Parametric Euler Sum Identities

Transcription

1 Parametric Euler Sum Identities David Borwein, Jonathan M. Borwein, and David M. Bradley September 23, 2004 Introduction A somewhat unlikely-looking identity is n n nn x m m x n n 2 n x, valid for all complex x not a positive integer. For x 0, becomes ζ2, ζ3 the most central Euler sum identity as discussed below. In this note we begin with an ab initio proof of, and then explore various consequences and extensions. We conclude by studying other generating function identities of which n n n 2 + y + n 2m 2 m 2 + y 2 m n 2 + 4y 2 is a pretty example. cothπy + coth2πy n 2πny n 2 + y 2 n 2 + 9y 2 Though we do not belabour the point, all our work was assisted by the use of computer algebra systems and some of it would have been impossible, at least for us, without such tools. This is true of both our discoveries and of our proofs. The joys of such symbolic and numeric computation... and much more are discussed in detail in [] and [2]. 2 2 A Parametric Euler sum We begin by defining various special functions which we shall exploit in the sequel. For Rex > 0, the gamma function and its logarithmic derivative are defined by Note that x nn x n Γx : 0 e t t x dt, Ψx : Γ x Γx. N Ψ x γ, where γ : lim N n log n n

2 is Euler s constant. Recall that the classical Riemann zeta function is defined by Correspondingly, ζs : ζs, t : n n>m>0, Res >. ns n s m t n n s n m m t 3 defines a double Euler sum. The two-place function 3 was first introduced by Euler, who noted the reflection formula ζs, t + ζt, s ζsζt ζs + t, Res >, Ret >, 4 and the reduction formula 6 below. An obvious extension of 3 is ζs, s 2,..., s N : n >n 2 > >n N >0 n s n s 2 2 n s, N N which defines an Euler sum of depth N and weight i s i. Some authors reverse the order of the variables. Euler sums may be studied through a profusion of methods: combinatorial, analytic and algebraic. The reader is referred to [2, Ch. 3] for a concise overview of Euler sums and their applications. We now prove identity directly. Theorem If x is any complex number not equal to a positive integer, then n n nn x m m x n n 2 n x. Proof. Fix x C \ Z +. Let S denote the left hand side. By partial fractions, S n n m m m m x nn mm x nn mn x nm+ mm x nn m nm+ n n m n 2 nn x n n m n m n nn x m m.

3 Now for fixed m Z +, nm+ n m lim n N m n, n N nm+ since m is fixed. Therefore, we have and S m n mm x n 2 n x. m n n n n m n n nn x m m n m n n lim N m n n nn x m For x, equation becomes the pair of tangent sum evaluations n n n mn nn 2 + m 2 + nn 2 + γ + Re Ψ + m n n nn 2 + m n + m m 2 + n π2 N n + n m n 2 n 2 + π2 6 Im Ψ + 6 π coth π. 2 m m Setting x 0 in equation gives ζ2, ζ3. More generally, differentiating k times with respect to x produces a corresponding formula for ζk + 3. For example, differentiating once and setting x 0 equivalently, comparing coefficients of x on both sides produces ζ4 ζ3, + ζ2, 2. Since, by the reflection formula 4, ζ2, 2 [ζ 2 2 ζ4]/2, we also evaluate ζ3, 4 ζ4 π This is the first case of a remarkable identity discovered by Zagier, and first proved in [4]. See below. Integrating equation produces the following corollary. 3

4 Corollary 2 For all complex x not equal to a positive integer, n n log x 2 n n n n m n m log For x, the imaginary part of 5 leads to n n m arctan m arctan /n n 2 mn + n 2 n m n x/m x/n n0. 5 n ζ2n + 3, 2n + where the last evaluation comes from writing arctan/n 0 nx2 + n 2 dx and exchanging the order of summation and integration. 3 Euler s Reduction Formula Euler s reduction formula is ζs, 2 sζs + s 2 ζk + ζs k, < s Z, 6 2 k which reduces the double Euler sum ζs, to a sum of products of classical Riemann ζ-values. Another marvellous fact is the sum formula ζ a + 2, a 2 +,, a r + ζr + s +, 7 Σa i s a i 0 valid for all integers s 0, r. It is easy to show that 7 is equivalent to r k j x k >k 2 > >k r >0 k j n n r n x, r Z+. 8 The first three non-trivial cases of 7 are ζ3 ζ2,, ζ4 ζ3, + ζ2, 2 and ζ2,, ζ4. The ordinary generating function of the sequence {ζs : < s Z} is Zx : ζs x s x Ψ x γ. nn x s2 n One can check that Euler s reduction 6 is equivalent to the ordinary generating function identity n { nn x k nn x x 2 2 n 2 n x + } nn x n k n n n 4

5 This relies on observing that the right hand side of 6 involves the square and the derivative of Zx. In turn, 9 is equivalent to. This equivalence may be demonstrated as follows: Let S : S 4 : n n n nn x k, S 2 : k nn x, S 5 : n n nn x 2, S 3 : n 2 n x, S 6 : n n n 2 n x 2, It suffices to show that S S 2 + S 3 + S 2 4x/2 S 6 S 5. Observe that S S 6 x n n nn x k x. k n nn x kk x, S 5 S 2 xs 3, k and S4 2 n S 3 2 nn x kk x. n k Combining these yields the desired equivalence. Correspondingly, equation is equivalent to ζs + 3 ζ2 + a, + b, 0 s Z, a+bs a,b 0 which is an inversion of Euler s reduction formula 6, and simultaneously recaptures the case r 2 of equation 7. Thus, Theorem has established all of these results directly, without using analytic methods, or sophisticated partial fraction decompositions, as is usual. The first case, once more, is ζ2, ζ3. 4 More Generating Functions Euler sums arise very naturally and it is perhaps a trick of time that ζ2n is viewed as more natural than n ζ{2} n : ζ2, 2,..., 2. }{{} k 2 n k >k 2 > >k n>0 j j Indeed, Euler s infinite product for the sine function is precisely equivalent to sinπx n ζ{2} n x 2n, πx n0 5

6 from which we may deduce that ζ{2} n π 2n 2n +!, 0 n Z. Similarly, with ω a primitive cube root of unity, ζ{3} n x 3n n0 + x3 n k 3 + ζ3x 3 + Γ + xγ + ωxγ + ω 2 x x 6 + O x 9, 2 ζ and allows one to show that ζ{3} n is always in the ring generated by the numbers ζ3k k, 2,..., n. By various methods, one can show that for all 0 n Z, while a proof of ζ{3} n ζ{2, } n ζ{2, } n? 2 3n ζ{2, } n, 0 n Z 0 remains elusive. The bar over the 2 on the right hand side of 0 signifies that in the summation, terms of the form /k 2 are to be multiplied by k. Such sums are called alternating Euler sums. Only the first case of 0, namely π 6 6! k k 2 k m m 8 k k k 2 k m m ζ3 has a self-contained proof [2, 4]. Indeed, the only other proven case is k k 2 k m m m p p 2 p q q 64 k k k 2 k m m p m p 2 p p q q ζ3, 3. There has been abundant evidence amassed to support 0 since it was first conjectured [3] in 996. For example, very recently Petr Lisoněk checked the first n 85 cases to 000 decimal places in about 4 hours with only the expected roundoff error. And he checked n 63 in ten hours. If true, 0 would be the only known identification thus far of an infinite parametrized class of alternating Euler sums with a corresponding class of non-alternating Euler sums. An intriguing reformulation of 0 is stated below. This is an outcome of a complete set of equations for MZV s of depth four. 6

7 Conjecture Define a sequence of polynomials a n a a 2 t 3 and a n t for positive integers n by nn + 2 a n+2 n2n + a n+ + n 3 + n+ t 3 a n, n. Then lim a n t 3 n + t3 n 8n 3. A more recondite generating function identity is ζ{3, } n x 4n coshπx cosπx, π 2 x 2 n0 which is equivalent to Zagier s conjecture subsequently proved: ζ{3, } n 2π4n, 0 n Z. 4n + 2! The proof of see [2, p. 60], [4], [5] devolves from a remarkable factorization of the generating function in terms of Gaussian hypergeometric functions: ζ{3, } n x 4n 2 F t, t; ; 2 F it, it; ;, n0 where t + ix/2. 5 Further extensions Relatedly and more centrally, Euler s reduction formula 6 has many extensions. For instance, with even a > 0 and odd b > with a + b 2N +, one has [3] ζa, b ζaζb + { } a + b ζa + b 2 a N { } 2 2r 2r + ζ2r + ζa + b 2r, a b r and hence by the reflection formula 4, ζb, a { a + b + 2 a N { 2r + + a r } ζa + b } 2r ζ2r + ζa + b 2r. b 3 7

8 Although ζ is undefined, Euler s reduction 6 implies that 2 also holds in the case b if we interpret ζ 0. Thus, 2 is indeed an extension of 6. We next recast 2 using generating functions, and then develop some further consequences and interesting special cases. We initially keep b 2t + fixed and want to express Nt+ ζa, bx 2N t Nt+ n x 2N t n 2N t n m m b n n 2 x 2 We suppose that x 2 is not a positive integer and set D : d. The requisite component dx generating functions are Nt+ Nt+ Nt+ a + b ζa + b b Nt+ ζax 2N t ζa + bx 2N t N x 2N t Nt+ n x 2N t x 2 r Nt+ n 2x 2 Nt+ x 2N t n 2N t n x 2N t n 2N t+b ζ2n + Db n b! x2n+ n 2 x 2, n m n b n 2 x 2, n n x + b+ n + x b+ n 2r ζ2r + ζ2n 2r b n n n 2 x 2 n n 2 x 2 n D b b! 2 m b. ζb x 2, x 2 nn 2 x 2 n x b + n + x b, and Nt+ N x 2N t r 2r ζ2r + ζ2n 2r a t ζ2mx D b 2m b 2m! n t ζ2mx 2 m m n x 2 nn 2 x 2 n x b+ 2m n + x b+ 2m 8

9 t ζb + 2mx m n 2 n x 2m n + x 2m Combining these, with a fair amount of care, yields a generating function for fixed b: Thus, n using n n 2 x 2 n 2 x 2 n m n m m ζ2s, bx 2s 2 b s m n. ζ2t + k 2 x 2 2 n 2 t+ n 2 x 2 k n + { n 2 2x 2 n x 2 t+2 + n + x 2 t+2 2 } n 2 t+2 n { } 2 n 2 x 2 n x + 2t+ n + x 2t+ n n t { } ζ2t + 2 2m 2x n x. 2m n + x 2m π cot πx mb 4x 2 n π cotπx 2x { n x 2 t+ + n + x 2 t+ 2 } n 2t+ n n 2t n 2 x 2 2x t+ m ζ2t + 2 2m { n 2x and ζ0 2 k 2 x 2 2. k n x 2m n + x 2m Now we sum over the odd parameter, b, and obtain a two-variable ordinary generating function expressible as: n n 2 x 2 n m my m 2 y 2 s>0,t 0 π 2 x cotπx ζ2s, 2t + x 2s 2 y 2t n + π 2 x cotπx n + π 2 y cotπy n 9 ny4y 2 x 2 /n 2 y 2 n + x 2 y 2 n x 2 y 2 ny n + x 2 y 2 n x 2 y 2 2ny n + x 2 y 2 n x 2 y 2 },

10 2 n ny n 2 y 2 n 2 x 2. 4 Attractive specializations come with x ±y, x ±2i, y, x 0 and, after dividing by y, for y 0. For example, at /2, /2 where the right hand side has a removable discontinuity, we deduce that n n m 4n 2 4m 2 π m Note that the second, third and fourth terms on the right hand side of 4 are expressible as Σ 2 : π Ψ + x + y Ψ + x y Ψ x + y + Ψ x y x cotπx, 4 4xy Σ 3 : π Ψ + x + y Ψ + x y Ψ x + y + Ψ x y y cotπy, 2 4xy {Ψ + x Ψ y + Ψx Ψy + π cotπx} y Σ 4 :, 4y 2 x 2 respectively while the first term has a corresponding evaluation. It is Σ : y 2 Ψ x y + Ψ + x y + Ψ + x + y + Ψ x + y π cot π 8xx 2 4 y 2 Ψ + x + y Ψ + x y Ψ x + y + Ψ x y yπ cotπx 6x 2 4 y 2 y 2 Ψy + + Ψ y π cotπx. 4xx 2 4 y 2 We finish by expressing a form of this generating function concisely as: Theorem 3 For all x and y with squares not equal to negative integers, n n 2 + y + n 2m 2 m 2 + y 2 n 2 + x 2 n m x 2 4y 2 n πx cothπx n 2 + y 2 {n 2 x 2 + y nx 2 } n n + 2πy cothπy + πx cothπx n 2 x 2 + y nx. 2 n 0

11 Proof. Replace x by ix and y by iy in 4, and then factor denominators and regroup the terms as needed. QED Setting x 2y pro- Letting x and y approach zero yields ζ2, ζ3 again. duces 2 the identity with which we began. To conclude, we challenge the reader to explicitly obtain the corresponding two variable generating function for ζ2t +, 2s. Acknowledgements. J. Borwein s research was supported by NSERC and the Canada Research Chair Programme, and D. Borwein s research was supported by NSERC. Thanks are due to John Zucker and John Boersma for the discussions which stimulated this work. Key Words. Euler sums, Zeta functions, Generating functions, Multiple zeta values. Classification Numbers. Primary 33C99 Secondary A99, M99

12 References [] J. M. BORWEIN and D. H. BAILEY, Mathematics by Experiment: Plausible Reasoning in the 2st Century, A. K. Peters Ltd., [2] J. M. BORWEIN, D. H. BAILEY, and R. GIRGENSOHN, Experimentation in Mathematics: Computational Paths to Discovery, A. K. Peters, [3] J. M. BORWEIN, D. J. BROADHURST, and D. M. BRADLEY, Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k, Electronic J. Combinatorics, 4 997, no. 2, #R5. Wilf Festschrift. [4] J. M. BORWEIN, D. M. BRADLEY, D. J. BROADHURST, and P. LISONĚK, Special values of multiple polylogarithms, Trans. Amer. Math. Soc., , no. 3, [5] D. BOWMAN, D. M. BRADLEY, and J. RYOO, Some multi-set inclusions associated with shuffle convolutions and multiple zeta values, European J. Combinatorics, , David Borwein Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 Canada Jonathan M. Borwein Faculty of Computer Science Dalhousie University Halifax, NS B3H W5 Canada David M. Bradley Department of Mathematics & Statistics University of Maine 5752 Neville Hall Orono, Maine U.S.A. dborwein@uwo.ca jborwein@cs.dal.ca dbradley@math.umaine.edu 2