Feng Qi

DETERMINANTAL EXPRESSIONS AND RECURRENCE RELATIONS FOR FUBINI AND EULERIAN POLYNOMIALS

Keywords: AMS subject classifications (2010): Primary 11B68, Secondary 11B37, 11B83, 26A24, 26C05, 33B10 generalized Fubini polynomial, Eulerian polynomial, determinantal expression, recurrence relation, Hessenberg determinant

In the paper, by virtue of an explicit formula for higher order derivatives of the ratio between two differentiable functions in terms of the Hessenberg determinant, the author presents determinantal expressions and recurrence relations for generalized Fubini polynomials. Hereafter, by virtue of relations between Eulerian polynomials and the (generalized) Fubini numbers and polynomials, the author derives a determinantal expression for Eulerian polynomials and several recurrence relations for the Fubini and Eulerian polynomials.

Introduction

For n ≥ 0, let

F n = n k=0 k!S(n, k), (1.1)
where S(n, k), which can be generated by the exponential function

(e x -1) k k! = ∞ n=k S(n, k) x n n!
and can be computed by the explicit formula

S(n, k) = 1 k! k =1 (-1) k-k n ,
denotes the Stirling numbers of the second kind [START_REF] Qi | Diagonal recurrence relations, inequalities, and monotonicity related to the Stirling numbers of the second kind[END_REF]. One calls these numbers F n the Fubini numbers [START_REF] Dasef | Some sums of some importance[END_REF], ordered Bell numbers [START_REF] Gross | Preferential arrangements[END_REF], or geometric numbers [START_REF] Tanny | On some numbers related to the Bell numbers[END_REF]. A Fubini number F n has been interpreted in [START_REF] Gross | Preferential arrangements[END_REF][START_REF] Tanny | On some numbers related to the Bell numbers[END_REF] combinatorially: it counts all the possible set partitions of an n element set such that the order of the blocks matters. In [START_REF] Gross | Preferential arrangements[END_REF], the Fubini numbers F n were connected with preference arrangements and the recursion for F n was derived. In [START_REF] Gross | Preferential arrangements[END_REF][START_REF] James | The factors of a square-free integer[END_REF], the exponential generating function

G(t) = 1 2 -e t = ∞ n=0 F n t n n! (1.2)
and an asymptotic estimate for F n were established.

In [START_REF] Tanny | On some numbers related to the Bell numbers[END_REF], the Fubini polynomials F n (y) were defined by

F n (y) = n k=0 S(n, k)k!y k . (1.3)
These polynomials can be generated [36, pp. 735-736] by

G(t, y) = 1 1 -y(e t -1) = ∞ n=0 F n (y) t n n! . (1.4)
It is clear that F n (1) = F n . Due to the relation

y d d y m 1 1 -y = ∞ k=0 k m y k = 1 1 -y F m y 1 -y , |y| < 1
in [START_REF] Boyadzhiev | A series transformation formula and related polynomials[END_REF], one also calls F n (y) the geometric polynomials.

In [12, Section 2], the Fubini polynomials F n (x, y) of two variables x, y are defined by means of the generating function

G(t, y, x) = e xt 1 -y(e t -1) = ∞ n=0 F n (x, y) t n n! .
It is apparent that F n (0, y) = F n (y).

In [START_REF] Kim | Two variable higher-order Fubini polynomials[END_REF], the bivariate Fubini polynomials F (r) n (x, y) of order r for r ∈ N, generated by

G(t, y, x, r) = e xt [1 -y(e t -1)] r = ∞ n=0 F (r) n (x, y) t n n! ,
were studied. It is obvious that F

n (x, y) = F n (x, y). The generating functions of F n , F n (y), F n (x, y), and F (r) n (x, y) remind us to consider the generating function

G(t, y, x, q, z) = e xt [z -y(e t -1)] q = ∞ n=0 F n (x, y, z, q) t n n! , x, q ∈ R (1.5)
and generalized Fubini polynomials F n (x, y, z, q) of four variables x, y, z, q. It is clear that, since

G(t, y, x, q, z) = 1 z q G t, y z , x, r ,
we have

F n (x, y, z, r) = F (r) n (x, y/z) z r .
In this paper, we will establish some properties of generalized Fubini polynomials F n (x, y, z, q) and, consequently, derive some known results.

A new determiantal expression of the Fubini polynomials

At first, we simply establish a new determinantal expressions of the Fubini polynomials and numbers F n (y) and F n respectively.

Theorem 2.1. The Fubini polynomials F n (y) for n ≥ 0 can be represented determinantally by

F n (y) = 1 -1 0 0 • • • 0 0 0 0 y -1 0 • • • 0 0 0 0 y 2 1 y -1 • • • 0 0 0 0 y 3 1 y 3 2 y • • • 0 0 0 • • • • • • • • • • • • . . . • • • • • • • • • 0 y n-2 1 y n-2 2 y • • • n-2 n-3 y -1 0 0 y n-1 1 y n-1 2 y • • • n-1 n-3 y n-1 n-2 y -1 0 y n 1 y n 2 y • • • n n-3 y n n-2 y n n-1 y . (2.1)
Consequently, the Fubini numbers F n can be represented determinantally by

F n = 1 -1 0 0 • • • 0 0 0 0 1 -1 0 • • • 0 0 0 0 1 2 1 -1 • • • 0 0 0 0 1 3 1 3 2 • • • 0 0 0 • • • • • • • • • • • • . . . • • • • • • • • • 0 1 n-2 1 n-2 2 • • • n-2 n-3 -1 0 0 1 n-1 1 n-1 2 • • • n-1 n-3 n-1 n-2 -1 0 1 n 1 n 2 • • • n n-3 n n-2 n n-1 .
Proof. Let u(t) and v(t) = 0 be differentiable functions, let U (n+1)×1 (t) be an (n + 1) × 1 matrix whose elements u k,1 (t) = u (k-1) (t) for 1 ≤ k ≤ n + 1, let V (n+1)×n (t) be an (n + 1) × n matrix whose elements

v i,j (t) =    i -1 j -1 v (i-j) (t), i -j ≥ 0; 0, i -j < 0 for 1 ≤ i ≤ n +
W (n+1)×(n+1) (t) = U (n+1)×1 (t) V (n+1)×n (t) .
Then the nth derivative of the ratio u(t) v(t) can be computed by

d n d x n u(t) v(t) = (-1) n W (n+1)×(n+1) (t) v n+1 (t) . (2.2)
See [1, p. 40, Exercise 5)], the papers [START_REF] Qi | Derivatives of tangent function and tangent numbers[END_REF][START_REF] Qi | Two closed forms for the Bernoulli polynomials[END_REF] and closely related references. Taking u(t) = 1 and

v(t) = 1 -y e t -1 in (2.2) yields d n d t n G(t, y) = (-1) n G n+1 (t, y) × 1 0 0 1 -y e t -1 0 • • • 0 0 1 0 -ye t 1 1 1 -y e t -1 • • • 0 0 2 0 -ye t 2 1 -ye t • • • 0 • • • • • • • • • . . . • • • 0 n-2 0 -ye t n-2 1 -ye t • • • 0 0 n-1 0 -ye t n-1 1 -ye t • • • n-1 n-1 1 -y e t -1 0 n 0 -ye t n 1 -ye t • • • n n-1 -ye t → (-1) n 1 0 0 0 • • • 0 0 1 0 (-y) 1 1 • • • 0 0 2 0 (-y) 2 1 (-y) • • • 0 • • • • • • • • • . . . • • • 0 n-2 0 (-y) n-2 1 (-y) • • • 0 0 n-1 0 (-y) n-1 1 (-y) • • • n-1 n-1 0 n 0 (-y) n 1 (-y) • • • n n-1 (-y) , t → 0 = 1 -1 0 • • • 0 0 1 0 y -1 • • • 0 0 2 0 y 2 1 y • • • 0 • • • • • • • • • . . . • • • 0 n-2 0 y n-2 1 y • • • 0 0 n-1 0 y n-1 1 y • • • -1 0 n 0 y n 1 y • • • n n-1 y .
The determinantal expression (2.1) is thus proved. The proof of Theorem 2.1 is complete.

Two recurrence relations of the Fubini polynomials

Basing on Theorem 2.1 and the binomial inversion theorem, we now derive two recurrence relations of the Fubini polynomials F n (y).

Theorem 3.1. The Fubini polynomials F n (y) satisfy

F n (y) = y n-1 r=0 n r F r (y), n ∈ N (3.1)
and

F n (y) = (-1) n+1 (1 + y) n-1 k=0 (-1) k n k F k (y) - 1 y , n ≥ 0. (3.2)
Proof. Let M 0 = 1 and

M n = m 1,1 m 1,2 0 . . . 0 0 m 2,1 m 2,2 m 2,3 . . . 0 0 m 3,1 m 3,2 m 3,3 . . . 0 0 . . . . . . . . . . . . . . . . . . m n-2,1 m n-2,2 m n-2,3 . . . m n-2,n-1 0 m n-1,1 m n-1,2 m n-1,3 . . . m n-1,n-1 m n-1,n m n,1 m n,2 m n,3 . . . m n,n-1 m n,n
for n ∈ N. Then the main theorem on page 222 in [START_REF] Cahill | Fibonacci determinants[END_REF] reads that the sequence M n for n ≥ 0 satisfies

M 1 = m 1,1 and M n = m n,n M n-1 + n-1 r=1 (-1) n-r m n,r n-1 j=r m j,j+1 M r-1 , n ≥ 2. ( 3
s n = n k=0 n k S k if and only if S n = n k=0 (-1) n-k n k s k
for n ≥ 0, where s n and S n are sequences of complex numbers. The relation (3.1) can be rearranged as

1 + 1 y F n+1 (y) = n+1 r=0 n + 1 r F r (y), n ≥ 0.
Therefore, we obtain

F n (y) = n k=0 (-1) n-k n k 1 + 1 y F k (y) - (-1) n y , n ≥ 0
which can be rewritten as (3.2). The proof of Theorem 3.1 is complete.

A determinantal expression for generalized Fubini polynomials

In this section, We establish a determinantal expression for generalized Fubini polynomials F n (x, y, z, q) of four variables x, y, z, q. Theorem 4.1. Generalized Fubini polynomials F n (x, y, z, q) can be determinantally expressed as

F n (x, y, z, q) = (-1) n z q(n+1) × 1 0 0 z q 0 • • • 0 x 1 0 m 1 (z, y, q) 1 1 z q • • • 0 x 2 2 0 m 2 (z, y, q) 2 1 m 1 (z, y, q) • • • 0 • • • • • • • • • . . . • • • x n-2 n-2 0 m n-2 (z, y, q) n-2 1 m n-3 (z, y, q) • • • 0 x n-1 n-1 0 m n-1 (z, y, q) n-1 1 m n-2 (z, y, q) • • • n-1 n-1 z q x n n 0 m n (z, y, q) n 1 m n-1 (z, y, q) • • • n n-1 m 1 (z, y, q) (4.1)
and satisfy the recurrence relation

F n (x, y, z, q) = 2 z q x n - n-1 r=0 n r m n-r (z, y, q)F r (x, y, z, q) , (4.2)
where

m k (z, y, q) = k =0
(-1) S(k, ) q z q-y , k ≥ 0.

Proof. In [6, p. 134, Theorem A] and [6, p. 139, Theorem C], the Bell polynomials of the second kind B n,k (x 1 , x 2 , . . . , x n-k+1 ) for n ≥ k ≥ 0 are defined by

B n,k (x 1 , x 2 , . . . , x n-k+1 ) = 1≤i≤n-k+1 i∈{0}∪N n-k+1 i=1 i i=n n-k+1 i=1 i=k n! n-k+1 i=1 i ! n-k+1 i=1 x i i! i .
The Faà di Bruno formula can be described in terms of the Bell polynomials of the second kind B n,k (x 1 , x 2 , . . . , x n-k+1 ) by

d n d t n f • h(t) = n k=0 f (k) (h(t)) B n,k h (t), h (t), . . . , h (n-k+1) (t) . (4.3)
In [6, p. 135], it is given that

B n,k abx 1 , ab 2 x 2 , . . . , ab n-k+1 x n-k+1 = a k b n B n,k (x 1 , x 2 , . . . , x n-k+1 ) (4.4)
and B n,k (1, 1, . . . , 1) = S(n, k), (4.5) where a and b are complex numbers.

By the Faà di Bruno formula (4.3) and the identities (4.4) and (4.5), we obtain

d k d t k z -y e t -1 q = k =0 d u q d u B k, -ye t , -ye t , . . . , -ye t = k =0 q u q--ye t B k, (1, 1, . . . , 1) = k =0 q z -y e t -1 q--ye t S(k, ) → k =0
(-1) S(k, ) q z q-y as t → 0, where u = u(t) = z -y e t -1 and

q n = n-1 k=0 (q -k) = q(q -1) • • • (q -n + 1), n ≥ 1 1, n = 0
is the falling factorial of the number q.

In the formula (2.2), taking u(t) = e xt and v(t) = z -y e t -1

q yields d n d t n G(t, y, x, q, z) = (-1) n [z -y(e t -1)] q(n+1) × e xt 0 0 z + y -ye t q 0 • • • 0 xe xt 1 0 z + y -ye t q 1 1 z + y -ye t q • • • 0 x 2 e xt 2 0 z + y -ye t q 2 1 z + y -ye t q • • • 0 • • • • • • • • • . . . • • • x n-2 e xt n-2 0 z + y -ye t q (n-2) n-2 1 z + y -ye t q (n-3) • • • 0 x n-1 e xt n-1 0 z + y -ye t q (n-1) n-1 1 z + y -ye t q (n-2) • • • n-1 n-1 z + y -ye t q x n e xt n 0 z + y -ye t q (n) n 1 z + y -ye t q (n-1) • • • n n-1 z + y -ye t q → (-1) n z q(n+1) 1 0 0 z q 0 • • • 0 x 1 0 m 1 (z, y, q) 1 1 z q • • • 0 x 2 2 0 m 2 (z, y, q) 2 1 m 1 (z, y, q) • • • 0 • • • • • • • • • . . . • • • x n-2 n-2 0 m n-2 (z, y, q) n-2 1 m n-3 (z, y, q) • • • 0 x n-1 n-1 0 m n-1 (z, y, q) n-1 1 m n-2 (z, y, q) • • • n-1 n-1 z q x n n 0 m n (z, y, q) n 1 m n-1 (z, y, q) • • • n n-1 m 1 (z,
y, q) as t tends to 0. The determinantal expression (4.1) is thus proved.

Applying (3.3) to the determinant in (4.1) gives

(-1) n z q(n+1) F n (x, y, z, q) = n n -1 m 1 (z, y, q)(-1) n-1 z qn F n-1 (x, y, z, q) +(-1) n-1+1
x n z qn (-1) 1-2 z q(1-1) F 1-2 (x, y, z, q)

+ n r=2 (-1) n-r+1 n r -2
m n-r+2 (z, y, q)z q(n-r+1) (-1) r-2 z q(r-1) F r-2 (x, y, z, q), where F -1 (x, y, z, q) = -1. This can be simplified as (4.2). The proof of Theorem 4.1 is complete.

Relations between Fubini and Eulerian polynomials

It is known [START_REF] Qi | Explicit formulas and recurrence relations for higher order Eulerian polynomials[END_REF][START_REF] Qi | Some identities related to Eulerian polynomials and involving the Stirling numbers[END_REF] that Eulerian polynomials A n (t) for n ≥ 0 can be generated by

1 -t e x(t-1) -t = ∞ n=0 A n (t) x n n! , t = 1 (5.1)
and that higher order Eulerian polynomials A (α) n (t) for n ≥ 0 and α ∈ R can be generated by

1 -t e x(t-1) -t α = ∞ n=0 A (α) n (t)
x n n! , t = 1.

(5.2)

Comparing (1.2) with (5.1) and (5.2) reveals that

A n (2) = A (1) n (2) = F n = F n (1) = F n (0, 1) = F (1)
n (0, 1) = yF n (0, y, y, 1), n ≥ 0.

(5.

3)

The equations (1.4) and (5.1) can be rewritten as

1 e w -1 + 1 y = -y ∞ n=0 F n (y) w n n! and 1 e w -t = - ∞ n=0 A n (t) (t -1) n+1 w n n! .
This means that

F n (y) = y n A n 1 + 1 y (5.4)
and

A n (t) = (t -1) n F n 1 t -1 . (5.5)
Similarly, the equations(1.5) and (5.2) can be rewritten as

1 e w -1 + z y q = y q e -xw ∞ n=0 F n (x, y, z, q) w n n! and 1 (e w -t) α = ∞ n=0 A (α) n (t) (1 -t) n+α w n n! .
Then we can conclude that

F n (0, y, z, q) = (-1) n+q y n z n+q A (q) n 1 + z y and A (α) n (t) = (1 -t) n+α F n (0, 1, t -1, α). (5.6)
In [23, Theorem 1], it was obtained that Eulerian polynomials A n (t) and higher order Eulerian polynomials A (α) n (t) for n ≥ 0 and α > 0 can be computed by

A n (t) = n k=0 k!S(n, k)(t -1) n-k (5.7)
and

A (α) n (t) = 1 Γ(α) n k=0 Γ(k + α)S(n, k)(t -1) n-k , (5.8)
where Γ(z) denotes the classical Euler gamma function [START_REF] Qi | Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities[END_REF][START_REF] Qi | Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function[END_REF]. Taking t = 2 in (5.7) and using (5.3) recover (1.1). Substituting (5.5) into (5.7) yields

(t -1) n F n 1 t -1 = n k=0 k!S(n, k)(t -1) n-k
which is equivalent to (1.3). Combining (5.6) and (5.8) results in

F n (0, 1, t -1, α) = 1 Γ(α) n k=0 (-1) n-k Γ(k + α)S(n, k) (1 -t) k+α .
Substituting (5.4) into (2.1), (3.1), and (3.2) arrives at a nice determinantal expression

A n (t) = 1 1 1-t 0 0 • • • 0 0 0 0 1 1 1-t 0 • • • 0 0 0 0 1 2 1 1 1-t • • • 0 0 0 0 1 3 1 3 2 • • • 0 0 0 • • • • • • • • • • • • . . . • • • • • • • • • 0 1 n-2 1 n-2 2 • • • n-2 n-3 1 1-t 0 0 1 n-1 1 n-1 2 • • • n-1 n-3 n-1 n-2 1 1-t 0 1 n 1 n 2 • • • n n-3 n n-2 n n-1
, n ≥ 0 and two recurrence relations

A n (t) = (t -1) n-1 n-1 r=0 n r A r (t) (t -1) r , n ∈ N, A n (t) = (-1) n+1 (t -1) n-1 t 1 -t + n-1 k=0 (-1) k n k A k (t) (t -1) k , n ≥ 0.
Theorem 2 in [START_REF] Qi | Explicit formulas and recurrence relations for higher order Eulerian polynomials[END_REF] reads that the function

F (t, x) = 1 e x(t-1) -t , t = 1
satisfies the nonlinear ordinary differential equation

∂ n F (t, x) ∂x n = (t -1) n n i=0 n k=i (-1) k k!S(n, k) k i t i F i+1 (t, x), n ≥ 0.
(5.9)

The equation (5.9) was simplified in [START_REF] Qi | Some identities related to Eulerian polynomials and involving the Stirling numbers[END_REF]Theorem 5] as

∂ n F (t, x) ∂x n = (1 -t) n n k=0 S(n + 1, k + 1)k!t k F k+1 (t, x), n ≥ 0.
(5.10)

As an inversion formula of (5.10), Theorem 4 in [START_REF] Qi | Some identities related to Eulerian polynomials and involving the Stirling numbers[END_REF] states that

F n+1 (t, x) = 1 n!t n n i=0 s(n + 1, i + 1) (1 -t) i ∂ i F (t, x) ∂x i , n ≥ 0, (5.11)
where s(n, k), which can be generated by

[ln(1 + x)] k k! = ∞ n=k s(n, k) x n n! , |x| < 1,
stands for the Stirling numbers of the first kind. Taking t = 2 in (5.10) and (5.11) leads to

G (n) (x) = (-1) n n k=0 (-1) k 2 k k!S(n + 1, k + 1)G k+1 (x), n ≥ 0 (5.12)
and

G n+1 (x) = (-1) n n!2 n n i=0 (-1) i s(n + 1, i + 1)G (i) (x), n ≥ 0.
(5.13)

The equations (5.12) and (5.13) were also established in [START_REF] Qi | Simplification of coefficients in two families of nonlinear ordinary differential equations[END_REF]Theorem 1] by three alternative approaches. By similar methods to those in [START_REF] Qi | Explicit formulas and recurrence relations for higher order Eulerian polynomials[END_REF][START_REF] Qi | Simplification of coefficients in two families of nonlinear ordinary differential equations[END_REF][START_REF] Qi | Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations[END_REF][START_REF] Qi | Some identities related to Eulerian polynomials and involving the Stirling numbers[END_REF][START_REF] Qi | Some identities for a sequence of unnamed polynomials connected with the Bell polynomials[END_REF] and closely related references therein, we can easily deduce

∂ n G(t, y) ∂t n = (-1) n n m=0 (-1) m (1 + y) m m!S(n + 1, m + 1)G m+1 (t, y), n ≥ 0 and G n+1 (t, y) = (-1) n n!(1 + y) n n m=0 (-1) m s(n + 1, m + 1) ∂ m G(t, y) ∂t m , n ≥ 0.
In [START_REF] Qi | Explicit formulas and recurrence relations for higher order Eulerian polynomials[END_REF]Theorem 3] and [START_REF] Qi | Some identities related to Eulerian polynomials and involving the Stirling numbers[END_REF]Theorem 6], it was obtained that

n k=0 (-1) k n k S(n -k, 0) (1 -t) k-1 + S(n -k, 1) (1 -t) k A k (t) = 0, n ∈ N (5.14) and n k=0 s(n, k) (t -1) k A k (t) = n! (t -1) n , n ≥ 0.
(5.15) Substituting (5.5) into (5.14) and (5.15) arrives at

n k=0 n k S(n -k, 1) - S(n -k, 0) t F k (t) = 0, n ∈ N and n k=0 s(n, k)F k (t) = n!t n , n ≥ 0.

Remarks

Finally, we list several remarks.

F n = n-1 r=0 n r F r , n ∈ N and F n = (-1) n 2 1 - n-1 k=0 (-1) k n k F k , n ≥ 0.
These two recurrence relations appeared in [START_REF] Dil | Investigating geometric and exponential polynomials with Euler-Seidel matrices[END_REF] in different forms. This recovers once again that the Fubini numbers and polynomials F n and F n (y) have generating functions 1 2-e t and 1 1-y(e t -1) .

Remark 6.3. On 24 July 2018, when commenting on the paper [START_REF] Qi | Some identities related to Eulerian polynomials and involving the Stirling numbers[END_REF] on ResearchGate, Professor Dr. Boyadzhiev (Ohio Northern University) recommended the website http://www.luschny.de/ math/euler/EulerianPolynomials.html and two papers [START_REF] Boyadzhiev | Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials[END_REF][START_REF] Foata | Eulerian polynomials: from Euler's time to the present[END_REF]. He said that, when writing about Eulerian polynomials, it is good to mention that they originate from Euler and their theory is very classical-generating function and other formulas have been known for very long time. He pointed out that the paper [START_REF] Boyadzhiev | Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials[END_REF] is a review where Eulerian polynomials are linked to geometric polynomials and Apostol-Bernoulli polynomials. On 15 May 2019, Professor Dr. Boyadzhiev (Ohio Northern University) commented on Research-Gate again that the author should take in consideration the paper [START_REF] Boyadzhiev | Geometric polynomials: properties and applications to series with zeta values[END_REF] and that the name "geometric polynomials" is established in the mathematics community while "Fubini polynomials" is not.

Remark 6.4. The ideas in this paper and the formulas (2.2) and (3.3) have been utilized in the papers [START_REF] Qi | Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials[END_REF][START_REF] Qi | A determinantal expression and a recurrence relation for the Euler polynomials[END_REF][START_REF] Qi | Expressing the generalized Fibonacci polynomials in terms of a tridiagonal determinant[END_REF][START_REF] Qi | Some determinantal expressions and recurrence relations of the Bernoulli polynomials[END_REF][START_REF] Qi | Two nice determinantal expressions and a recurrence relation for the Apostol-Bernoulli polynomials[END_REF][START_REF] Qi | Alternative proofs of some formulas for two tridiagonal determinants[END_REF][START_REF] Qi | A determinantal expression for the Fibonacci polynomials in terms of a tridiagonal determinant[END_REF][START_REF] Qi | A representation for derangement numbers in terms of a tridiagonal determinant[END_REF][START_REF] Qi | Some properties of the Bernoulli numbers of the second kind and their generating function[END_REF][START_REF] Qi | Closed forms for derangement numbers in terms of the Hessenberg determinants[END_REF] and closely related references therein. Remark 6.5. This paper is an extension of the preprint [START_REF] Qi | On generalized Fubini polynomials[END_REF] and a slight revision of the preprint [START_REF] Qi | Determinantal expressions and recurrence relations for Fubini and Eulerian polynomials[END_REF].

Remark 6 . 1 .

 61 When y = 1, the recurrence relations (3.1) and (3.2) becomes

Remark 6 . 2 .

 62 Let u = u(t) = y(e t -1). Then, by the Faà di Bruno formula (4.3) and the identities (4.4) and (4.5)B n,k ye t , ye t , . . . , ye t u) k+1 ye t k B n,k (1, 1, . . . , 1) = n k=0 S(n, k)k!y k = F n (y).


M Matallah
email: matallah@mail.univ-tlemcen.dz

M Farah

F Grondin

J Saliba

A Loukili

FRACTURE ENERGY OF CONCRETE AT VERY EARLY AGES BY INVERSE ANALYSIS

Keywords: Fracture energy, early age, inverse analysis

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INTRODUCTION

The fracture energy is one of the most important fracture characteristics of concrete. The determination of the evolution of the fracture energy at early-ages has been already investigated by some authors [START_REF] Wittmann | Influence of age of loading, water-cement ratio and rate of loading of fracture energy of concrete[END_REF][START_REF] Østergaard | Early age stress-crack opening relationships for high performance concrete[END_REF][START_REF] Schutter | Fracture Energy of concrete at early ages[END_REF][START_REF] Kim | Fracture characteristics of concrete at early ages[END_REF]. It was found that the fracture energy increases with age. Based on the three-point bending test or the wedge splitting test, the fracture energy could be evaluated directly from the load-displacement curves [START_REF] Guinea | Measurement of the fracture energy using 3-point bend tests .1. Influence of experimental procedures[END_REF][START_REF] Planas | Measurement of the fracture energy using 3-point bend tests .2. Influence of bulk energy-dissipation[END_REF][START_REF] Elices | Measurement of the fracture energy using 3-point bend tests .3. Influence of cutting the p-delta tail[END_REF] according to the RILEM recommendation, or indirectly, by numerical inverse analysis. When using inverse analysis, experimental results are approximated by repeated numerical simulations of the tests. Thus, a function is needed to describe concrete softening. Discrete models (Fictitious crack models [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements Cement[END_REF]) or smeared crack models based on the crack band theory [START_REF] Bazant | Crack band theory for fracture of concrete[END_REF] are often used for modelling crack initiation and propagation in concrete. However these models require a stress-crack opening relationship. The area under the tensile softening curve, described by this relationship, determines the specific fracture energy G f .

As defined by the RILEM technical committee, the specific fracture energy measurement is based on the load-displacement (pδ) curve and expressed as :

G f = 1 B(W -a) p∂δ ( 1
)
Where B is the specimen thickness, W its width in the crack direction and a the initial notch or crack length. The RILEM energy definition is based on Hillerborg's fictitious crack model. So, the motivation of the RILEM was to provide (from load and load-displacements curves of concrete) a constant parameter G f representing the area under the tensile softening curve which is supposed unique. However, as it is widely agreed, size effect on the RILEM G f exists [START_REF] Bazant | Size effect on blunt fracture : concrete, rock, metal[END_REF][START_REF] Bazant | Size dependence of concrete fracture energy determined by RILEM workof-fracture[END_REF][START_REF] Duan | Size effect on specific fracture energy of concrete[END_REF]. G f depends not only on the specimen size W but also on the crack length a even if W is constant. There is a size/ligament effect on G f [START_REF] Duan | Boundary effect on concrete fracture and non-constant fracture energy distribution[END_REF][START_REF] Hu | Size effect : Influence of proximity of fracture process zone to specimens boundary[END_REF]. Eq 1 gives only an average value where the fracture energy is assumed to be uniform over the crack length. Definition of G f from a complete pδ curve includes the boundary region which explains why the RILEM G f is ligament dependent. Inverse analysis obtained without considering boundary effect implies a constant σdw (stress-crack opening) relation. If the inverse analysis is performed with a model that does not take into account boundaries effects, the solution is to fit only a part of the Pδ (or p -CMOD) curve where the distribution of the local fracture energy is uniform. In this region (the inner zone), a good estimation of the size independent fracture energy (G F ) could be obtained. The objective of the present study is to contribute to the identification of the fracture energy at very early ages using numerical inverse analysis. Experimentally, three point bending tests are performed on rectangular notched specimens at ages between 12 hours and 24 hours [START_REF] Grondin | Improving the characterization of early age cracking of concrete: engineering issues and scientific answers[END_REF]. The Load-CMOD curves are obtained. In order to estimate the fracture energy a damage based model is used to represent the softening behaviour of concrete. A fracture energy regularization is applied to this model in order to avoid spurious mesh sensitivity. Therefore, the energy dissipation due to the fracture per unit length (or unit width) described by the stress-strain curves is constant. As the softening behaviour is non linear, the Levenberg-Marquardt [START_REF] Levenberg | A Method for the solution of certain non-linear problems in least squares[END_REF][START_REF] Marquardt | An Algorithm for least-squares estimation of nonlinear parameters[END_REF] algorithm (LMA) is used to perform inverse analysis. Only a part of the P -CMOD curves is fitted in order to estimate the size independent fracture energy.

Experimental program 2.1 Materials and Methods

Concrete formulation

The materials constituting the CEOS concrete [START_REF]fr 2008 Comportement et évaluation des ouvrages spéciaux fissuration retrait[END_REF] are: semi-crushed gravel of class 4/20 mm, alluvial sand of class 0/4 mm and Cement CEMI 52.2 N . A small amount of super-plasticizer (Optima 206) was adjusted manually for workability. The formulation used is given in table 2 . The water to cement ratio (W/C) is 0.42 and the gravel to sand ration (G / S) is 1.25.

Experimental procedure

Concrete was poured into moulds of dimensions 10 * 15 * 70cm 3 . A Teflon plate with a thickness of 3mm was placed in the mould before casting for the notch position. It was removed at the time of the test. To better interpret the repeatability tests, the slump test of the Abrams cone was carried out . Concrete was poured into moulds in two layers and vibrated with a plate vibrator. Specimens are then covered with an impermeable plastic sheet and kept in an air conditioned room at a temperature of 20 C and a relative humidity of 50%. The formwork was removed with care one hour before the test to glue the plates (which were used to hold the CMOD gauge) and the reflector for the measurement of the deflection. The loading frame consists of an Instron make closed loop universal testing machine of 160 kN capacity. The load was applied with a circular jack to ensure a point load. A rubber pad was placed between the load jack and the beam to take care of the surface unevenness and to avoid damage under the load. The beam was simply supported on two circular supports. The notch mouth opening displacement was measured with a CMOD gauge. The two blades of the CMOD gauge, each 10 mm in length, were attached to two metallic plates 10 mm apart, one on either side of the notch at the bottom face of the beam. These metal plates were firmly attached to other plates (which were inserted into the mould before casting concrete) with high strength glue that ensures perfect stability of plates. All the tests were performed under crack mouth opening displacement control with a slow rate of 0.5 µm/sec. This allowed to reach the peak load in about one minute. Through these tests, the load was obtained as a function of time, CMOD and deflection. However, the load-deflection curve is not always reliable compared to the load-CMOD which is in most case good. The data were recorded every one second. The deflection was measured by a laser sensor (Figure 2) which allows to calculate the optic deflection of the reflector (corresponding to that of the beam) placed in the middle part of the beam on the neutral axis of the UN-cracked part.

Experimental results

Figure 3 shows the development of the Young's modulus with age. The Young's modulus increase is due to the formation of hydration products. The products actually continue to form and the amount of water continues to drop which makes the cohesion between the aggregates and the cement paste stronger and contributes to the increase of concrete strength. Thus, the stiffness increases over time. Young's Modulus (GPa)

Ages (H) Figure 4 shows the Load-CMOD curves at the age of 12 h-14 h-18 h-24 h ages. The curves show that the failure load and the slope of the elastic part increases with age while the postpeak part decreases rapidly. Increase in the elas-tic slope can be attributed to the evolution of the Young's modulus with time. The rapid fall of the curve after the peak shows that at certain ages concrete become more brittle.

Stress-strain softening curve

In order to perform inverse analysis, the stress-crack opening curve (σdw) is needed. A damage based model is chosen to represent the softening behaviour of concrete. Therefore, the σdw softening curve is implicitly introduced. The stress-strain relationship is given by [START_REF] Fichant | Isotropic and anisotropic description of damage in concrete structures[END_REF]

σ ij = (1 -d)C 0 ijkl ε kl = (1 -d)σ ij (2)
Where C 0 ijkl is the initial stiffness tensor, d the damage variable and σ the effective stress.

The softening behaviour is represented by an exponential damage evolution law

d = 1 - ε d0 ε exp(B(ε d0 -ε)) (3)
B is a parameter which commands the slope of the softening curve defined by the exponential expression and ε d0 is the strain threshold.

According to the Crack Band Theory, instead of treating cracks as lines as in the Fictitious crack model [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements Cement[END_REF], Bazant and Oh [START_REF] Bazant | Crack band theory for fracture of concrete[END_REF] consider the fracture zone to have a certain width h over which micro-cracks are uniformly distributed. The energy dissipation due to fracture per unit length (or unit width) is therefore a constant. The fracture energy is given by :

G f = ∞ 0 σδw (4)
Where w is the crack opening displacement

δw = ε f h (5)
with ε f the fracture strain.

To apply this to the damage model, the softening behaviour must be changed such the same energy dissipation is imposed whatever the element size [START_REF] Matallah | A practical Method to estimate crack openings in concrete structures[END_REF] :

G f h = ∞ 0 ((1 -d)Eε) dε (6)
Where h is the characteristic length related to the element size. So, the local energy in each element g f elem is assumed to be uniformly smeared over an element G f elem = h * g f leme . This implies that micro-cracks are assumed to be uniformly distributed over a finite element as prescribed by the crack band theory. Using the stress-strain relationship, the fracture energy is therefore given by :

G f h = f t ε d0 2 + f t B (7)
With f t the tensile strength and E the Young's modulus f t = E * ε d0 . An example of a stress-crack opening curve obtained by this model is given in figure 5 (more details are given in [START_REF] Matallah | Inelasticity-damage-based model to numerical modelling of concrete cracking[END_REF]. The crack opening displacement is calculated using the procedure developed by Matallah et al [START_REF] Matallah | Inelasticity-damage-based model to numerical modelling of concrete cracking[END_REF][START_REF] Matallah | A practical Method to estimate crack openings in concrete structures[END_REF]

Inverse Analysis Algorithm

In order to find a trend of the evolution of the fracture energy with age, inverse analysis is carried out. As the stress-strain relationship describing the tensile softening curve is non linear, the Levenberg-Marquardt Algorithm (LMA) is used. Also known as the damped least-squares (DLS), it is one of the most important algorithms for solving systems of non linear equations. It is widely adopted for solving curvefitting problems in many research area. To obtain the optimum solution of the experimental P -CMOD curves, we minimise the sum of weighted squares of the errors between the measured data F (u i ) and the curve-fit function F (u i , p), where vector p is the vector of parameters. The inverse analysis procedure involves two major steps :

• formulation of the mechanical model,

• estimation of the coefficient of this model by inverse analysis.

Using the damage model exposed above, only two fracture parameters are used as input parameters p=(f t ,G f ) (the tensile strength and the fracture energy).

Results and Discussion

The damage model used above does not take into account boundary effects. As it has been outlined before, the solutions proposed was to fit partially the Load-CMOD curves. Hence an estimation of the size independent fracture energy is obtained. Figure 6 shows the development of the size independent fracture energy with age. Overall, The data indicate an "upward trend" (increasing of the fracture energy with age) which is in agreement with the most experimental results. Fracture Energy (N/m) Ages (H)

Conclusion

In this study, the fracture characteristics of concrete at very early-ages were investigated. An experimental procedure was carried out to perform three point bending tests on notched beams of concrete specimens at very-early ages (before 24H). The Load-CMOD curves were established in order to determine the fracture characteristics. To this end, an inverse analysis procedure was developed. The stress-crack opening curve which is needed in such analysis is implicitly taken into account by a damage model through the stress-strain relationship. Only two physical parameters (G F and f t ) are used as input parameters. At very-early ages, an "upward trend" was found regarding the evolution of the fracture energy. Summing up, the numerical inverse analysis developed in this study has shown its applicability for the identification of the fracture characteristics of concrete. The numerical results as the experimental ones confirm the basis trend already stated in the literature.

Figure 1 :Figure 2 :

 12 Figure 1: Schematic representation of the three-point bending test setup

Figure 3 :

 3 Figure 3: Young's Modulus evolution

Figure 4 :

 4 Figure 4: Load-CMOD curves for different ages

Figure 5 :

 5 Figure 5: Stress-Crack opening curve

Figure 6 :

 6 Figure 6: Development of fracture energy with age

Table 1 :

 1 Characteristics of used aggregates

		Sand 0/4 Gravel 4/20
	Water content %	3.5	2.1
	Absorption %	0.7	1.39
	Density (Kg/l)	2.6	2.57

Table 2 :

 2 Concrete formulation

		Dosage (Kg/m 3 )
	Gravel 4/20	980
	Sand 0/4	785
	CEMI 52.2N	400
	Super plasticizer	1.8
	Added water	154.1
	Effective Water	165.9


Feng Qi

AND Dongkyu Lim

Yong-Hong Yao

NOTES ON TWO KINDS OF SPECIAL VALUES FOR THE BELL POLYNOMIALS OF THE SECOND KIND

In the paper, by methods and techniques in combinatorial analysis and the theory of special functions, the authors discuss two kinds of special values for the Bell polynomials of the second kind for two special sequences, find a relation between these two kinds of special values for the Bell polynomials of the second kind, and derive an identity involving the combinatorial numbers.

Motivation

In [START_REF] Charalambides | Enumerative Combinatorics[END_REF]Definition 11.2] and [2, p. 134, Theorem A], the Bell polynomials of the second kind, denoted by B n,k (x 1 , x 2 , . . . , x n-k+1 ) for n ≥ k ≥ 0, are defined by

B n,k (x 1 , x 2 , . . . , x n-k+1 ) = 1≤i≤n-k+1 i ∈{0}∪N n-k+1 i=1 i i =n n-k+1 i=1 i =k n! n-k+1 i=1 i ! n-k+1 i=1 x i i! i .
For more information on the Bell polynomials of the second kind B n,k , please refer to the monographs and handbooks [START_REF] Charalambides | Enumerative Combinatorics[END_REF][START_REF] Comtet | Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition[END_REF][START_REF]NIST Handbook of Mathematical Functions[END_REF] and closely related references therin.

In [1, p. 451], the formulas B 2r,k (0, 2!, . . . , 0, (2r

)!) = (2r)! k! r -1 k -1 , B 2r-1,k (0,
B 2r,2s (1!, 0, . . . , (2r -1)!, 0) = (2r)! (2s)! r + s -1 2s -1 , B 2r,2s-1 (1!, 0, . . . , (2r -1)!, 0) = 0, B 2r-1,2s-1 (1!, 0, . . . , (2r -1)!, 0) = (2r -1)! (2s -1)! r + s -2 2s -2 , B 2r-1,2s (1!
, 0, . . . , 0, (2r -1)!) = 0 were stated, but no proof was supplied for them there.

For simplicity, we denote

λ(n, k) = B n,k 0, 2!, 0, 4!, . . . , (n -k + 1)! 1 + (-1) n-k+1 2 and µ(n, k) = B n,k 1!, 0, 3!, 0, . . . , (n -k + 1)! 1 -(-1) n-k+1 2 .
In terms of these notations, the above claims in [1, p. 451] can be restated as

λ(2r, k) = (2r)! k! r -1 k -1 , λ(2r -1, k) = 0, µ(2r, 2s) = (2r)! (2s)! r + s -1 2s -1 , µ(2r, 2s -1) = 0, µ(2r -1, 2s -1) = (2r -1)! (2s -1)! r + s -2 2s -2 , µ(2r -1, 2s = 0.
In this paper, by methods and techniques in combinatorial analysis and the theory of special functions, we will provide alternative proofs for the above six formulas, find a relation between them, and derive an identity involving combinatorial numbers.

Main reults

We first derive an identity involving combinatorial numbers, which will be useful in next proofs of our main results.

Theorem 1. For k ≥ 1 and n ≥ 0, we have

n q=0 (-1) q k + q n q = 1 k k+n k . (2.1) Proof. Let f (x) = n q=0 (-1) q k + q n q x k+q , x ∈ [-1, 1].
Then f (0) = 0 and

f (x) = n q=0 (-1) q n q x k+q-1 = x k-1 n q=0 (-1) q n q x q = x k-1 (1 -x) n .
Integrating from 0 to t ∈ (0, 1] on both sides of the above equality yields

f (t) = t 0 x k-1 (1 -x) n d x = B(t; k, n + 1),
where B(z; a, b) denotes the incomplete beta function, see [3, p. 183]. Therefore, we obtain

f (1) = B(1; k, n + 1) = B(k, n + 1) = Γ(k)Γ(n + 1) Γ(k + n + 1) = (k -1)!n! (k + n)! = 1 k k!n! (k + n)! = 1 k n+k k ,
where B(a, b) and Γ(z) denote the beta function and the classical Euler gamma function respectively, see [3, p. 142] and [START_REF]NIST Handbook of Mathematical Functions[END_REF]Chapter 5]. The formula (2.1) is thus proved. The proof of Theorem 1 is complete.

We are now in a position to state and prove our main results.

Theorem 2. For n ≥ k ≥ 0, we have the relation µ(n, k) n! = λ(n + k, k) (n + k)! (2.2)
and two explicit formulas

λ(n, k) = 1 + (-1) n 2 n! k! n 2 -1 k -1 (2.3) and µ(n, k) = 1 + (-1) n+k 2 n! k! n+k 2 -1 k -1 . (2.4)
Proof. It is known [START_REF] Comtet | Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition[END_REF][START_REF] Qi | Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers[END_REF] that the quantities

x n = n-1 =0 (x -) = x(x -1) • • • (x -n + 1), n ≥ 1 1, n = 0 and (x) n = n-1 =0 (x + ) = x(x + 1) • • • (x + n -1), n ≥ 1 1, n = 0
are called the falling and rising factorials respectively. In [2, p. 133], it was listed that

1 k! ∞ m=1 x m t m m! k = ∞ n=k B n,k (x 1 , x 2 , . . . , x n-k+1 ) t n n! (2.5)
for k ≥ 0. Hence, we have

∞ n=k λ(n, k) t n n! = 1 k! ∞ m=1 m! 1 + (-1) m 2 t m m! k = 1 k! t 2 1 -t 2 k which is equivalent to ∞ n=0 λ(n + k, k) 1 n+k n t n n! = t 1 -t 2 k . Since the function t 1-t 2 is odd, we derive λ(2r-1, k) = 0. Further computation yields λ(n + k, k) = n + k n lim t→0 d n d t n t 1 -t 2 k = n + k n 1 2 k lim t→0 1 1 -t - 1 1 + t k (n) = n + k n 1 2 k lim t→0 k =0 k 1 1 -t (-1) k- 1 1 + t k-(n) = n + k n 1 2 k k =0 (-1) k-k lim t→0 n q=0 n q 1 1 -t (q) 1 1 + t k-(n-q) = n + k n 1 2 k k =0 (-1) k-k lim t→0 n q=0 n q × -q (-1) q 1 1 -t +q -(k -) n-q 1 1 + t k-+(n-q) = (-1) k 2 k n + k n k =0 (-1) k n q=0 (-1) q n q -q -k n-q = (-1) n+k 2 k n + k n k =0 (-1) k n q=0 (-1) q n q ( ) q (k -) n-q .
In summary, we obtain

λ(n, k) = (-1) k 2 k n k k =0 (-1) k n-k q=0 (-1) q n -k q -q -k n-k-q = (-1) n 2 k n k k =0 (-1) k n-k q=0 (-1) q n -k q ( ) q (k -) n-k-q . (2.6)
By similar argument as above, it follows that

∞ n=k µ(n, k) t n n! = 1 k! ∞ m=1 m! 1 -(-1) m 2 t m m! k = (-1) k k!2 k 1 t -1 + 1 t + 1 k which is equivalent to ∞ n=0 µ(n + k, k) 1 n+k n t n n! = (-1) k 2 k 1 t k 1 t -1 + 1 t + 1 k = 1 1 -t 2 k . (2.7)
Since 1 1-t 2 is even, we deduce immediately that µ(2r-1, 2s) = 0 and µ(2r, 2s-1) = 0. By the L'Hôspital rule and the identity (2.1) in Theorem 1, it follows that

µ(n + k, k) = n + k n (-1) k 2 k lim t→0 d n d t n 1 t k 1 t -1 + 1 t + 1 k = n + k n (-1) k 2 k lim t→0 n q=0 n q -k q t k+q 1 t -1 + 1 t + 1 k (n-q) = n + k n (-1) k 2 k n q=0 n q -k q (k + q)! lim t→0 1 t -1 + 1 t + 1 k (n+k) = n + k n (-1) k 2 k n q=0 n q (-1) q (k + q -1)! (k -1)!(k + q)! × lim t→0 k =0 k 1 t -1 1 t + 1 k-(n+k) = n + k n (-1) k 2 k (k -1)! n q=0 (-1) q k + q n q k =0 k × lim t→0 n+k m=0 n + k m 1 t -1 (m) 1 t + 1 k-(n+k-m) = (-1) k 2 k k! k =0 k lim t→0 n+k m=0 n + k m × -m 1 t -1 +m -k n+k-m 1 t + 1 (k-)+(n+k-m) = (-1) k 2 k k! k =0 (-1) k n+k m=0 (-1) m n + k m -m -k n+k-m .
In short, we obtain

µ(n, k) = (-1) k 2 k k! k =0 (-1) k n m=0 (-1) m n m -m -k n-m = (-1) n+k 2 k k! k =0 (-1) k n m=0 (-1) m n m ( ) m (k -) n-m . (2.8)
Comparing between (2.6) and (2.8) reveals

µ(n, k) = (-1) n+k 2 k k! 2 k (-1) n+k 1 n+k k λ(n + k, k)
which can be rearranged as (2.2). The Faà di Bruno formula, see [START_REF] Charalambides | Enumerative Combinatorics[END_REF]Theorem 11.4] and [2, p. 139, Theorem C], can be described in terms of B n,k (x 1 , x 2 , . . . , x n-k+1 ) by

d n d x n f • h(x) = n k=0 f (k) (h(x)) B n,k h (x), h (x), . . . , h (n-k+1) (x) .
(2.9)

In [1, p. 412] and [2, p. 135], one can find the identity

B n,k abx 1 , ab 2 x 2 , . . . , ab n-k+1 x n-k+1 = a k b n B n,k (x 1 , x 2 , . . . , x n-k+1 ) (2.
B n,k (x, 1, 0, . . . , 0) = 1 2 n-k n! k! k n -k x 2k-n , (2.11)
where 0 0 = 1 and p q = 0 for q > p ≥ 0. For detailed information on applications of the formula (2.11), please refer to the papers [4-8, 12, 14-17] and closely related references therein. Then it follows from (2.7), (2.9), (2.10), and (2.11) that, when denoting u = u

(t) = 1 -t 2 , µ(n + k, k) = n + k n lim t→0 d n d t n 1 1 -t 2 k = n + k n lim t→0 n =0 1 u k ( ) B n, (-2t, -2, 0, . . . , 0) = n + k n lim t→0 n =0 -k 1 u k+ (-2) B n, (t, 1, 0, . . . , 0) = n + k n lim t→0 n =0 -k 1 (1 -t 2 ) k+ (-2) 1 2 n- n! ! n - t 2 -n = n + k n lim t→0 n =0 (k) n! ! n - (2t) 2 -n (1 -t 2 ) k+ =    0, n = 2m -1 2m + k 2m (k) m (2m)! m! m 2m -m , n = 2m =    0, n = 2m -1 (2m + k)! k! m + k -1 k -1 , n = 2m =    0, n = 2m -1 (n + k)! k! n+2k 2 -1 k -1 , n = 2m
for m ∈ N. The formula in (2.4) is thus proved. Substituting the formula in (2.4) into (2.2) and simplifying lead readily to the formula in (2.3). The proof of Theorem 2 is complete.

Remarks

In this section, we state several remarks on something related. Remark 1. In [8, Theorem 2.1], it was proved that B n,k 1, 0, 1, . . . ,

1 -(-1) n-k+1 2 = 1 2 k k! k =0 (-1) k (k -2 ) n and B n,k 0, 1, 0, . . . , 1 + (-1) n-k+1 2 = 1 2 k k! 2k =0 (-1) 2k (k -) n
for n ≥ k ≥ 0, where 0 0 is regarded as 1. In [8, Section 3], basing on numerical calculation, the authors guessed that B 2 -1,k 0, 1, 0, . . . ,

1 + (-1) (2 -1)-k+1 2 = 0, 2 -1 ≥ k ≥ 0, (3.1)
B 2 ,k 0, 1, 0, . . . ,

1 -(-1) 2 -k+1 2 = 0, 2 > k > , (3.2)
B 2 ,k 0, 1, 0, . . . ,

1 -(-1) 2 -k+1 2 = 0, ≥ k ≥ 1, (3.3) B k+2 ,k (1, 0, 1, . . . , 1, 0, 1) = 0, k, ∈ N, (3.4)
and B k+2 -1,k (1, 0, 1, . . . , 0, 1, 0) = 0, k, ∈ N.

(3.5)

In [8, Theorem 3.1], the identity (3.5) was proved to be true. In fact, the identities (3.1) and (3.5) can be concluded readily from the proof of [8, Theorem 2.1] as follows. From the formula (2.5), it followed that Remark 2. The first formula in (2.3) of Theorem 2 was also discussed in [9, Lemma 2.5], [START_REF] Qi | Special values of the Bell polynomials of the second kind for some sequences and functions[END_REF] and closely related references therein. summable?

∞ n=k B n,k 0, 1, 0, . . . , 1 + (-1) n-k+1 2
t n n! = (cosh t -1) k k! (3.6) and ∞ n=k B n,k 1, 0, 1, . . . , 1 -(-1) n-k+1 2
t n n! = sinh k t k! . ( 3
Remark 4. This paper is a slightly revised version of the preprint [START_REF] Qi | Notes on two kinds of special values for the Bell polynomials of the second kind[END_REF].

  10) for n ≥ k ≥ 0 and a, b ∈ C. In [8, Theorem 5.1] and [15, Section 3], it was set up little by little that

. 7 )

 7 In(3.6), the function cosh t -1 is even, so the identity (3.1) is clearly valid. Since the function sinh t in (3.7) is odd, then B 2n,2k-1 (1, 0, 1, . . . , 1, 0) = 0 and B 2n-1,2k (1, 0, 1, . . . , 1, 0) = 0 which are equivalent to the identity (3.5). However, the validity of the identities (3.2), (3.3), and (3.4) has not been verified yet.

Remark 3 . 1 .

 31 The formula (2.8) can be rewritten asµ(n, k) = (-1) n+k n!2Comparing these identities with the Vandermonde convolution formula n k=0 n k x k y n-k = x + y n in [1, Theorem 3.1] and [2, p. 44] motivates us to ask a question: is the quantity n k=0 (-1) k n k x k y n-k

Acknowledgements. The second author was supported by the National Research Foundation of Korea (Grant No. 2018R1D1A1B07041846).

Authors' addresses