PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 9, September 2012, Pages 2945–2952 S 0002-9939(2012)11747-2 Article electronically published on April 30, 2012 ALGEBRAIC STRUCTURES OF EULER NUMBERS I-CHIAU HUANG (Communicated by Matthew A. Papanikolas) Abstract. We define a finitely generated Q-algebra E with a module structure over the universal enveloping algebra of a Lie algebra. Identities of Euler numbers are investigated using the algebraic structures of E. 1. Introduction We would like to introduce an algebraic method to study classical numbers. Such a method has been applied to Bernoulli numbers [3]. In this paper, we extend the T ∞ i idea to Euler numbers. Let e := i=0 T /i! be an element in the power series ring Q[[T ]] with coefficients in the field Q of rational numbers. Recall that the i-th Bernoulli number Bi and the i-th Euler number Ei are defined by power series ∞ T B B := = i T i eT − 1 i! i=0 and ∞ 2eT E E := = i T i. e2T +1 i! i=0 We are interested in identities involving classical numbers. One of the most famous is perhaps Euler’s identity for the sum of products of two Bernoulli numbers: − n1 2n B B − = −(2n +1)B , 2i 2i 2n 2i 2n i=1 where n ≥ 2. Euler’s identity has been generalized to sums of products of more Bernoulli numbers [7, 6, 8, 2]. For instance, 2n 1 B B B =(n + 1)(2n +1)B + n(2n − 1)B − , 2i, 2j, 2k 2i 2j 2k 2n 2 2n 2 i+j+k=n where n ≥ 3. An interpretation of these formulae in terms of complete sums is given in [4]. Euler’s identity has other variations such as n n 1 1 B B − = (n − 1)B − B − (n +3)B . i 1+i 1+n i 6 n n+1 6 n+2 i=0 See [1] for more similar identities. The new viewpoint of [3] on the above identities is that they all come from algebraic relations of elements in a commutative ring Received by the editors November 9, 2010. 2010 Mathematics Subject Classification. Primary 11B68; Secondary 16S30. Key words and phrases. Euler number, identity, Lie algebra, skew polynomial ring, universal enveloping algebra. c 2012 American Mathematical Society Reverts to public domain 28 years from publication 2945 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2946 I-CHIAU HUANG with a module structure over the Weyl algebra QT, d/dT . We remark that the Weyl algebra is a skew polynomial ring. For the definitions and basic properties of Weyl algebras and skew polynomial rings, the reader is referred to [5, Chapter 1]. In this paper, we define a subring E of Q[[T ]] containing E with a module struc- ture over another skew polynomial ring, which is the universal enveloping algebra of a two-dimensional Lie algebra. See [5, Chapter 1] for definitions and construc- tions of universal enveloping algebras and Lie algebras. Via algebraic relations of elements in E, we reproduce formulae in [9] for the sum of products of 2m+1 Euler numbers such as 2n 1 1 E E E = E − E . 2i, 2j, 2k 2i 2j 2k 2 2n 2 2n+2 i+j+k=n See [9] also for explicit formulae of the sum of products of five, seven and nine Euler numbers. Analyzing the algebraic structures of E, we show that such a type of formula does not exist for the sum of products of 2m Euler numbers. Our method gives numerous identities of Euler numbers; some are elegant and some can be complicated. Among them, we have n 1 1 1 E E E = E − E − E i, j, k i 1+j 1+k 8 n 12 n+2 24 n+4 i+j+k=n and n 1 1 1 E E E = − E + E − E . i, j, k 2+i j k 4 n 3 n+2 12 n+4 i+j+k=n As in [3] for Bernoulli polynomials, one may extend the algebraic structures considered in this paper to include Euler polynomials. It is also plausible that there exist algebraic structures accommodating two or more classical numbers (for instance Bernoulli numbers and Euler numbers) simultaneously. We leave the pos- sibility to the reader. 2. Algebraic structures Definition 2.1. We define E to be the Q-subalgebra of Q[[T ]] generated by x := eT and y := E. To understand the ring structure of E, we consider the Q-algebra homomorphism π : Q[X, Y ] → Q[[T ]] given by π(X)=eT and π(Y )=E. Clearly, π(X2Y + Y − 2X)=0. Proposition 2.2. The Q-algebra E is isomorphic to Q[X, Y ]/(X2Y + Y − 2X). Proof. It suffices to prove that the kernel of π is generated by X2Y + Y − 2X.The image of π is contained in the quotient field of Q[eT ]. Since eT is transcendental over Q, to prove the proposition, it suffices to prove that the kernel of the Q[X]-algebra homomorphism π˜ : Q[X, Y ] → Q(X) sending Y to 2X/(X2 + 1) is generated by X2Y + Y − 2X as well. Consider an element in the kernel ofπ ˜. Replacing X2Y by 2X − Y repeatedly, we can write the element in the form f(X)X + g(Y )YX+ h(Y ) for some one-variable polynomials License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EULER NUMBERS 2947 f, g and h over Q. It suffices to show that f(X)X + g(Y )YX+ h(Y )=0.Wework on the field Q((X−1)) of Laurent series in the variable X−1:In 2X−1 2 2X−1 f(X)X + g + h =0, 1+X−2 1+X−2 1+X−2 only f(X)X consists of positive powers of X. Hence f(X) = 0. Now we work on the field Q((X)) of Laurent series in the variable X: The constant term on the left-hand side of 2X 2X2 2X g + h =0 X2 +1 X2 +1 X2 +1 is the constant term of h(Y ). Hence h(0) = 0. Back to the field Q((X−1)): The constant term on the left-hand side of 2X−1 2 2X−1 g + h =0 1+X−2 1+X−2 1+X−2 is twice the constant term of g(Y ). Hence g(0) = 0. So we can factor out Y from g(Y )andh(Y ). Induction on the degree of h(Y )showsthath(Y )=0.Now g(Y )YX ∈ kerπ ˜; hence g(2X/(X2 + 1)) = 0. This implies that g(Y )=0,since 2X/(X2 + 1) is transcendental over Q. Definition 2.3. Let U be the ring of endomorphisms of Q[[T ]] generated by the multiplication map x := eT and the derivation δ := d/dT . We denote the action of ϕ ∈ U on f ∈ Q[[T ]] by ϕ · f. Here we use the notation x both for an element in E andanelementinU without confusion. Hence a polynomial in x may stand for an element in E or an element in U depending on the context. The relations δ · x = x, δ · y = y − xy2 in E imply that E is stable under the action of U.Inotherwords,E is a U-module. As a special case of the Leibniz rule, we have the relation δx − xδ = x in U. In general, one can show that (2.1) δnxm = xm(m + δ)n, (2.2) xmδn =(δ − m)nxm by induction. Consider fn,fn+1,...,fn+m ∈ Q[x]. If fn does not vanish, then n n+1 n+m neither does the linear combination ϕ = fnδ + fn+1δ + ···+ fn+mδ ,since n n!fn = ϕ · T . Thus every element in U can be written uniquely as a Q-linear combination of elements of the form xmδn. Using relation (2.1), one shows also that every element in U can be written uniquely as a Q-linear combination of elements of the form δnxm. Using relation (2.2) for n = 1, one sees that U is isomorphic to the skew polynomial ring Q[x][δ; −x(d/dx)]. Note that U is also isomorphic to the universal enveloping algebra of the two-dimensional Lie algebra g with basis {δ, x} and Lie bracket [δ, x]=x. See [5, 1.7.6]. The ideal Ey of E is also a U-submodule of E. To understand algebraic meanings of identities of Euler numbers, we need to find generators and relations of Ey as a U-module. We begin with a comparison between the E-ideal and the U-submodule generated by a power of y. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2948 I-CHIAU HUANG Proposition 2.4. Eym = U · ym for m ≥ 1. Proof. Clearly, xn · ym = xnym for any n and δ · ym = m(1 − xy)ym. Induction on n shows that δn · ym ∈ Eym.ThusU · ym ⊂ Eym.ToproveEym ⊂ U · ym,we consider the derivation xδ on E.Notethat (xδ) · ym = mym−1x(y − xy2)=mxym − m(2x − y)ym = mym+1 − mxym, which can also be written as m!ym+1 =(m − 1)!x(δ + m) · ym. Indeed the above relation is a special case of (2.3) (n!)yn+1 =(m − 1)!xn−m+1(δ + m)(δ + m +2)···(δ +2n − m) · ym for n ≥ m ≥ 1, which implies that Eym ⊂ U · ym.