Euler-MacLaurin using the ZETA-matrix

Transcription

1 Idetities ivolvig biomial-coeiciets, Beroulli- ad Stirligumbers Gottried Helms - Uiv Kassel 2' Euler-MacLauri usig the ZETA-matrix Abstract: The represetatio o the Euler-MacLauri-summatio ormula i terms o the ZETA-matrix (modiied Beroulli-polyomials) is give. It is show, that the Euler-MacLauri-ormula ca be see as aother variat o summatio usig the ZETA-matrix, where simply the order o summatio is chaged. I the text I use the term "matrixoperator". I called some o my matrices this way beore I became aware, that this type o matrices was already kow as "Carlema"-matrices, or, i a actorially similarity-scaled orm, as "Bell"- matrix ad are already well studied as tools or the expressio o iteratio o uctios whe cosidered i terms o their ormal power series. I've let my ow amig covetio here because o reasos o coveiece or mysel, perhaps I'll adapt this later. (Vers.4). Sum o values whe a uctio is evaluated at cosecutive argumets 2.. The Euler-MacLauri-ormula 2.2. The otatio i terms o matrixoperators ad dotproducts 3.3. The use o the ZETA-matrix/beroulli-polyomials 3.4. Chagig order o computatio: the ZETA-matrix ad the dot-product with F 5.5. The Euler-MacLauri sum S i terms o V(), ZETA ad F 6 2. Reereces 8 Gottried Helms, irst versio 7'2, helms@ui-kassel.de

2 EulerMacLauri by ZETA-Matrix S Sum o values whe a uctio is evaluated at cosecutive argumets.. The Euler-MacLauri-ormula We begi with the Euler-/MacLauri-ormula where we also simply the otatio rom () to as give i [Kopp]'s book: b () 2 b = ( t) dt + ( + ) + ( ' ') + ( ) + Rk 2 2! 4! where the b k are the beroulli-umbers ad R k relects the remaiig error, i that series is trucated at the k'th term. The ormula ca be made more smooth by cacellig o (): this also adapts sig i the secod summad. Also we iterpret +/2 by b to coect this term with the sequece o beroulli-umbers: (b) b b2 b = ( t) dt ( ) + ( ' ') + ( ) + R! 2! 4! The we replace the Beroulli-umbers by zeta-uctio represetatios: b = ζ ( ) b2 = 2 ζ ( 2) b3 = 3 ζ ( 3) see or istace [Woo] 98) k... write the traslated versio, () () ( ) ( ) ( ) = ( t) dt ζ () ζ ( ) ζ ( 3) +!! 3! R k assume the limit where r->i (ad R r ->) ad the get the rewritte expressio to which I'll reer i the ollowig: ( (2) S = ( k) = ( t) dt ζ ( k) ( k ) ( ) k = k = k! ( k ) ()) take rom K. Kopp, "iiite series" Idetities with biomials, Beroulli- ad other umbertheoretical umbers Mathematical Miiatures

3 EulerMacLauri by ZETA-Matrix S The otatio i terms o matrixoperators/carlemamatrices ad dotproducts Now, i the otatio o matrixoperators/carlemamatrices this gets the ollowig represetatio: First we assume a columvector F o iiite size which cotais the coeiciets o the ormal powerseries or (x), say: (x) = K + ax + bx 2 + cx 3 + such that F = [K,a,b,c,d, ], Next we itroduce the otatio o a Vadermodevector V(x) havig a ormal idetermiate argumet ad has the orm: V(x)=[, x, x 2, x 3, ]. The the uctio (x) ca ormally be writte as dot-product o V(x) ad F: (x) = V(x)~ F Sice our goal is to represet the Euler-MacLauri-ormula i this ramework, we restate aaloguously (3) S = () + (2) + (3) + + () = (V() + V(2) + V(3) + + V())~ F.3. The use o the ZETA-matrix/beroulli-polyomials Now we use a modiied versio o the beroulli-polyomials to express the sum V()+ +V(). For this I itroduced i "sums-o-like-powers" [H 27] the ZETA-matrix 2, which cotais the coeiciets o the itegrals o the beroulli-polyomials, ad ca immediately be used or the problem o sums-o-like-powers i the ollowig way: ZETA (V(a) V(b)) = V(a+)+ V(a+2) + + V(b) The ZETA-matrix is o iiite size, triagular with oe upper subdiagoal illed ad has the ollowig aspect (top-let segmet is show): The structure is simple; it cotais just the values o the zeta at egative itegers coactored by biomials (like i the Pascalmatrix P): (4) ZETA = ζ( ) ζ(-) ζ( ) -/2.... ζ(-2) ζ(-) 2 ζ( ) -/3... ζ(-3) ζ(-2) 3 ζ(-) 3 ζ( )... ζ(-4) ζ(-3) 4 ζ(-2) 6 ζ(-) 4... ζ(-5) ζ(-4) 5 ζ(-3) ζ(-2) Note, that the egative reciprocals i the upper subdiagoal ca cosistetly be uderstood as cotaiig the limit s-> o ζ( s)/γ(s-) 2 which is a extesio o the origial matrix o Faulhaber by the leadig colum cotaiig zeta-values (see the grey-shaded colum Idetities with biomials, Beroulli- ad other umbertheoretical umbers Mathematical Miiatures

4 EulerMacLauri by ZETA-Matrix S. -4- The expressio, or istace rom a+ to b is the ideed (usig ZETA or the sum o like powers) ZETA (V(a) - V(b)) = V(a+)+V(a+2)+ +V(b) or rom to ZETA (V() - V()) = V()+V(2)+ +V() example V(a)-V(b) - - a - b - ZETA (V(a) - V(b)) = V(a+)+V(a+2)+ +V(b) a 2 - b 2-2 a 3 - b 3-3 a 4 - b 4-4 ZETA (V() - V()) = V()+V(2)+ +V() a - b a 6 - b ζ( ) (a+) +(a+2) +...+b example V()-V() ζ(-) ζ( ) -/2.... (a+) +(a+2) +...+b ζ(-) 2 ζ( ) -/3... (a+) 2 +(a+2) b ζ(-3) ζ(-) 3 ζ( )... (a+) 3 +(a+2) b ζ(-3) 4. ζ(-) 4... (a+) 4 +(a+2) b ζ(-5) ζ(-3)... (a+) 5 +(a+2) b ζ(-5) 6 ζ(-3) 2... (a+) 6 +(a+2) b The same i terms o beroulli umbers: (b = /2) arbitrary -b..... (a+) +(a+2) +...+b arbitrary b -/2 b.... (a+) +(a+2) +...+b arbitrary -b 2 2/2 b - /3 b... (a+) 2 +(a+2) b arbitrary -3/2 b 2 3/3 b... (a+) 3 +(a+2) b arbitrary -b 4-6/3 b 2... (a+) 4 +(a+2) b arbitrary -5/2 b 4... (a+) 5 +(a+2) b arbitrary -b 6-5/3 b 4... (a+) 6 +(a+2) b arbitrary Note: "arbitrary" meas here that values are ot covered by the commo deiitio o/asatz with the Beroulli-polyomials resp. their itegrals. Note: the Beroulli-umber b is sometimes deied havig the positive value istead Idetities with biomials, Beroulli- ad other umbertheoretical umbers Mathematical Miiatures

5 EulerMacLauri by ZETA-Matrix S Chagig order o computatio: the ZETA-matrix ad the dot-product with F For the ollowig, to have the otatio compatible with my other recet articles, we use the trasposed o the above ZETA-equatio: (V() - V(x))~ ZETA~ = V()~+V(2)~+...+V()~ The that expressio shall be multiplied by the coeiciets-vector or the uctio (x) = K + ax + bx² +... : (V()~+V(2)~+...+V()~ ) F = () + (2) + (3) () = ((V() - V())~ ZETA~) F... K a b c d e (-) ζ() (-) ζ(-) (-) ζ(-3) (-) ζ(-5)... (-) (-) ζ() (-) ζ(-) 2 - ζ(-3) 4 - ζ(-5) (- 2 ) /2 (- 2 ) ζ() - 2 ζ(-) 3-2 ζ(-3)..... (- 3 ) /3-3 ζ() - 3 ζ(-) 4-3 ζ(-3) The same i terms o beroulli-umbers (b =-/2) b... - (-) b b 2 b 4 b /2-2 b 2 b 2 3/2 2 b 4 5/ /3-3 b 3 b 2 6/3 3 b 4 5/ Idetities with biomials, Beroulli- ad other umbertheoretical umbers Mathematical Miiatures

6 EulerMacLauri by ZETA-Matrix S The Euler-MacLauri sum S i terms o V(), ZETA ad F The expressio or the Euler-MacLauri ormula occurs the by reorderig o summatio. I we do ot compute the dot-product o the rows o ZETA with F, but evaluate usig the subdiagoals (which cotai the same zeta-value) separately: (-) ζ() (-) ζ(-) (-) ζ(-3) (-) ζ(-5) ζ() - ζ(-) 2 - ζ(-3) 4 - ζ(-5) /2-2 ζ() - 2 ζ(-) 3-2 ζ(-3) /3-3 ζ() - 3 ζ(-) 4-3 ζ(-3) K a b c d e... the we get the ollowig, orgaized alog the sums over the subdiagoals : - K/ (- ) K ζ() a/2 (- ) a ζ() (-) a ζ(-) b/3 (- 2 ) b ζ() - 2 b ζ(-) b c/4 (- 3 ) c ζ() c ζ(-) c (-) c ζ(-3) d/5 (- 4 ) d ζ() d ζ(-) d - 4 d ζ(-3) d. - 6 e/6 (- 5 ) e ζ() e ζ(-) e - 2 e ζ(-3) e (-) e ζ(-5) - 7 /7 (- 6 ) ζ() ζ(-) ζ(-3) - 6 ζ(-5) which, whe summed up columwise, gives ( ) ( ) ζ () ( t) dt! () () ζ ( ) ζ! ( 3) ζ 3! (5) (5) ( 5) ζ 5! The sum o all that colum-sums ( k ) ( k ) () ( ) ζ ( k) k! i ( t) dt + k = relects precisely the Euler-MacLauri sum as give i (2). Idetities with biomials, Beroulli- ad other umbertheoretical umbers Mathematical Miiatures

7 The same i terms o the beroulli-umbers looks like: EulerMacLauri by ZETA-Matrix S K/ () K b a/2 ( - ) a b () a b 2 / b/3 ( - 2 ) b b 2 b b 2 /2 b c/4 ( - 3 ) c b 2 3 c b 2 /2 c () c b 4 / d/5 ( - 4 ) d b 3 4 d b 2 /2 d 4 d b 4 /4 d. - 6 e/6 ( - 5 ) e b 4 5 e b 2 /2 e 2 e b 4 /4 e () e b 6 /6-7 /7 ( - 6 ) b 5 6 b 2 /2 3 2 b 4 /4 6 b 6 /6 Whe summed up columwise this gives t) dt ( ( ) b () () b2 2! b4 4! (5) (5) b6 6! The sum o that colum-sums i ( k ) ( k ) b (2k ) (2k ) ( () ( ) ) + ( ( ) ()) b2 k (2k)! ( t) dt +! k = relects the Euler-MacLauri sum as give i (.b) ater sigs are adapted. Idetities with biomials, Beroulli- ad other umbertheoretical umbers Mathematical Miiatures

8 EulerMacLauri by ZETA-Matrix S Reereces [H 27] Gottried Helms Summig o like powers see pg 4..9 [Kopp] [Woo] Korad Kopp Theorie ud Awedug der uedliche Reihe 5 th editio, Spriger (964) (Olie available at Digiceter Uiversity Göttige) see pg 542, ormula # 298 S.C.Woo Geeralizatio o a relatio betwee the Riema zeta uctio Idetities with biomials, Beroulli- ad other umbertheoretical umbers Mathematical Miiatures