Appendix Curious And Exotic Identities For Bernoulli Numbers

AppendixCurious and Exotic Identities for BernoulliNumbersDon ZagierBernoulli numbers, which are ubiquitous in mathematics, typically appear either asthe Taylor coefficients of x tan x or else, very closely related to this, as specialvalues of the Riemann zeta function. But they also sometimes appear in other guisesand in other combinations. In this appendix we want to describe some of the lessstandard properties of these fascinating numbers.In Sect. A.1, which is the foundation for most of the rest, we show that, as wellas the familiar (and convergent) exponential generating series11X Bnx2x4x6xxnDxC C D1 ex 1nŠ212 72030240nD0(A.1)defining the Bernoulli numbers, the less familiar (and divergent) ordinary generating seriesˇ.x/ D1XnD0Bn x n D 1 x2 x4x6xC C 263042(A.2)also has many virtues and is often just as useful as, or even more useful than, itsbetter-known counterpart (A.1). As a first application, in Sect. A.2 we discuss the“modified Bernoulli numbers”!nX nCrBr Bn Dn 1 :(A.3)nCr2rrD0Here, and throughout this appendix, we use the convention B1 D 1 2, rather than theconvention B1 D 1 2 used in the main text of the book.1T. Arakawa et al., Bernoulli Numbers and Zeta Functions, Springer Monographsin Mathematics, DOI 10.1007/978-4-431-54919-2, Springer Japan 2014239

240Appendix: Curious and Exotic Identities for Bernoulli NumbersThese numbers, which arose in connection with the trace formula for the Heckeoperators acting on modular forms on SL.2; Z/, have several unexpected properties,including the surprising periodicity BnC12D Bn (n odd)(A.4)and a modified form of the classical von Staudt–Clausen formula for the valueof Bn modulo 1. The following section is devoted to an identity discovered byMiki [A10] (and a generalization due to Gessel P[A4]) which has n strikingPtheBr Bn r ,property of involving Bernoulli sums both of typeBr Bn r andri.e., sums related to both the generating functions (A.1) and (A.2). In Sect. A.4 welook at products of Bernoulli numbers and Bernoulli polynomials in more detail.In particular, we prove the result (discovered by Nielsen) that when a productof two Bernoulli polynomials is expressed as a linear combination of Bernoullipolynomials, then the coefficients are themselves multiples of Bernoulli numbers.This generalizes to a formula for the product of two Bernoulli polynomials in twodifferent arguments, and leads to a further proof, due to I. Artamkin, of the Miki–Gessel identities. Finally, in Sect. A.5 we discuss the continued fraction expansionsof various power series related to both (A.1) and (A.2) and, as an extra titbit,describe an unexpected appearance of one of these continued fraction expansionsin connection with some recent and amazing discoveries of Yu. Matiyasevichconcerning the non-trivial zeros of the Riemann zeta function.This appendix can be read independently of the main text and we will recallall facts and notations needed. We should also add a warning: if you don’t likegenerating functions, don’t read this appendix!A.1 The “Other” Generating Function(s)for the Bernoulli NumbersGiven a sequence of interesting numbers fan gn 0 , one often tries to understandthem by using the properties of the corresponding generatingThe twoP1functions.nmost popular choices for thesegeneratingfunctionsareax(“ordinarynD0 nP1ngenerating function”) andnD0 an x nŠ (“exponential generating function”).Usually, of course, at most one of these turns out to have useful properties. For theBernoulli numbers the standard choice is the exponential generating function (A.1)because it has an expression “in closed form.” What is not so well known is that theordinary generating function of the Bernoulli numbers, i.e., the power series (A.2),even though it is divergent for all non-zero complex values of x, also has extremelyattractive properties and many nice applications. The key property that makes ituseful, despite its being divergent and not being expressible as an elementaryfunction, is the following functional equation:

A.1 The “Other” Generating Function(s) for the Bernoulli Numbers241Proposition A.1. The power series (A.2) is the unique solution in QŒŒx of theequation x 1ˇ ˇ.x/ D x :1 x1 x(A.5)Proof. Let fBn g be unspecified numbers and define ˇ.x/ by the first equalityin (A.2). Then comparing the coefficients of x m in both sides of (A.5) gives!m1Bn Dn0m 1XnD0if m D 1;if m 1:(A.6)This is the same as the standard recursion for the Bernoulli numbers obtained bymultiplying both sides of (A.1) by e x 1 and comparing the coefficients of x m mŠon both sides.tuThe functional equation (A.5) can be rewritten in a slightly prettier form bysettingˇ1 .x/ D x ˇ.x/ D1XBn x nC1 ;nD0in which case it becomes simplyˇ1 x ˇ1 .x/ D x 2 :1 x(A.7)A generalization of this is given by the following proposition.Proposition A.2. For each integer r 1, the power series!1XnCr 1ˇr .x/ DBn x nCrnnD0(A.8)satisfies the functional equationˇr x ˇr .x/ D r x rC11 x(A.9)and is the unique power series having this property.Proof. Equation (A.9) for any fixed value of r 1 is equivalent to the recursion (A.6), by the calculation

242Appendix: Curious and Exotic Identities for Bernoulli Numbers!11 x XXnCr 1Bnˇr ˇr .x/ Dn1 xnD0D1X Dr DnCr! x C1nCr 1! ! rX r C1 C1xBn D r x rC1 :r 1nnD0!Alternatively, we can deduce (A.9) from (A.7) by induction on r by using the easilychecked identityx 2 ˇr0 .x/ D r ˇrC1 .x/.r 1/(A.10)and the fact thatx2d x x 2 0 x FDFdx1 x1 x1 x(A.11)tufor any power series F .x/.We observe next that the definition (A.8) makes sense for any r in Z,2 and thatthe properties (A.9) and (A.10) still hold. But this extension is not particularlyinteresting since ˇ k .x/ for k 2 Z 0 is just a known polynomial in 1 x :!!1kXXn k 1Bnn kn kˇ k .x/ DD. 1/Bn xnn x k nnD0nD0D Bk 1 xCkx k 1D Bk 1 1 C 1 D . 1/k Bk ;xxwhere Bk .X / is the kth Bernoulli polynomial. (One can also prove these identitiesby induction on k, using either (A.10) or else (A.9) together with the uniquenessstatement in Proposition A.2 and the corresponding well-known functional equationfor the Bernoulli polynomials.) However, there is a different and more interestingway to extend the definition of ˇr to non-positive integral values of r. For k 2 Z,define k .x/ DXn max.1; k/.n 1/ŠBnCk x n.n C k/Š2 x QŒŒx :Then one easily checks that r .x/ D .r 1/Š ˇr .x/ for r 0, so that thenegative-index power series k are just renormalized versions of the positive-indexpower series ˇr . But now we do get interesting power series (rather than merelypolynomials) when k 0, e.g.2Or even in C if we work formally in x r QŒŒx .

A.1 The “Other” Generating Function(s) for the Bernoulli Numbers11XXBnC1 x nBnC2 x n; 2 .x/ D:nn.n C 1/n.n C 1/.n C 2/nD1nD1nD1(A.12)The properties of these new functions corresponding to (A.10) and (A.9) aregiven by: 0 .x/ D1XBn x n243; 1 .x/ DProposition A.3. The power series k .x/ satisfy the differential recursionx 2 k0 .x/ D k 1 .x/ BkxkŠ.k 0/.with 1 .x/ D ˇ1 .x// as well as the functional equations x 0 .x/ D log.1 x/ C x ; 01 x x 1 1 1 .x/ D log.1 x/ 1 ; 11 xx 2(A.13)(A.14)and more generally for k 1 x 1 1. 1/k k;Bk k .x/ Dlog.1 x/ C Pk 11 xkŠxx(A.15)where Pk 1 .X / is a polynomial of degree k 1, the first few values of which are1P0 .X / D 1, P1 .X / D X 12 , P2 .X / D X 2 X C 12, P3 .X / D X 3 32 X 2 C113113X C 12 and P4 .X / D X 4 2X 3 C 4 X 2 C 4 X 360 .3Proof. Equation (A.13) follows directly from the definitions, and then Eqs. (A.14)and (A.15) (by induction over k) follow successively from (A.7) using the generalidentity (A.11).tuWe end this section with the observation that, although ˇ.x/ and the relatedpower series ˇr .x/ and k .x/ that we have discussed are divergent and do not givethe Taylor or Laurent expansion of any elementary functions, they are related tothe asymptotic expansions of very familiar, “nearly elementary” functions. Indeed,Stirling’s formula in its logarithmic form says that the logarithm of Euler’s Gammafunction has the asymptotic expansion1 X11 Bnlog X X C log.2 / CX nC1X 22n.n 1/nD2log .X /as X ! 1, and hence that its derivativeexpansion.X / WD0.X /.X /.X / (“digamma function”) has the 1 X Bn1log X X n D log X 0 2X nD2 nX1

244Appendix: Curious and Exotic Identities for Bernoulli Numbersas X ! 1, with 0 .x/ defined as in Eq. (A.12), and the functions ˇr .x/ correspondsimilarly to the derivatives of .x/ (“polygamma functions”). The transformationx 7! x .1 x/ occurring in the functional equations (A.5), (A.9), (A.14) and (A.15)corresponds under the substitution X D 1 x to the translation X 7! X C1, and thecompatibility equation (A.11) simply to the fact that this translation commutes withthe differential operator d dX , while the functional equations themselves reflectthe defining functional equation .X C 1/ D X .X / of the Gamma function.A.2 An Application: Periodicity of Modified BernoulliNumbersThe “modified Bernoulli numbers” defined by (A.3) were introduced in [A14].These numbers, as already mentioned in the introduction, occurred naturally in acertain elementary derivation of the formula for the traces of Hecke operators actingon modular forms for the full modular group [A15]. They have two surprisingproperties which are parallel to the two following well-known properties of theordinary Bernoulli numbers:n 1 odd)n 0 even)Bn D 0 ;X 1Bn pp prime(A.16).mod 1/(A.17).p 1/jn(von Staudt–Clausen theorem). These properties are given by:Proposition A.4. Let Bn .n 0/ be the numbers defined by (A.3). Then for n oddwe haveBn D 3 4 if n 1 .mod 12/,1 4 if n 3 or 5 .mod 12/,(A.18)and for n even we have the modified von Staudt–Clausen formula2nBn Bn X 1pp prime.mod 1/ :(A.19).pC1/jnRemark. The modulo 12 periodicity in (A.18) is related, via the above-mentionedconnection with modular forms on the full modular group SL.2; Z/, with the wellknown fact that the space of these modular forms of even weight k 2 is the sumof k 12 and a number that depends only on k .mod 12/.

A.2 An Application: Periodicity of Modified Bernoulli Numbers245Proof. The second assertion is an easy consequence of the correspondingproperty (A.17) of the ordinary Bernoulli numbers and we omit the proof. (It isgiven in [A15].) To prove the first, we use the generating functions for Bernoullinumbers introduced in Sect. A.1. Specifically, for 2 Q we define a power seriesg .t/ 2 QŒŒt by the formulag .t/ D 0where 0 .x/ Dspecializes toPn 0 Bn xg2 .t/ Dn t log.1 t C t 2 / ;1 t C t2 n is the power series defined in (A.12). For1XBrrD11Xtr 2log.1 t/D2Bn t n :r .1 t/2rnD1D 2 this(A.20)with Bn as in (A.3). On the other hand, the functional equation (A.14) applied tox D t .1 t C t 2 /, together with the parity property 0 .x/ C x D 0 . x/, whichis a restatement of (A.16), implies the two functional equationsgC1 .t/D g .t/ CtD g . t/1 t C t2for the power series g . From this we deduce g2 .t/ g2 . t/ D g2 .t/ g1 .t/ C g1 .t/ g0 .t/ C g0 .t/ g 1 .t/Dttt3t t 3 t 5 C t 7 C t 9 3t 11CCD;1 t C t21 C t21 C t C t21 t 12and comparing this with (A.20) immediately gives the desired formula (A.18) forBn , n odd.utWe mention one further result about the modified Bernoulli numbers from [A15].The ordinary Bernoulli numbers satisfy the asymptotic formulaBn . 1/.n 2/ 22 nŠ.2 /n(n ! 1, n even).(A.21)As one might expect, the modified ones have asymptotics given by a very similarformula:Bn . 1/.n 2/ 2.n 1/Š.2 /n(n ! 1, n even).(A.22)The (small) surprise is that, while the asymptotic formula (A.21) holds to all ordersin 1 n (because the ratio of the two sides equals .n/ D 1 C O.2 n /), this is not