EXPLICIT FORMULAS FOR SPECIAL VALUES OF

EXPLICIT FORMULAS FOR SPECIAL VALUES OF THE BELL POLYNOMIALS OF THE SECOND KIND AND FOR THE EULER NUMBERS AND POLYNOMIALS FENG QI AND BAI-NI GUO

Abstract. In the paper, the authors establish by two approaches several explicit formulas for special values of the Bell polynomials of the second kind, derive explicit formulas for the Euler numbers and polynomials in terms of double sums, and find a property for special values of the Bell polynomials of the second kind.

1. Introduction In combinatorics, the Bell polynomials of the second kind, or say, the partial Bell polynomials, Bn,k (x1 , x2 , . . . , xn−k+1 ) can be defined by Bn,k (x1 , x2 , . . . , xn−k+1 ) =

n−k+1 Y 

n!

X 1≤i≤n,` Pn i ∈{0}∪N i`i =n Pi=1 n i=1 `i =k

Qn−k+1 i=1

`i !

i=1

x i  `i i!

for n ≥ k ≥ 0. The well-known Fa`a di Bruno formula can be described in terms of the Bell polynomials of the second kind Bn,k (x1 , x2 , . . . , xn−k+1 ) by (1.1)

n X
dn f ◦ g(x) = f (k) (g(x))Bn,k g 0 (x), g 00 (x), . . . , g (n−k+1) (x) n dx k=0

for n ≥ 0. See [5, p. 139, Theorem C]. It is well known that the Euler numbers En can be defined by ∞ X 1 tn En , = cosh t n=0 n!

|t| < π.

1 Since the function cosh t is even on (−π, π), we see that E2k−1 = 0 for all k ∈ N. The absolute values of the Euler numbers with even indexes, |E2k | for all k ∈ N, are also called secant numbers, due to

sec z =

∞ X

(−1)n E2n

n=0

z 2n , (2n)!

|z|