New Multiple Harmonic Sum Identities

Transcription

1 New Multiple Harmonic Sum Identities Helmut Prodinger Department of Mathematics University of Stellenbosch 760 Stellenbosch, South Africa Roberto Tauraso Dipartimento di Matematica Università di Roma Tor Vergata 0033 Roma, Italy Submitted: Dec 6, 03; Accepted: May 9, 04; Published: May 8, 04 Mathematics Subject Classifications: 05A9, A07, M3, B65 Abstract We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given. Keywords: multiple harmonic sums, partial fraction decomposition, Bernoulli numbers, congruences, zeta values Introduction The binomial transform of a sequence, {a n } n 0, is the sequence defined by n n a. 0 This ind of transform has been widely studied, see for example the pioneering paper [. In it, the following integral representation Nörlund-Rice is used: n f n πi 0 C n! fz dz, zz z n where fz is an analytic extension of the sequence f, and the curve C includes the poles 0,,..., n and no others. Enlarging the contour of integration leads to an asymptotic expansion, but often also to identities, in particular when fz is a rational function. In this case the method is equivalent to considering the partial fraction expansion of n! zz z n fz. This is also the point of view that we adopt in the sequel. Our terms the electronic journal of combinatorics 04, #P.43

2 f are not exactly rational functions in, but not too far away. We consider for example harmonic numbers H n s with s N +, which can be written as H n s j s j, s n + j s and for fixed j, this is a rational function, leading to an identity, and these identities will be summed over all j. In this paper we are interested in a variation of the binomial transform, that is n ± n n + a, in the special case where the generic term a n involves harmonic numbers. As we will see in a moment, these types of sums can be evaluated in terms of multiple harmonic sums which are defined by H n s, s,..., s l < << l n i l sgns i i for l N +, s s, s,..., s l Z l. The integers ls : l and s : l i s i are called the length or depth and the weight of a multiple harmonic sum. By {s, s,..., s j } m we denote the set formed by repeating m times s, s,..., s j. The following three identities appear in [6, Th. and Th : for any integers n, r > 0 and d >, n n + n n + n n + r H n H n, H d n s {} r s {} d, ls H n s, s i i ls H n s,..., s ls, s ls, where s t t, t,..., t m N + m means that the sums are taen over all the compositions of t, i. e. any s s, s,..., s l such that s i j i < j i t for some 0 j 0 < j < < < j l m. So, for example s,,, if and only if s {,,,,,,,,,,,, 3,, 3,, 4, 3,, 5}. Moreover, let S m,n s,..., s l s m l n s, l r the electronic journal of combinatorics 04, #P.43

3 for s s, s,..., s l N l, with the convention that S n s S,n s. The following four identities appear in [4, Cor.. and Th..3: for any positive integers n, a, b, n n + S b n { b }, n n + S b n, { b }, n n + H b a + + H a + S n { a }, 3, { b }, n n + H b a + H a S n { a },, { b }. Note that also in this case the right-hand side is a combination of multiple harmonic sums because S n t s t H n s. In the next two sections we provide three new identities which extend the pattern. The proofs are based on the elementary method of the partial fraction decomposition see [5 and [6 for more applications of this technique. In the final section we give some applications to infinite series and to congruences. In particular we show the following exotic sums: where R, R, R n a, b for positive integers a, b. j < j < S, {} a S,n {} b Results: first part 3 8 ζ5 7 4 ζ3ζ, Theorem. For n, r and d n n + H r n n + H r d 49 3 ζ7 + ζ3ζ ζ5ζ, j j a< b n s,{} r s {} d,3,{} r a b ls H n s, s,..., s ls, ls H n s. the electronic journal of combinatorics 04, #P.43 3

4 Proof. For n, j, let factorials are defined via Gamma functions [ z + n!z n! fz : z!z! z r+ j, d z + j d and consider its partial fraction decomposition n n + fz n r [ j d + j d z + C rn, j + + C n, j + D dn, j z r z z + j + + D n, j d z + j. Multiply by z and let z. [ n n + 0 n r j + C d + j d n, j + D n, j where [ C n, j [z z + n!z n! z!z! z r+ j d z + j [ d [z r z + n!z n! z!z! j d z + j [ d [z r [w d z + n!z n! z!z! w j w z j [ D n, j [z + j z + n!z n! z!z! z r+ j d z + j d [z + j d z + n!z n! z!z! z r+ [w d w j + n!w j n! w j!w j! w j r+ [z r [w d w j + n!w j n! w j!w j! z w j [ [z r [w d w j + n!w j n! w j!w j! Define Sn via [z r [w d Sn j C n, j + D n, j, w j w z j. then by summing over j, we obtain [z r [w d n Sn n r n + H d. the electronic journal of combinatorics 04, #P.43 4

5 Now set and F n, j z + n!z n! + z!z! Gn, j Then, as is easy to chec, [ w j + n!w j n! w j!w j! w j + n +!w j n! w j!w j! w j w z j z n +. n + zf n +, j + n + + zf n, j Gn, j + Gn, j. Summing over j, we get n + zsn + + n + + zsn lim Gn, j + Gn, z Gn,. j n + Let then, since we have Therefore and Gn, n n z! T n n + z! Sn w + n!w n! w!w! z n +, n n z! z T n + T n n + z! n + + z n + n n z! n + + z! T n n Sn T T n + z! n z! z! + z! z z/ z! + z! j+ + z/j z/j + Gn, w + n!w n! w!w! z n +. w +!w! z w!w!, w +!w! z w!w! w 3 w/ + w/j w/j,. the electronic journal of combinatorics 04, #P.43 5

6 If d then is given by n r n + H [z r [w 0 n Sn [z r z z/ j+ [z r + s,{} r + z/j z/j s z s s n j+ ls H n s, s,..., s ls. + s z s j s If d then is given by n r n + H d [z r [w d n Sn [z r [w d z z/ w 3 w/ [z r [w d + 3 s {} d,3,{} r + s j+ + w/j w/j + z/j z/j w s s s z s s ls H n s. n j+ + + w s j s s s z s j s 3 Results: second part Theorem. For n, a, b n b n + H a H a R n a, b. 3 the electronic journal of combinatorics 04, #P.43 6

7 Proof. For n, j, let fz : Then its partial fraction decomposition is [ z!z n! z + n!z! z r+ j. d z + j d [ n fz n +!n! r j d + j d z [ + r n n +!n! r j d z + + r, j Multiply by z and let z. [ n n +!n! r + j d z + + C rn, j + + C n, j + D d+n, j z r z z + j + + D n, j d+ z + j. [ n 0 n +!n! r j d + j d [ + r n n +!n! r j d [ + r n n +!n! r + j d, j + C n, j + D n, j. Moreover [ C n, j [z r [w d z!z n! z + n!z! w j w z + j and [ D n, j [z r [w d w j!w j n! w j + n!w j! w j w z + j. Define Sn via [z r [w d Sn n! j C n, j + D n, j the electronic journal of combinatorics 04, #P.43 7

8 then by summing over j, we obtain Let [z r [w d n n n + Sn H r d + r n n + ζd r j r n n + r j d r n n + r j d j+ r n n + r j d jn+ n n + H r d r+d H d. F n, j n! z!z n! z + n!z! and [ w j!w j n! + w j + n!w j! w j w z + j w j n! z + n + z Gn, j n! w j + z + w j + n +! n + n +. Then [z r [w d F n, j n! C n, j + D n, j, z n + and by summing over j, we get z n + F n +, j + F n, j Gn, j + Gn, j Sn + + Sn lim j Gn, j + Gn, Gn,. Let then, T n n T n + T n n z Sn z n Gn,. the electronic journal of combinatorics 04, #P.43 8

9 Since w n! Gn, n! w + n! n n w we have that n Sn z + n + n + z w + z + z n + z T n z z j z w n + w + z T T z w j + z z n + ww n, z G, z + w/ w + z w w / z w w z w zw 3. Hence n n + H r d r+d H d [z r [w d z w j w z w zw 3. 4 the electronic journal of combinatorics 04, #P.43 9

10 If r b and d a then we obtain n n + H b a + H a [z b [w a z w j + w. Since n n + H b a + H a [z b [w a z w j and H a + H a H a + H a H a H a it follows that b n n + H a H a [z b [w a j z w S, {} a S,n {} b R n a, b. Note that if r b and d then 4 yields, b n n + [z b [w z S n{} b. j w w the electronic journal of combinatorics 04, #P.43 0

11 Moreover, if r b and d a + then 4 yields, b n n + H a + + H a + [z b [w a+ j S n{} a, 3, {} b. z w zw 3 4 Two applications of identity 3 Our first application involves infinite series. Evaluations for Euler sums of length two and of odd weight m + n, H m, n and H m, n in terms of zeta values are well nown see, for example, [, Th. 7.: H m, n ζm + n m ζmζn m+n / r + m ζr + ζm + n r n r m+n / m r r ζr + ζm + n r, 5 m H n, m H m, n ζmζn + ζm + n, 6 where ζ0 /, ζ ln, ζn n ζn, and ζn /n, for n >. Theorem 3. Let a, b positive integers with b >. Then R a, b 4 a+b r r r ζr + ζa + b r b a r+ Proof. By taing the limit n in 3, Then use 5 and 6. R a, b H a, b H a, b. For the second application of 3, we first recall some congruences. Let m, n be positive integers and let p be any prime p > w + where w m + n, i if w is even then [7, Th. 3. H p m, n w + m n m w + m m n pbp w w w + mod p, the electronic journal of combinatorics 04, #P.43

12 ii if m is even and n is even then [4, Lemma 3. m n w w H p m, n m + n + m iii if w is odd then [3, Lemma H p m, n n w m w w + + w Bp w w iv if m is even and n is odd then [4, Lemma 3. w w B p w H p m, n + w m w pb p w mod p. mod p, mod p. Lemma 4. Let m, n, r be positive integers. If n + r is odd and m is even then for any prime p > maxm, n +, p H m H mh n + H n r 0 mod p. Proof. Let a be a positive integer or a negative even integer then, by Wolstenholme s theorem, for p > a +, H p a 0 mod p. Thus Hence L : p p p H p a p sgna j p sgna p j j a p j a j p a sgna j sgna j j a a sgnah p a H a a sgnah a mod p. H m H mh n + H n r H p m p H p mh p n + H p n p r H m + H m n H n H n r r p n+r L + n+r L mod p, H n H n r+m the electronic journal of combinatorics 04, #P.43

13 which implies that L 0 mod p. Theorem 5. For any prime p > a + b, R p / a, b a+b a + b B p a+b a + b a R p a, b a b + R p /a, b mod p. a + mod p, Proof. Setting n p, we note that n n + p 3 p + p 3 + p Therefore, by i and ii, R p / a, b Now, setting n p we have that p + p b H a H a mod p. H p / a, b H p / a, b a+b a + b B p a+b mod p. a + b a n n + p p j + p j ph ph p p H + H mod p, and by the previous lemma, by iii and iv, it follows that R p / a, b b p H + H H a H a H a H a p b Hp a, b p a b + R p /a, b a + H p a, b p mod p. the electronic journal of combinatorics 04, #P.43 3

14 References [ P. Flajolet and B. Salvy. Euler sums and contour integral representations. Exp. Math., 7:5 35, 998. [ P. Flajolet and R. Sedgewic. Mellin transforms and asymptotics: Finite differences and Rice s integrals. Theor. Comput. Sci., 44-:0 4, 995. [3 Kh. Hessami Pilehrood, T. Hessami Pilehrood, and R. Tauraso. Congruences concerning Jacobi polynomials and Apéry-lie formulae. Int. J. Number Theory, 87:789 8, 0. [4 Kh. Hessami Pilehrood, T. Hessami Pilehrood, and R. Tauraso. New properties of multiple harmonic sums modulo p and p-analogues of Leshchiner s series. Trans. Amer. Math. Soc., 3666:33 359, 04. [5 H. Prodinger. Human proofs of identities by Osburn and Schneider. Integers, 8:Art. A0, 8 pp. electronic, 008. [6 H. Prodinger. Identities involving harmonic numbers that are of interest for physicists. Util. Math., 83:9 99, 00. [7 J. Zhao. Wolstenholme type theorem for multiple harmonic sums. Int. J. Number Theory, 4: 73 06, 008. the electronic journal of combinatorics 04, #P.43 4