Closed formulas and determinantal expressions for higher-order Bernoulli and Euler polynomials in terms of Stirling numbers Muhammet Cihat DAGLI˘ Department of Mathematics, Akdeniz University, 07058-Antalya, Turkey E-mail: [email protected] Abstract In this paper, applying the Fa`adi Bruno formula and some prop- erties of Bell polynomials, several closed formulas and determinantal expressions involving Stirling numbers of the second kind for higher- order Bernoulli and Euler polynomials are presented. Keywords: Bernoulli and Euler polynomials, Stirling numbers, Closed forms, Determinantal expression. Mathematics Subject Classification 2010: 11B68, 05A19, 11B83, 11C20, 11Y55. 1 Introduction arXiv:2005.03921v1 [math.NT] 8 May 2020 The classical Bernoulli polynomials Bn(x) and Euler polynomials En(x) are usually defined by means of the following generating functions: text ∞ tn 2ext ∞ tn = B (x) (|t| < 2π) and = E (x) (|t| < π) . et − 1 n n! et +1 n n! n=0 n=0 X X n In particular, the rational numbers Bn = Bn(0) and integers En =2 En(1/2) are called classical Bernoulli numbers and Euler numbers, respectively. 1 As is well known, the classical Bernoulli and Euler polynomials play im- portant roles in different areas of mathematics such as number theory, com- binatorics, special functions and analysis. Numerous generalizations of these polynomials and numbers are defined and many properties are studied in a variety of context. One of them can be (α) traced back to N¨orlund [7]: The higher-order Bernoulli polynomials Bn (x) (α) and higher-order Euler polynomials En (x), each of degree n in x and in α, are defined by means of the generating functions t α ∞ tn ext = B(α)(x) (1.1) et − 1 n n! n=0 X and 2 α ∞ tn ext = E(α)(x) , (1.2) et +1 n n! n=0 X (1) (1) respectively. For α = 1, we have Bn (x)= Bn(x) and En (x)= En(x). According to Wiki [2], ”In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of oper- ations. It may contain constants, variables, certain ‘well-known’ operations (e.g.,+ − ×÷), and functions (e.g., nth root, exponent, logarithm, trigono- metric functions, and inverse hyperbolic functions), but usually no limit.” From this point of view, Wei and Qi [15] studied several closed form expressions for Euler polynomials in terms of determinant and the Stirling numbers of the second kind. Also, Qi and Chapman [8] established two closed forms for the Bernoulli polynomials and numbers involving the Stirling numbers of the second kind and in terms of a determinant of combinatorial numbers. Moreover, some special determinantal expressions and recursive formulas for Bernoulli polynomials and numbers can be found in [9]. In 2018, Hu and Kim [6] presented two closed formulas for Apostol- Bernoulli polynomials by aid of the properties of the Bell polynomials of the second kind Bn,k (x1, x2, ..., xn−k+1) (see Lemma 2.1, below). Very recently, Dai and Pan [4] have obtained the closed forms for degenerate Bernoulli polynomials. In this paper, we focus on higher-order Bernoulli and Euler polynomials in those respects, mentioned above. Firstly, we find some novel closed for- mulas for higher-order Bernoulli and Euler polynomials in terms of Stirling numbers of the second kind S(n, k) via the Fa`adi Bruno formula for the 2 Bell polynomials of the second kind and the generating function methods. Secondly, we establish some determinantal expressions by applying a formula of higher order derivative for the ratio of two differentiable functions. Conse- quently, taking some special cases in our results provides the further formulas for classical Bernoulli and Euler polynomials, and numbers. 2 Some Lemmas In order to prove our main results, we recall several lemmas below. Lemma 2.1 ( [3, p. 134 and 139]) The Bell polynomials of the second kind, or say, partial Bell polynomials, denoted by Bn,k (x1, x2, ..., xn−k+1) for n ≥ k ≥ 0, defined by ∞ l−k+1 li n! xi Bn,k (x1, x2, ..., xn−k+1)= l−k+1 . l ! i=1 i! 1≤i≤n, li∈{0}∪N i=1 i n X n Q i=1ili=n, i=1li=k Q The Fa`adi Bruno formula canP be describedP in terms of the Bell polynomials of the second kind Bn,k (x1, x2, ..., xn−k+1) by n dn f ◦ h (t)= f (k) (h (t)) B h′ (t) , h′′ (t) , ..., h(n−k+1) (t) . (2.1) dtn n,k k=0 X Lemma 2.2 ( [3, p. 135]) For n ≥ k ≥ 0, we have 2 n−k+1 k n Bn,k abx1, ab x2, ..., ab xn−k+1 = a b Bn,k (x1, x2, ..., xn−k+1) , (2.2) where a and b are any complex number. Lemma 2.3 ( [3, p. 135])For n ≥ k ≥ 0, we have Bn,k (1, 1, ..., 1) = S (n, k) , (2.3) where S (n, k) is the Stirling numbers of the second kind, defined by [3, p. 206] (et − 1)k ∞ tn = S (n, k) . k! n! n k X= 3 Lemma 2.4 ( [5])For n ≥ k ≥ 1, we have k 1 1 1 n! n + k B , , ..., = (−1)k−i S (n + i, i) . n,k 2 3 n − k +2 (n + k)! k − i i=0 X (2.4) Lemma 2.5 ([1, p. 40, Entry 5]) For two differentiable functions p (x) and q (x) =06 , we have for k ≥ 0 dk p (x) dxk q (x) p q 0 ... 0 0 p′ q′ q . 0 0 ′′ ′′ 2 ′ p q q ... 0 0 1 k ........ (−1) = ........ (2.5) qk+1 ........ (k−2) (k−2) k−2 (k−3) p q 1 q ... q 0 k k p(k−1) q(k−1) −1 q(k−2) ... −1 q′ q 1 k−2 p(k) q(k) k q(k−1) ... k q′′ k q′ 1 k−2 k−1 In other words, the formula (2.5 ) can be represented as dk p (x) (−1)k = W (x) , dxk q (x) qk+1 (k+1)×(k+1) where W(k+1)×(k+1) (x) denotes the determinant of the matrix W(k+1)×(k +1) (x)= U(k+1)×1 (x) V(k+1)×k (x) . (l−1) Here U(k+1)×1 (x) has the elements ul,1 (x)= p (x) for 1 ≤ l ≤ k +1 and V(k+1)×k (x) has the entries of the form i−1 (i−j) j−1 q (x) , if i − j ≥ 0; vi,j (x)= (0, if i − j < 0, for 1 ≤ i ≤ k +1 and 1 ≤ j ≤ k. 4 3 Closed formulas In this section, we give closed formulas for higher-order Bernoulli and Euler polynomials. (α) Theorem 3.1 The higher-order Bernoulli polynomials Bn (x) can be ex- pressed as n k i n k! k + i B(α)(x)= h−αi (−1)i−j S(k + j, j)xn−k, n k i (k + i)! i − j k i=0 j=0 X=0 X X where S(n, k) is the Stirling numbers of the second kind and hxin denotes the falling factorial, defined for x ∈ R by n −1 x (x − 1) ...(x − n + 1), if n ≥ 1; hxin = (x − k)= k (1, if n =0. Y=0 (α) In particular x = 0, the higher-order Bernoulli numbers Bn possess the following form n i n! n + i B(α) = h−αi (−1)i−j S(n + j, j). (3.1) n i (n + i)! i − j i=0 j=0 X X et − 1 α e α Proof. Let us begin by writing = st−1ds . From (2.1) and t 1 (2.4), we have Z dk et − 1 −α dtk t k et − 1 −α−i e e e = h−αi B st−1 ln sds, st−1 ln2 sds, ..., st−1 lnk−i+1 sds i t k,i i=0 Z1 Z1 Z1 Xk 1 1 1 → h−αi B , , ..., , as t → 0 i k,i 2 3 n − i +2 i=0 Xk i k! k + i = h−αi (−1)i−j S (k + j, j) , i (k + i)! i − j i=0 j=0 X X 5 Also (ext)(k) = xkext → xk, as t → 0. So, using the Leibnitz’s formula for the nth derivative of the product of two functions yields that dn et − 1 −α lim ext t→0 dtn t " # n k i n k! k + i = h−αi (−1)i−j S (k + j, j) xn−k, k i (k + i)! i − j k i=0 j=0 X=0 X X (α) which is equal to Bn (x) from the generating function (1.1). For x = 0, we immediately get the identity (3.1). i Remark 3.2 For α =1, noting the fact h−1ii =(−1) i!, the equation (3.1) becomes n i n! n + i B = i! (−1)j S(n + j, j) n (n + i)! i − j i=0 j=0 Xn X n S(n + j, j) i = (−1)j n+j j j=0 j i=j X X n n+1 j j+1 = (−1) n+j S(n + j, j), j=0 j X which is [8, Eq. 1.3]. (α) Theorem 3.3 The higher-order Euler polynomials En (x) can be repre- sented as n k n S(k, i) E(α)(x)= h−αi xn−k. n k i 2i k i=0 X=0 X (α) In particular, the higher-order Euler numbers En have the following form n k n E(α) = h−αi S(k, i)2k−i.