SUMS OF GENERALIZED HARMONIC SERIES. Michael E. Ho man Department of Mathematics, U. S. Naval Academy, Annapolis, Maryland

Transcription

1 #A46 INTEGERS 4 (204) SUMS OF GENERALIZED HARMONIC SERIES Michael E. Ho ma Departmet of Mathematics, U. S. Naval Academy, Aapolis, Marylad meh@usa.edu Courtey Moe Departmet of Mathematics, U. S. Naval Academy, Aapolis, Marylad chm@usa.edu Received: 9/3/3, Accepted: 6/2/4, Published: 8/27/4 Abstract We prove two results o sums of geeralized harmoic series.. Itroductio For oegative itegers a, a 2,..., a k with a a k 2, defie H(a, a 2..., a k ) a ( ) a2 ( k ) a. () k It is easy to show that this series coverges, ad i fact (by cosiderig the partialfractios decompositio of a ( ) a2 ( k ) a k ) that it ca be writte as a ratioal umber plus a sum of values (m) of the Riema zeta fuctio for 2 apple m apple max{a,..., a k }. For example, H(2, 3) 3 apple 2 ( ) 3 apple ( ) 2 ( ) ( ) 2 ( ) 3 63 (2) (3). We call the uatity () a H-series of legth k ad weight a a k. I 2 below we establish various formulas for H-series, i which Stirlig umbers of the first kid are promietly ivolved. These are used i 3 to prove the followig result.

2 INTEGERS: 4 (204) 2 Theorem. For ay positive itegers k, m with m 2, H(a,..., a k ) k (m), a a 2 a k m where the sum is over k-tuples of oegative itegers. The H-series ca be put ito the framework of Shitai zeta fuctios [5]. For a m k matrix A (a i ) ad a k-vector ~x (x,..., x k ), the Shitai zeta fuctio of (s,..., s k ) is (s,..., s k ; A; ~x),..., m 0 (x P m i a i i ) s (x k P m i a. (2) ik i ) s k The a H-series is ust the special case of (2) with m, A [ ], ad ~x (, 2,..., k). Several other special cases of Shitai zeta fuctios have bee extesively studied. These iclude the Bares zeta fuctios [], which is the case k ; ad the multiple zeta values [4, 7], which is the case where m k, A is the k k matrix with s o ad below the mai diagoal ad 0 s above, ad ~x (k, k,..., ). I the latter case the series (2) becomes (s,..., s k ),..., k 0 (k P k i i) s (k P k i2 i) s2 ( k ) s k m s ms2 2. ms k k m >m 2> >m k I fact, Theorem is superficially similar to the famous sum coecture for multiple zeta values, i.e., (a, a 2,..., a k ) (m) (3) a a k m, a i, a > for m 2. The multiple zeta values sum coecture (3) was made by the secod author (see [4]), ad was proved i full geerality by A. Graville [2]. Note, however, that (a,..., a k ) is a k-fold sum while H(a,..., a k ) is a sigle sum. Also, the sum o the left-had side of (3) is over k-tuples of positive itegers rather tha oegative itegers. It is also atural to ask about the sum of all H-series H(a,..., a k ) of weight m over all k-tuples (a,..., a k ) of positive itegers of fixed weight. As idicated by the example H(, ) ( ), the aswer is a ratioal umber. More precisely, we have the followig result, which follows easily from the machiery developed i 2.

3 INTEGERS: 4 (204) 3 Theorem 2. For m k 2, a a k m, a i H(a,..., a k ) (k )! k 2 k 2 i i0 ( ) i (i ) m k. 2. H-Series ad Stirlig Numbers of the First Kid Heceforth we assume m to be a positive iteger greater tha. followig result. We have the Lemma. Let apple i < apple p, ad let a,..., a i, a i,..., a, a,..., a p be fixed oegative itegers. The m k H(a,..., a i, k, a i,..., a, m k, a,..., a p ) i [H(a,..., a i, m, a i,..., a, 0, a,..., a p ) H(a,..., a i, 0, a i,..., a, m, a,..., a p )]. Proof. Evidetly so that ( i )( ) i apple i, H(a,..., a i, k, a i,..., a, m k, a,..., a p ) i [H(a,..., a i, k, a i,..., a, m k, a,..., a p ) Now sum o k to obtai the coclusio. H(a,..., a i, k, a i,..., a, m k, a,..., a p )]. Usig the precedig result, we ca obtai a formula for the sum of all H-series of fixed weight ad legth havig ozero etries at specified locatios. Lemma 2. For k ad apple i 0 < i < < i k apple, a i0 a i a ik m, a 6 0 i i for some k H(a,..., a ) ( ) H (m k) i 0,i (i i 0 )(i i ) (i i )(i i ) (i k i ), (4)

4 INTEGERS: 4 (204) 4 where for p apple H () p, p. Proof. For coveiece, deote by S(i 0, i,..., i k ; m) the left-had side of euatio (4). We proceed by iductio o k, usig Lemma. First, ote that the case k of the result, i.e., ) H(m i S(i 0, i ; m) 0,i, i i 0 follows immediately from Lemma. Now suppose the result holds for k, ad cosider S(i 0,..., i k, i k ; m), i.e., the sum of all H(a,..., a ) with ozero etries a i0, a i,..., a ik addig up to m. Fix a i0,..., a ik, ad sum the H(a,..., a ) from a ik to a ik m a i0 a ik, keepig the sum of a ik ad a ik eual to m a i0 a ik. By Lemma this gives us, after summatio o a i0,..., a ik, i k i k [S(i 0,..., i k, i k ; m ) S(i 0,..., i k, i k ; m )]. (5) By the iductio hypothesis, S(i 0,..., i k ; m ) is k ( ) (m k ) H i 0,i (i i 0 ) (i i )(i i ) (i k i ) ( )k H ad S(i 0,..., i k, i k ; m ) is k ( ) (m k ) H i 0,i (i i 0 ) (i i )(i i ) (i k i )(i k i ) Hece the uatity (5) is k ( ) (m k ) H i 0,i (i i 0 ) (i i )(i i ) (i k i ) (m k ) i 0,i k (i k i 0 ) (i k i k ) ( ) k (m k ) H i 0,i k (i k i 0 ) (i k i k ). i k i i k i k i k ( ) k (m k ) H i 0,i k (i k i 0 ) (i k i k )(i k i k ) ( ) k H (i k i 0 ) (i k i k )(i k i k ) k i (m k ) i 0,i k ( ) H (m k ) i 0,i (i i 0 ) (i i )(i i ) (i k i ).

5 INTEGERS: 4 (204) 5 For 0 apple k apple, let C(k, ; m) be the sum of all H(a,..., a ) of weight m (m k) with exactly k of the a i ozero. Sice each of the H i 0,i i the precedig result is a sum of uatities with apple i p m k 0 apple p apple i apple, for k we ca write k, C(k, ; m) m k (6) where k, property. is ratioal. The the umbers c() k, Lemma 3. For apple k, apple, ( )k k,. have a symmetry/atisymmetry Proof. Borrowig the otatio used i the proof of Lemma 2, we have C(k, ; m) S(i 0, i..., i k ; m). applei 0<i < <i k apple Now ote that Lemma 2 implies that if the S(i 0, i,..., i k ; m) p p 2 2 m k p ( ) m k, S( i k, i k,..., i 0 ; m) ( ) k p p 2 2 m k p ( ) m k. The first (ad last) colums of the umbers k, ca be writte i terms of (usiged) Stirlig umbers of the first kid: we write k for the umber of permutatios of {, 2..., } with exactly k disoit cycles. This otatio follows [3], which is also a useful referece o these umbers. Lemma 4. For apple k apple, k, ( )k ( )! Proof. It su ces to prove the formula for Lemma 3. Now Lemma 2 says that applei 0<i < <i k < apple k., as that for c() k, the follows from ( ) k ( i 0 )( i ) ( i k ),

6 INTEGERS: 4 (204) 6 so to prove the result we eed to show applei 0<i < <i k < ( )! ( i 0 )( i ) ( i k ) apple k. The left-had side is evidetly the sum of all products of k distict factors from the set {, 2,..., }, ad that this is the Stirlig umber k follows from cosideratio of the geeratig fuctio x(x ) (x ) k apple k xk,. (7) The precedig result geeralizes as follows. Lemma 5. For apple k, apple, k, Proof. We eed to show that k, p k, k2 p ( ) p p ( )!( )! p k2 p ( ) p p ( )!( )!. (8). (9) Suppose first that apple k. Usig Lemma 2, all terms that cotribute to cotribute to are of the forms, ad the terms that cotribute to c() (m k) also ad ot to c() ( ) k H i 0, ( i 0 ) ( i k ), i 0 < i < < i k <, (m k) ( ) k 2 H i 0, ( i 0 ) ( i k 2 )(i k ), i 0 < i < < i k 2 < < i k,... (m k) ( ) k H i 0, ( i 0 ) ( i k )(i k 2 ) (i k ), i 0 < i < < i k < < i k 2 < < i k (where we assume throughout that i 0,..., i k are itegers betwee ad ), which

7 INTEGERS: 4 (204) 7 we refer to as classes, 2,.... The cotributio from class is i 0< <i k < The cotributio from class 2 is i <i k k s s ( ) k ( i 0 )( i ) ( i k ) ( ) k k ( )! ( )k k ( )! ( )!. i 0< <i k 2 < i 0< <i k 2 < ( ) k 2 ( i 0 ) ( i k 2 ) ( ) k 2 ( i 0 ) ( i k 2 ) ( ) k ( ) 2 k ( )! ( )!, ad so forth. Note that for class we must have i k 2 2, i k 3 3,..., i k ad the cotributio from this class is i 0< <i k < ( ) k ( i 0 )( i ) ( i k )( )! ( ) ( k ) k 2 ( )! ( )!. Addig together the cotributios, we see that 2 ( ) k k k ( ) ( )!( )! is k 2, (0) ad euatio (9) follows from Lemma 3. Now if > k, the classes k, k 2,..., are evidetly empty. correspodig terms i (0) are zero, except for ( ) ( )k k k k ( )!( )! ( )!. All the This uatity is accouted for by terms that cotribute to ad ot to

8 INTEGERS: 4 (204) 8, provided <i < <i k p k ( ) p (i p )(i p i ) (i p i p )(i p i p ) (i k i p ) k ( )!. () By euatig the expressios give for c () k, by Lemmas 2 ad 4 respectively, we have k <i < <i k apple p from which euatio () follows. ( ) p (i p )(i p i ) (i p i p )(i p i p ) (i k i p ) k ( )!, 3. Proofs of the Theorems First we eed oe more lemma. Lemma 6. For positive itegers ad, Proof. We use the sake oil method of H. Wilf [6]. Let F (x) The F (x) ca be rewritte ( ) ad euatio (2) follows. x ( ). (2) ( ) x x ( x) x ( x) x ( x) ( x) Now we prove Theorems ad 2. ( ). ( ) m p0 p m x m ( x) p x ( x) 2 x 2x2 3x 3,

9 INTEGERS: 4 (204) 9 Proof of Theorem. We have Now H(a,..., a ) C(k, ; m). a a 2 a m k0 ad C(0, ; m) H(m, 0, 0..., 0) H(0, m, 0,..., 0) H(0,..., 0, m) (m) (m) (m) 2 m ( ) m 2 (m) ( ) 2 m ( ) m so to prove the result we eed for all apple apple C(k, ; m) i k, m k, k, i i,. By Lemma 5 this meas we must show ( i )p p i2 p. (3) ( )!( )! i p We rewrite the left-had side of euatio (3) as ( ) p p apple ( )! ( )! i 2 p i. (4) p If p, the ier sum i (4) is apple i i i apple i i2 where we have used euatio (7). Hece sice i i i apple i ( )! i ( )! ( )! ( )!. If p 2, the the ier sum i (4) is apple i 2 p i apple i i i apple ip 2 p 2 ( )! p ( )!, ( )!!,

10 INTEGERS: 4 (204) 0 agai usig euatio (7), from which follows ( )! apple i i 2 p i p. Thus, the sum (4) ca be writte apple 2 p2 If we use euatio (7), this becomes ( ) p p ( )! ( )! 2 The euatio (3) follows from Lemma 6. p apple ( ) p ( )! p. p2 apple ( ) ( ) 2 ( ) apple ( ) ( ). Proof of Theorem 2. It su ces to show that, ( ) 2. ( )! From Lemma 5 we have, p p ( ) p p ( )!( )! ( )! ( ) ( )!( )! ( ) ( ) 2. ( )!

11 INTEGERS: 4 (204) Refereces [] E. W. Bares, O the theory of multiple gamma fuctios, Tras. Cambridge Philos. Soc. 9 (904), [2] A. Graville, A decompositio of Riema s zeta-fuctio, i Aalytic Number Theory, Y. Motohashi (ed.), Lodo Math. Soc. Lect. Note Ser. 247, Cambridge Uiversity Press, New York, 997, pp [3] R. Graham, D. Kuth, ad O. Patashik, Cocrete Mathematics, 2d ed., Addiso-Wesley, New York, 994. [4] M. E. Ho ma, Multiple harmoic series, Pacific J. Math. 52 (992), [5] T. Shitai, O evaluatio of zeta fuctios of totally real algebraic umber fields at opositive itegers, J. Fac. Sci. Uiv. Tokyo Sect. IA Math. 23 (976), [6] H. Wilf, Geeratigfuctioology, 3rd ed., A K Peters, Wellesley, MA, [7] D. Zagier, Values of zeta fuctios ad their applicatios, i First Europea Cogress of Mathematics (Paris, 992), Vol. II, A. Joseph et. al. (eds.), Birkhäuser, Basel, 994, pp