Bernoulli Numbers Jeff Morton

Transcription

1 Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f is entire, Tylor s d k theorem sys tht this expnsion s power series in t bout converges to the vlue of f + t. This being so for every C, we hve tht e t f f + t. 2. The difference opertor is defined s f f+ f e f f, for ll f,. So s n opertor, e e. 3. If F f, we hve n i fi n i F i n i F i + F i. This is telescoping sum, in which every vlue in the sum except the first nd lst ppers twice, with opposite signs first positive, then negtive with the next vlue of of the summtion index. Cncelltion leves the F i + term for the cse i n nd the F i term for i. Thus, we hve fi F n F. n i 4. Given ny entire function f E, we hve f d d d f d fudu As complex-vlued function on the rel line, the Fundmentl Theorem of Clculus mens tht the derivtive of fudu with respect to t is just the vlue of f there, f. Since f is entire s function f : C C, the complex derivtive exists everywhere nd is just the sme s the derivtive s function f : R C. So in fct we hve for ll tht f f, so f f nd this is true f E, hence in fct s n opertor Id E. On the other hnd, since the integrl of function is only defined up to constnt, is not right inverse of : if f k for ny constnt k, we hve: d f fudu du k du du This is, in generl, not f, which is k for ll. Thus, is left inverse only for, so clled. In ctegoricl terms, both mps re endomorphisms of E. Seen this wy, is n epimorphism indeed, split epi since it hs right inverse - nd in fct is surjective s set-mp since every function in E is the derivtive of something in E nd likewise is monomorphism indeed, spilt mono since it hs left inverse - nd in fct is injective s set mp, since there is exctly one function in E whose integrl ny given f is, nmely its derivtive. 5. The Bernoulli numbers re the coefficients B k in the expression e k B k k, which is n entire function. The function e is lso entire.

2 nd only ero t, where it hs ero of first order hence e being entire. The power series for this function is e j j j j. Since both of these re entire functions, the power series converge everywhere in C, nd the product is lso entire, so we cn write the product of the two functions which is just, of course s the product of the two power series: j j k B k k Now, to see tht the sme pplies when we replce the complex vrible by the differentil opertor d d, we cn note tht the Fourier trnsform of this differentil opertor is multipliction by i.e. if we tke functions to their Fourier trnsforms, the ction of is to tke the trnsform of function f, sy f, to f, nd k cting on f tkes f to k f. Thus, by the bove nd by linerity of the Fourier trnsform, the effect of on f is: Now, by definition of nd j j e k B k k, this just sys tht e. 6. To see tht is right-inverse of, note tht for ny f E, we hve f e f definition of e fudu definition of fudu by prt 5 fudu d d f So in fct f f. On the other hnd, the converse need not be so: f e f e e f e j j fdu k B k k j j f du k B k k j j f du This lst step mkes sense by linerity nd since f is entire, so every derivtive exists everywhere: the sum converges since e f is lso entire. Now, since the integrl is only right-inverse of, this will not necessrily be the sme s f. If the integrl were left-inverse of, we could pss the integrl through the derivtives in front of f nd get bck f by prt 5. However, this is not gurnteed to work, nd we mke get constnt of integrtion. Thus, f my not be equl to f. 7. We hd seen tht j j k B k k nd if we equte coefficients of powers of, we find tht every coefficient of the right hnd side is except 2

3 for the coefficient of, which is. Now, the coefficient of i on the right hnd side will be j + k i j > Bk i j B i j i If we expnd this sum for ny i, so tht the whole sum is, we find:! B i i! i! B +! i! B! 8. The reltions we found in prt 7 give expressions in the B j which sum to, one for ech vlue of i greter thn. In ech cse, we hve frctionl coefficients which cn be clered by multiplying the whole expression on the right hnd side by i!, in which cse we get the reltions: j + k i j > i! B i j j + k i j > i j B i j Notice tht the coefficients of the B j re the sme s the binomil coefficients from Pscl s tringle, s we hd hoped. 9. To find out B, recll tht we defined the B k to be the coefficients in of the power series for the function e extended to equl t x. B is the constnt coefficient for the power series bout, nd is therefore. Using the reltions from prt 8, this implies tht: + 2B, hence B B B 2, hence B B 3 + 4B 3, hence B B B 4, hence B B 5, hence B 5. We hd seen tht n i ip p p+ k B p+ k k n + p+ k Applying this to the sitution where p 4, we find n i i4 4 5 k B 5 k k n + 5 k [ 5 n n n ] 3 5n + n+5 5 n n+3 3 n+ 3. The binomil expnsion for B + n + p+ is p+ p+ k k B k n + p+ k. Identifying B k with B k nd dividing by p + gives the expression bove. 2. The first nd most obvious reson it s difficult to ctegorify this business is the presence of negtive coefficients, which mens we cn t ctegorify using 3

4 ordinry species. Cubicl species could help here, though they re not necessry t first. Another problem is tht A nd will not be ssocited with nturl trnsformtions between structure types, since they necessrily involves n rbitrry choice of elements to remove, in ech cse. Here follow some comments on ctegorifying the results from vrious prts of this computtion: ta k,. If we define E ta to be the opertion on structure types E T A k this mounts to n opertion tking structure type nd producing the sum union of types which pply T A some number k of times, over ll k. For ech prticulr cse k, in the cse t, this is simply tking the derivtive k times, which gives new structure whose effect on set S is to put the originl structure on the set S + k - the denomintor reduces be the ction of permuttion group, mening these elements re unordered. When T is generl, we interpret this s mening tht the elements we dd re T -coloured. The result here is tht E T A F Z F Z + T - tht is, putting the structure E ta F on set is the sme s putting n F structure on set of things which re either one-element sets or members of T, the set of colours we could pint the new elements we dd in the definition of E T A. Tht is, we think of these not s elements of set contributing to the crdinlity of the set S on which we put the E T A -structure, but s just colours. Here we re using the interprettion of composition tht F Z + T structure is n F structure on sets of Z + T structures, i.e. things which re either one-element set or colour from T. 2. The opertor, pplied to structure type F, should stisfy F Z F Z + F Z, which s n equivlence of structure types mens tht F structure on set S is n F structure on set consisting of either single elements, or the empty set tht is, n F structure on ny set lrger thn or equl to thn S, since we re simply not counting some of the points towrd the crdinlity, with the exception tht it cnnot be simply n F structure on S tht is, those on sets whose elements re just one-point sets re removed. This mens F -structure on S is n F structure on ny set bigger thn S. Wht we re sying here E A, is tht the nturl trnsformtion between structure types which re functors is the sme s the nt. trns. which tkes the derivtive ny number of times other thn - tht is, which dds ny number of points surreptitiously into our set before putting the F -structure on it where F is whtever structure is cting on. This is obvious from the description in the lst prgrph. 3. We wnt to sy tht if G F, then n i F i Gn G. The first sys tht F structures on S re G structures on nything strictly It seems there should be correction in the crdinlity ccounting for the denomintor, something to the effect tht ll the dded elements re interchngeble... Not cler to me t the moment wht exctly this should be, though. 4

5 contining S. The second sys tht when we tke the groupoid we get by evluting G t the n-element set nd removing from it the sub-groupoid which is the sme s wht we get evluting G t the empty set, we should get the sme s if we tck together ll the groupoids obtined by evluting F t sets of sie smller thn n. When we proved this in the power series cse, we hd telescoping series - similir effect should occur here - ech of the F i groupoids will be esily describble s some groupoid Gi + with Gi removed the strict inclusion in our description of F, so tking ll these together will fill in ll the missing prts of Gn from F n except the prt where we evlute t the empty set Now we re defining n inverse to the derivtive. This A is clerly nonunique, since ny given set S cn be written in S different wys s some smller set with single element djoined. So when we tke A of some structure type F nd put this new type on set S, we get sets of F structures on S with one element removed nonuniqueness coming from the fct tht we could tke out different elements, so there is no nturl wy to do this. This is not relly nturl trnsformtion of species, which presumbly hs something to do with the extr constnt tht comes in when we integrte e.g. integrting the structure type being 5-element set, Z 5 gives 6 Z6 - this frctionl coefficient pprently counting the number of wys we could hve done this, suggesting tht it mesures the degree of nonuniqueness of A. Tht AA F F, if we swllow this problem nd keep going, is due to the fct tht putting n element in, once we hve removed one, gives set tht cn be nturlly identified with the originl by clling these the sme element, which we re putting bck in. Tht A AF is not nturlly equivlent to F is due to the fct tht if we remove the element AFTER putting one it, it my not be the sme one. 5. Getting n inverse for, the trnsformtion which, pplied to F gives F -structures on bigger sets thn S, is problemtic for similr resons to the problems we encountered in ctegorifying prt 4, only more so. The even more so is visible in the fct tht we would need to ctegorify the differentil-opertor power series we hd for the difference opertor, which, however, hs coefficients which re not only frctionl which we might could hndle by some clever trick with groupoids, but lso negtive. This might could be hndled by re-csting this whole cry ffir in the setting of cubicl species, but I won t be doing tht here. Well, from here on in these problems will only get worse, so let s tke this opportunity to stop ctegorifying for the moment, mentioning only tht to do this properly would require working in ctegory in which we cn hndle negtive coefficients, which by itself would be oky since such ctegory exists. We lso lso would hve to somehow del suitbly with the non-nturlity of 2 I m not quite sure how to put this better. Describing the groupoids involved here is still bit mysterious to me. So it goes. 5

6 some of the trnsformtions involved - nd the method for doing this ought to give entities rther like structure types, but with frctionl coefficients, where the denomintors hndle the sie of the collection of different choices we might mke t certin key points. Since we lredy hve non-integrl coefficients turning up when we consider groupoid crdinlities, nd these re relted to utomorphisms of objects in groupoid, this might be relevnt tool, using those non-unique choices to give those utomorphisms. Then gin, it might not. 6