MATRIX REPRESENTATIONS FOR MULTIPLICATIVE NESTED SUMS. 1. Introduction. The harmonic sums, defined by [BK99, eq. 4, p. 1] sign (i 1 ) n 1 (N) :=

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1 MATRIX REPRESENTATIONS FOR MULTIPLICATIVE NESTED SUMS LIN JIU AND DIANE YAHUI SHI* Abstract We study the multiplicative nested sums which are generalizations of the harmonic sums and provide a calculation through multiplication of index matrices Special cases interpret the index matrices as stochastic transition matrices of random wals on a finite number of sites Relations among multiplicative nested sums which are generalizations of relations between harmonic series and multiple zeta functions can be easily derived from identities of the index matrices Combinatorial identities and their generalizations can also be derived from this computation Introduction The harmonic sums defined by [BK99 eq 4 p ] sign i n S i i N := sign i n and [DMH7 p 68] 2 H i i N := N n n N>n > >n n i sign i n n i n i sign i n are naturally connected to zeta functions For instance taing = i = x > 0 and N in either or 2 gives the Riemann zeta-function ζx; 2 when i i > 0 and N 2 becomes the multiple zeta value ζi i Applications of harmonic sums appear in various areas such as [A2 p ] perturbative calculations of massless or massive single scale problems in quantum field theory Ablinger [A2 Chpt 6] implemented the Mathematica pacage HarmonicSumsm based on the recurrence [B04 eq 2 p 2] that is inherited from the quasi-shuffle relations [H00 eq p 5] for calculation of harmonic sums 200 Mathematics Subject Classification Primary C20; Secondary 05A9 Key words and phrases harmonic sum multiple zeta function random wal combinatorial identity *Corresponding Author n i

2 2 L JIU AND DYH SHI The current wor here is to present an alternative calculation for not only harmonic sums but also for the general sums defined as follows Definition We consider the multiplicative nested sums MNS: for m N N 3 S f f ; N m := and 4 A f f ; N m := N n n m N>n > >n m f n f n f n f n That is the summand is multiplicative f n n = f n f n and the summation indices are nested Here for all l = f l can be any function defined on {m m + N} unless N = when convergence needs to be taen into consideration Remar If f l x = signi l x /x i l for l = then 3 gives and 4 gives 2 In Section 2 we present the main theorem ie the calculation for MNS by associating to each function f l an index matrix and then considering the multiplications Since MNS are generalizations of harmonic sums this method naturally wors for harmonic sums Rather than recursively applying the quasi-shuffle relations in [A2] this matrix calculation is more direct and also simultaneously calculate for multiple pairs of N and m Properties of the index matrix such as inverse identities and eigenvalues eigenvectors and diagonalization follow after the main theorem Applications of this matrix calculation presented in Section 3 connect different fields Originally this idea was inspired by constructing random wals for special harmonic sums Different types of random wals appear in and connect to various fields For example the coefficients connecting Euler polynomials and generalized Euler polynomials [JMV4 eq 38 p 78] appear in a random wal over a finite number of sites [JMV4 Note 48 p 787] In Subsection 3 the special sum when f = = f = x a for a is interpreted as the probability of a certain random wal while the index matrix is exactly the corresponding stochastic matrix Consider the limit case of harmonic sums and 2 as N and further assuming i i > 0 ie S /x i /x i ; and

3 MATRIX REPRESENTATION 3 A /x i /x i ; The relation between them are of great importance and interest For instance the fact S x 2 x ; = 2A x ; = 2ζ 3 3 has been well studied and rediscovered many times Hoffman [H92 Thm 2 p 277 Thm 22 p 278] obtained the symmetric summation expressions in terms of the Riemann zeta-function ζ for both S/x i /x i ; and A/x i /x i ; In particular the direct relations for = 2 and = 3 between S /x i /x i ; and A /x i /x i ; see [H92 p 276] or 35 and 36 below are also easy to obtain In Subsection 32 we provide the truncated and generalized version of these relations easily derived from the identities of the index matrices Finally we focus on combinatorial identities where harmonic sums also appear For instance Dilcher [D95 Cor 3 p 93] established for special harmonic sum the relation 5 S N = N l= N l l l from q-series of divisor functions In Subsection 33 we present examples of combinatorial identities including a generalization of 5 Here those identities are obtained by applying calculations and properties of the index matrices Therefore all the examples can be viewed as alternative proofs 2 Matrix representations and properties In this section we provide the matrix representations of MNS The ey is to associate to each function f l an index matrix 2 Index matrices and calculation for MNS Definition 2 Given a positive integer N and a function f on { N} we define the following N N lower triangular index matrices: f f 2 f S f := f N f N f N f N

4 4 L JIU AND DYH SHI and 22 A f := f f 2 f f N f N f N f N 0 Remars Shifting S f downward by one row gives A f ie { if a = b; 23 A f = δ i j N N S f where δ ab = 0 otherwise For simplicity we further denote := δ i j N N so that A f = S f 2 When the dimensions of index matrices need to be clarified we use S N f and A N f Through index matrices defined above we could express MNS through matrix multiplications as follows Theorem 22 Let Then we have P = 24 S f f ; N m = 25 A f f ; N m = P P N N S fl l= A fl where M ij denotes the entry located at the i th row and j th column of a matrix M Proof Since the proof for Af f ; N m is similar we shall only prove a stronger result for Sf f ; N m ie for integers i j N S f f ; i j = P S fl When = it is easy to see that P S f ij = i f l = S f ; i j l= l= l=j Nm Nm ij

5 2 Suppose S f f ; i j = MATRIX REPRESENTATION 5 P S fl l= + S f f + ; i j = P S fl = = = l= ij = ij Then P S fl S f+ l= i S f f ; i l f + l l=j i f + l l=j i n n l j i n n l = S f f + ; i j f n f n f n f n f + l ij 22 Properties of the index matrix S Now we study some properties of the index matrix To simplify expressions in this section we denote S a := a a 2 a a N a N a N a N and A a := S a In other words we assume in 2 and 22 fl = a l for all l = N so that we replace the lower index f by a Next we give some properties of S a Proposition 23 Assume all a i 0 The inverse of S a is given by /a /a /a S a = 0 /a 2 /a /a N /a N 2 We have the matrix identities 26 S a S ab S b = I 27 S a S b S c + S ab S c + S a S bc + S abc = S a S b S c

6 6 L JIU AND DYH SHI 3 S a has eigenvalues {a a N } If all the a j s are distinct define D a = d ij N N and E a := e ij N N by d ij := a i a N N =i+ a a j and e ij := a N a i N =j i a ai if i j and d ij = 0 = e ij otherwise It follows that d j d Nj T is an eigenvector of S a with respect to a j and D a = E a implying 28 S a = D a diag a a N E a where diag a a N means the diagonal matrix with entries {a a n } on the diagonal Proof We omit the straightforward computation and only setch the idea here The inverse can be easily computed 2 Denote I a := diag /a /a N and a = I a so that S a = I a a Since I a S ab = S b but S ab I b S a and a S ab = A b = S b one easily obtains S a S ab S b = I a a S ab S b = S b S b S b S b = I Similarly multiplying by S a 27 we obtain S b + S a S ab + S bc S c from the left and by S c + S a S abc S c = S b from the right on which reduces to I = S b S bc S c ie 26 3 The eigenvalues are easy to see To verify the eigenvectors it is equivalent to prove that for all i = j N we have i a i N 29 a l a a i = a j a N a j a N l=j =l+ N =i+ a a j which can be directly computed by induction on i The inverse D a is equivalent to 20 δ ij = which reduces to 2 i t=j a i a t N t=j =i+ a a t N =j t i i = δ ij a t a =j t a at = E a

7 MATRIX REPRESENTATION 7 Now consider the partial fraction decomposition [Z05 eq p 33] i a j z a i z = i a t a t z a t a l t=j By multiplying both sides by z and then letting z we obtain 2 Thus the proof is complete Remar The diagonalization leads to an easy computation of powers of S a namely =j t 22 S a = D a diag a a N Ea 3 Applications Applications of the index matrix 2 and matrix expressions of MNS 24 appear in different areas We shall study related topics in random wals relations of harmonic series and multiple zeta values and combinatorial identities where one could see: some index matrices are stochastic transition matrices of random wals; 2 identities of index matrices directly lead to relations among MNS which are generalizations of that between harmonic series and multiple zeta values; 3 some combinatorial identities can be proven and generalized through matrix representations of MNS 3 Random wals In this subsection we let f l x H a x := /x a for l = where a Assume a = then we have 3 S H = Now label N sites as follows: N N N N 2 3 N N and consider a random wal starting from site N with the rules: one can only jump to sites that are NOT to the right of the current site with equal probabilities; steps are independent Let P i j denote the probability from site i to site j For example suppose we are at site 6 : here N

8 8 L JIU AND DYH SHI then the next step only allows to wal to sites { } with probabilities: P 6 6 = = P 6 = 6 Therefore a typical wal is as follows: STEP : wal from N to some site n N with P N n = N ; STEP 2: wal from n to n 2 n with P n n 2 = n ; STEP + : wal n n + n with P n n + = n We consider the event that after + steps we arrive the site m ie P n + = m Since the steps are independent 32 P n + = m = = Nn n N S H H ; N m N n n m Meanwhile the stochastic transition matrix is exactly given by S H namely S H = P i j N N Thus 33 S H + = P n + = m = Nm N S H H ; N m which is a probabilistic interpretation of 24 with the slight difference that P in 24 is replaced by S H As a result this probabilistic interpretation of harmonic sums easily leads to the following limit Proposition 3 lim SH H H ; N = lim Proof From 32 we see that N n n n n = N SH H H ; N = N P n + = Evidently for a random wal among finite number of sites the probability that it reaches the sin ie eventually this wal ends is More precisely for each site l = 2 N the probability for the wal not moving to site is l N Thus P n + N + 0 as

9 MATRIX REPRESENTATION 9 In other words lim P n + = = which completes the proof Remar When a > we could also form a similar random wal by adding another sin R to the right of N with P R n = δ Rn for n { N R}; defining for l = 2 N 0 if l < j N; P l j = if j l; l a if j = R l a Now for the stochastic transition matrix + Sa Sa S + a = A similar calculation for P n + = shows that 34 S H a H a ; N m = N a S Ha + Nm The analogue of Prop 3 fails due to more than one sin ie lim Pn + = < 32 Relations between S and A Now we consider the relations between the harmonic series S /x i /x i ; and the multiple zeta values A /x i /x i ; For example when = 2 we have 35 S x i x ; = A i 2 x i x ; + A i 2 x i +i 2 ; ; and when = 3 S x i x i 2 x ; i 3 36 =A + A x ; i 3 x i x i 2 x i x i 2+i 3 ; + A x i +i 2 + A x ; i 3 x i +i 2 +i 3 ; Both the identities above can be found eg in [H92 p 276] Next we will establish the truncated and generalized versions of 35 and 36 in the sense that we truncate the series from both above and below into sums which at the same time allows flexibility for general summands not restricted to negative powers

10 0 L JIU AND DYH SHI Theorem 32 For positive integers N and m with N > m we have 37 S f g; N m = A f g; N m + A fg; N m and 38 S f g h; N m =A f g h; N m + A fg h; N m + A f gh; N m + A fgh; N m Proof By Theorem 22 the right-hand side of 37 is given by PA f A g Nm + PA fg Nm = P S f S g + S fg Nm From 26 we see that I = S f S fg S g S f S g + S fg = S f S g Noticing the different dimensions an easy observation shows that PN N S N f S N g Nm = P N S N f S N g = S f g; N m N m Similarly 27 implies 38 by the replacement a b c f g h 33 Combinatorial identities The matrix computations in Section 2 especially the diagonalization for computing a matrix power lead to alternative proofs for some combinatorial identities and their generalizations Example 33 Butler and Karasi [BK0 Thm 4 p 7] showed that if G n satisfies G n n = G n = 0 and for G n = G n + a G n then S a a N := a n a n = G N + N N n n based on a proof related to Stirling numbers of the second ind Here we provide an alternative proof without using Stirling numbers as follows When = an induction on N shows directly that a n = a N G N N + G N N = G N + N N n 2 For the inductive step in similarly to 34 we see by recurrence S a a N = a N S a = a N S a S a l= N l= N = N a N a m G m + m a N m= = G N + N

11 MATRIX REPRESENTATION Example 34 Suppose the a m N m= are all distinct An alternative expression of the previous example can be obtained by diagonalization S a a N = DHa diag { } a + a + N EHa a N N = N a + j a N N a N a j am a j = j= j= N N m= m j am a j m= m j a j This recovers a general result [Z05 eq 2 p 33] which when we tae a j = a bq j+i /c zq j+i and N = n i + turns out to be a common source of several q-identities [Z05 p 34] The special case a m = m a yields N N n a 39 Sa a N = n a l a l a l= which gives 5 when a = Remar When a = m Z and m > consider the factorization n= n l n m l m = n l n ξ m l n ξ m m l where ξ m := exp 2πi/m and i 2 = We could obtain the following binomial-type expression N m N π ξ t Sa a N = m l ξ t l= t=0 ml sin πξml t l m which is similar to 5 and the usual binomial coefficient is generalized as x y := Γ x + / Γ x + Γ x y + Acnowledgment The corresponding author was supported by the National Science Foundation of China No 4049 This wor was initiated when the first author was a postdoc at Research Institute for Symbolic Computation Johannes Kepler University supported by SFB F50 F5006-N5 and F5009-N5 grant and continued when he moved as a postdoc to Johann Radon Institute for Applied and Computational Mathematics Austrian Academy of Science supported by Austrian Science Fund FWF grant FWF-Projet Now he is supported by the Killam Postdoctoral Fellowship at Dalhousie University He also would

12 2 L JIU AND DYH SHI lie to than his current supervisor Dr Karl Dilcher for his careful reading and valuable suggestions on this wor The authors finally appreciate the referee for valuable suggestion that leads to Prop 3 References [A2] J Ablinger Computer Algebra Algorithms for Special Functions in Particle Physics PhD Thesis Research Institute for Symbolic Computation Johannes Kepler University 202 [B04] J Blümlein Algebraic relations between harmonic sums and associated quantities Comput Phys Commun [BK99] J Blümlein and S Kurth Harmonic sums and Mellin transforms up to two-loop order Phys Rev D Article 0408 [BK0] S Butler and P Karasi A note on nested sums J Integer Seq Article 044 [D95] K Dilcher Some q-series identities related to divisor functions Discrete Math [DMH7] G H E Duchamp V H N Minh and N Q Hoan Harmonic sums and polylogarithms at non-positive multi-indices J Symbolic Comput [H92] M E Hoffman Multiple harmonic series Pacific J Math [H00] M E Hoffman Quasi-shuffle products J Algebraic Combin [JMV4] L Jiu V H Moll and C Vignat Identities for generalized Euler polynomials Integral Transforms Spec Funct [V99] J A M Vermaseren Harmonic sums Mellin transforms and integrals Internat J Modern Phys A [Z05] J Zeng On some q-identities related to divisor functions Adv Appl Math Department of Mathematics and Statistics Dalhousie University 636 Coburg Road PO BOX 5000 Halifax Nova Scotia Canada B3H 4R2 address: LinJiu@dalca School of Mathematics Tianjin University No 92 Weijin Road Nanai District Tianjin P R China address: shiyahui@tjuedcn