(1 +x)]k+1 , i≥2,

or

_x

ln(1 +x) (i)

= 1

(1 +x)i i

X

k=0

(−1)k_{k![xs(i, k) +}_{i(1 +}_x)s(i₋_{1, k)]}

[ln(1 +x)]k+1 , i≥1,

where an,2= (n−1)! forn∈N,

an,i= (i−1)!(n−1)! n−1

X

`1=1

1 `1

`1−1

X

`2=1

1 `2

· · ·

`i−4−1

X

`i−3=1

1 `i−3

`i−3−1

X

`i−2=1

1 `i−2

forn+ 1≥i≥3, and

an,i= (−1)n+i−1(i−1)!s(n, i−1)

for 2≤i≤n+ 1.

Remark 3. This paper is a slightly revised and extended version of the preprint [19].

Acknowledgments. The first author thanks Professors Taekyun Kim and Lee-Chae Jang at Kwangwoon University and Konkuk University in South Korea for their academic visit and nice lectures between 22–26 February 2016 at Tianjin Polytechnic University in China and for their recommending the papers [11, 12].

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