Bernoulli numbers and generalized factorial sums

Size: px

Start display at page:

Download "Bernoulli numbers and generalized factorial sums"

Erik Peters
6 years ago
Views:

1 Bernoulli nubers and generalized factorial sus Paul Thoas Young Departent of Matheatics, College of Charleston Charleston, SC June 25, 2010 Abstract We prove a pair of identities expressing Bernoulli nubers and Bernoulli nubers of the second kind as sus of generalized falling factorials. These are derived fro an expression for the Mahler coefficients of degenerate Bernoulli nubers. As corollaries several unusual identities and congruences are derived. 1 Introduction The generalized falling factorial (x λ) n with increent λ is defined for positive integers n by n 1 (x λ) n (x iλ) (1.1) with the convention (x λ) 0 1; clearly (n 1) n (1 1) n n! is the usual factorial. In this note we establish a pair of identities ( ) ( 1 n + 1 j 1 B n (i j) n, (1.2) j j (n + 1)! n! b n ( 1) n+1 j ( n + 1 j 1 (i j) n, (1.3)

2 for the Bernoulli nubers B n and Bernoulli nubers of the second kind b n. As consequences we deduce congruences such as 12 p r +3 j2 ( 1) j j 1 j! (p r + 2)! (p r + 3 j)! (i j) p r +2 p 2r (od p 2r+1 Z p ) (1.4) for odd pries p, where Z p denotes the ring of p-adic integers. We also prove that if n > 1 is odd and d is any divisor of n 2, then j 1 2 j2 ( 1) j (1 d) (i j) n d ( j)! j! 0. (1.5) These well-known sequences of rational nubers are defined [11] by generating functions t e t 1 t n B n n!, (1.6) n0 t log(1 + t) b n t n. (1.7) The nubers n!b n have also been called Cauchy nubers in [7]. We will deduce these identities fro their relation to the degenerate Bernoulli nubers β n (λ) which are defined [6, 11] for λ 0 by eans of the generating function n0 t (1 + λt) µ 1 β n (λ) tn n! n0 (1.8) where λµ 1. These are polynoials in λ with rational coefficients; since (1+λt) µ e t as λ 0 it is evident that β n (0) B n ; since ((1+t) µ 1)/µ log(1 + t) as µ 0 we have li λ λ n β n (λ) n! b n. Thus β n (λ) is a polynoial of degree n in λ whose constant ter is B n and whose leading coefficient is n! b n. The nubers b n and β n (λ) have been interpreted as divided differences of binoial coefficients [1] and are related to gae theory [4]. 2 Proof of Identities The identities (1.2), (1.3) are deduced fro the observation ([11], eq.(3.15)) that aβ n (a) σ n (a, a 1) (2.1) 2

3 for positive integers a, where σ n (λ, c) c (i λ) n (2.2) is a generalized falling factorial su. This equation ay be given the following interpretation: for integers i, n 0 one ay consider that the generalized falling factorial (i a) n is the product of all eleents in the coset i + (a) of the ideal (a) in the factor ring Z/anZ, a product which is well-defined odulo anz. Therefore, the integer aβ n (a) is, up to a ultiple of an, equal to the su of all coset products over cosets of (a) in Z/anZ. It follows that the rational nuber β n (a) is, up to an integer ultiple of n, the average coset product of (a) in Z/anZ. If f Z[x] is a polynoial of degree N then it ay be expressed in the for N ( ) x f(x) a, (2.3) 0 where the Mahler coefficients a Z are uniquely deterined by a ( ( 1) j j j0 ) f(j) (2.4) (cf. [9], 52). Since the function λ λβ n (λ) is a polynoial of degree n + 1, we have ( ) ( ) λ λβ n (λ) ( 1) j σ n (j, j 1) (2.5) j 1 as an identity of polynoials in Z[λ], and it siilarly follows that ( ( 1) j j (i j) n 0 when > n + 1. (2.6) Theore 1. For all nonnegative integers n we have B n ( ( 1 n + 1 j j (i j) n 3

4 and Proof. Fro (2.5) we have where ( n + 1 (n + 1)! n! b n ( 1) n+1 j j a λβ n (λ) 1 ( ( 1) j j a ( λ ) (i j) n. (2.7) ) σ n (j, j 1) (2.8) as an identity of polynoials in λ. Equating coefficients of λ n+1 in (2.7) gives n! b n a n+1 /(n + 1)!, (2.9) giving the second stateent. For the first stateent, we divide both sides of (2.7) by λ to obtain ( ) a λ 1 β n (λ) (2.10) 1 1 with a as in (2.8). Since ( 1) ( 1) 1, evaluating (2.10) at λ 0 then yields B n 1 1 ( 1) 1 a ( 1 ( ) j 1 j (i j) n (2.11) ( ( 1 1 (i j) n. j j 1 1 Fro the failiar property ( ( n k 1) + n ) ( k n+1 ) k we have ( ) ( 1 n + 1 j 1 j 1 ), (2.12) and therefore the triple su of (2.11) becoes the double su of the first stateent of the theore. 4

5 Reark. In these identities the sus over index i run fro 0 to j 1 so as to ake the valid for n 0; when n > 0 the i 0 ter contributes nothing, so the inner su ay be indexed fro i 1 to j 1, and in turn the outer su ay be indexed fro j 2 to n + 1 when n > 0. 3 Bernoulli nubers The above theore expresses the Bernoulli nuber B n as a su of integer ultiples of 1/j for 1 j n + 1. As such it ay be copared to identities such as and B n B n n j0 1 j + 1 j ( ) j i n (3.1) i ( ) j ( 1 n + 1 i n (3.2) j j (cf. [3]), although it gives a different such expression. The following result ay also be deduced fro the above theore and well-known fact that B 2k+1 0 for positive integers k. Corollary 1. If n > 1 is odd then j2 ( ( 1 n + 1 j j (i j) n 0. We ay also derive the following congruences for the Bernoulli nubers. These are generalizations of the well-known fact that pb p 1 1 (od p) for odd pries p. Corollary 2. If p is an odd prie and n/p k, then pb n k+1 ( ( 1 n + 1 j jp ) jp 1 (i jp) n (od p k+1 Z p ). For the case n 1 (od p) the congruence ay be iproved to pb kp 1 k ( ) ( 1 jp 1 kp (i jp) kp 1 (od kp k+1 Z p ), j jp 5

6 and for p > 3 and k > 1 we also have pb kp 1 k ( ) ( 1 jp 1 k (i jp) kp 1 (od kp 3 Z p ). j j Proof. If j is not a ultiple of p then each falling factorial (i j) n with 1 i < j lies in p k Z, so the first stateent follows directly fro the theore. When n kp 1 and j is not a ultiple of p the binoial coefficient ( ) kp j lies in kpz p, while each falling factorial (i j) kp 1 with 1 i < j lies in p k 1 Z. Therefore, pb kp 1 k ( ) ( 1 jp 1 kp (i jp) kp 1 (od kp k+1 Z p ). (3.3) j jp Invoking the Kazandzidis supercongruence ( ) ( ) kp k (od jk(k j)p 3 Z p ) (3.4) jp j (cf. [8]) for p > 3 gives the second result. 4 Bernoulli nubers of the second kind The following corollary expresses the well-known fact [5, 12] that b n 0 for all n. Corollary 3. For all integers n 0, ( n + 1 ( 1) j j (i j) n 0. The rational nubers b n have recently been expressed as p-adically convergent sus of traces of algebraic integers [12, 13]. Here we present a congruence result which illustrates this description. Corollary 4. For all integers n 0 and all pries p, p n/(p 1) ( 1) n+1 j j 1 (i j) n is p integral j! n! (n + 1 j)! 6

7 and ( 1) n+1 j j 1 j! n! (n + 1 j)! (i j) n Tr(ζ p (ζ p 1) n ) (od p 1 (n+p 1)/(p2 p) Z p ), where ζ p is any priitive p-th root of unity and Tr denotes the trace ap fro Q(ζ p ) to Q. Proof. By our theore, the left eber of the above congruence is siply b n. For odd pries p these stateents follow fro [13], Corollary 1. For p 2 the first stateent follows fro [2], Theore 2 and the second follows fro [12], Theore 1. Rearks. It is well-known and easy to see that p n/(p 1) /n! is always p- integral, but there sees to be no obvious reason why the given expression is p-integral. In the above congruence, both ebers are rational nubers which lie in p n/(p 1) Z p generically and the congruence says that they agree to approxiately 1+((n 1)/p) digits p-adically. This congruence is unusual in that the left eber, which is independent of the prie p, is siultaneously congruent to the right eber for all pries p. We conclude this section with another congruence illustrating the unusual arithetic properties of the sequence {b n }; the left eber ostensibly should have a factor of p 2r in its denoinator, rather than its nuerator. Corollary 5. For all odd pries p and all r > 0, 12 p r +3 j2 ( 1) j j 1 j! (p r + 2)! (p r + 3 j)! (i j) p r +2 p 2r (od p 2r+1 Z p ). Proof. This is a restateent of the congruence 12b p r +2 p 2r (od p 2r+1 Z p ) given in [11], Theore Other generalizations of Bernoulli nubers The Bernoulli nubers B n (w) of order w are defined by the generating function ( ) w t B (w) t n e t n 1 n!, (5.1) 7 n0

8 and the nubers B n (n), where the order equals the degree, are called Nörlund nubers [5, 12]. Coparing the theore with the identity [5] yields the triple su identity B (n) n n! B (n) n n! n k+1 k0 n ( 1) n k b k (5.2) k0 ( 1) n+1 j j 1 j! k! (k + 1 j)! (i j) k (5.3) for the Nörlund nubers. The next corollary gives a regular Kuer-type congruence for the degenerate Bernoulli nubers. It applies to β rather than to the divided sequence β /, and also differs fro other such congruences for Bernoulli nubers in that the case p 1 is not excluded (cf. e.g. [9], 61). Corollary 6. Suppose p is a prie nuber and λ p k Z p with k > 0. If p is odd let n (od (p 1)p e ) and if p 2 let n (od 2 e+1 ). Then where B in{, n, e + 1} k. Proof. The generating function β (λ) β n (λ) (od p B Z p ) (1 + λt) (c+1)µ 1 (1 + λt) µ 1 σ (λ, c) t! ; (5.4) [11], eq. (2.3) reveals that {σ (λ, c)} is the degenerate nuber sequence associated to the polynoial h(t ) (T c+1 1)/(T 1) Z[T ] in the terinology of [10]. Therefore by [10], Theore 1.2 we have 0 σ (λ, c) σ n (λ, c) (od p A Z p ) (5.5) under the stated hypotheses on, n, and λ, where A in{, n, e + 1}. Taking λ to be a positive integer ultiple of p and c λ 1 yields λβ (λ) λβ n (λ) (od p A Z p ) (5.6) via (2.1). Dividing by λ gives the stated congruence for positive integers λ; since the set of positive integer ultiples of p is dense in pz p, the result follows for all λ pz p. 8

9 We conclude with the following triple su identity, for which we would like to find a cobinatorial explanation. Corollary 7. If n > 1 is odd and d is any divisor of n 2, positive or negative, then j 1 ( 1) j (1 d) (i j) n 0. d ( j)! j! 2 j2 Proof. We recently proved [11], Theore 4.7 that if n > 1 is odd and d is any divisor of n 2, then β n (1/d) 0. Fro (2.5) we have 0 β n (1/d)/d ( ) 1/d ( ( 1) j j (1/d 1)! (1 d) d! j 1 1 ( ( 1) j j ( ( 1) j j (i j) n (i j) n (i j) n (5.7) ( 1) j (1 d) (i j) n. d ( j)! j! The ters corresponding to j 1 or to 1 contribute nothing, so the corollary is proved. References [1] A. Adelberg, A finite difference approach to degenerate Bernoulli and Stirling polynoials, Discrete Math. 140 (1995), [2] A. Adelberg, 2-adic congruences of Nörlund nubers and of Bernoulli nubers of the second kind, J. Nuber Theory 73 (1998), [3] H. W. Gould, Explicit frorulas for Bernoulli nubers, Aer. Math. Monthly 79 (1972),

10 [4] G. Hetyei, Enueration by kernel positions, Adv. Appl. Math. 42 (2009), [5] F. T. Howard, Nörlund s nuber B n (n), in Applications of Fibonacci Nubers, Vol. 5, Kluwer, Dordrecht, 1993, pp [6] F. T. Howard, Explicit forulas for degenerate Bernoulli nubers, Discrete Math. 162 (1996), [7] D. Merlini, R. Sprugnoli, M.C. Verri, The Cauchy nubers, Discrete Math. 306 (2006), [8] A. Robert, M. Zuber, The Kazandzidis supercongruence. A siple proof and an application, Rend. Se. Mat. Univ. Padova 94 (1995), [9] W. H. Schikhof, Ultraetric calculus. An introduction to p-adic analysis, Cabridge University Press, London, [10] P. T. Young, Congruences for degenerate nuber sequences, Discrete Math. 270 (2003), [11] P. T. Young, Degenerate Bernoulli polynoials, generalized factorial sus, and their applications, J. Nuber Theory (2008), [12] P. T. Young, A 2-adic forula for Bernoulli nubers of the second kind and for the Nörlund nubers, J. Nuber Theory (2008), [13] P. T. Young, A p-adic forula for the Nörlund nubers and for Bernoulli nubers of the second kind, Congressus Nuerantiu 201 (2010),

EXPLICIT CONGRUENCES FOR EULER POLYNOMIALS

EXPLICIT CONGRUENCES FOR EULER POLYNOMIALS Zhi-Wei Sun Departent of Matheatics, Nanjing University Nanjing 10093, People s Republic of China zwsun@nju.edu.cn Abstract In this paper we establish soe explicit