Bernoulli Number Identities via Euler-Maclaurin Summation

Transcription

1 = Jacob eroulli eroulli Nuber Idetities via Euler-Maclauri Suatio Hieu D. Nguye Math Dept Colloquiu Septeber 4, 8

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3 Sus of Powers () 5,5 ()() 6 333,833,5 3...? ( ) 3... ( ) (Pythagoreas) ( )( )... ( 3 ) (Archiedes)

4 eroulli Challege (October 6) Prize Proble: Fid the su of the th powers of the first atural ubers, i.e Rules: Studets oly. No techology allowed. Hit: The aswer is a 3-digit uber. (Jacob eroulli did it i less tha half of a quarter of a hour ) 4

5 "With the help of this table it too e less tha half of a quarter of a hour to fid that the teth powers of the first ubers beig added together will yield the su Fro this it will becoe clear how useless was the wor of Isael ullialdus spet o the copilatio of his voluious Arithetica Ifiitoru i which he did othig ore tha copute with iese labor the sus of the first six powers, which is oly a part of what we have accoplished i the space of a sigle page." Jacob eroulli, Ars Cojectadi (73) 5

6 Sus of Powers Forula p! p! p!!( p )!! p!!( p )! p p p p eroulli ubers: p! 3 3!( p )! p...,,, 3, 4, 5, !!!!3!!!!! p p p p ( 3 3 ) 6 6

7 Geeratig fuctio Recursive forula eroulli Polyoials ( x), ( x) x, 3 3 ( x) x x, 3( x) x x x, 6... te t e xt t ( x)! ( x) x!( )!! The eroulli ubers are defied by () : t t e t t t t... 7

8 Soe Properties of eroulli Nubers ad Polyoials. ( ) ( ) ( ) ( )! 6. ( x) x!!( )! 5 5!!3! ( x) ( ) ( x) ( () ( ) ()) ( x) x x x( x ) 6 6 ( x) ( x)( x) x( x ) ( x) 6 6 r ( ) r r r 8

9 eroulli Nuber Idetities Euler coied the ter eroulli ubers i his textboo E, Istitutioes calculi differetialis, Part II, Chapter 5, (755)

10 Quadratic recurrece: ( ) ( ) ( ) :

11 Euler s proof (trigooetric idetity): ( ) ( ) s cot u u u ( )! u ( )! ds ( ) ( ) csc du 4 u ( )! u u ds 4 4ss du (cot u csc u) 4 ( ) ( ) 4 ( ) u u u u ( )! ( )! ( ) ( ) ( ) u u ( )! ( )!(( ))! 4 ( ) u ( ) ( ) ( ) u ( )! ( )!(( ))! 4 ( ) u

12 ( ) ( ) ( ) ( )! ( )!(( ))! ( ) ( ) Power Rule: ( ) ( ) ( ) Alterate Proof (Euler-Maclauri Suatio) p ( x) x! p '( x) x p ( x) ( )!! p ''( ) ( ) x x p ( x) ( )! 3! p '''( ) ( )( ) x x p ( 3)! ( )!! p ( x) x p ( x) ( )! ( )! 3 ( x)

13 Appell Sequeces 3 3 ( x), ( x) x, ( x) x x, 3 ( x) x x x 6 '( x) ( x) '( x) x ( x / ) ( x) x x x 3 '( ) 3 3 / 3( x x / 6) 3 ( x) ( x) ( x) ( x) ( x) 3.5 3

14 eroulli polyoials as Appell sequece: (i) ( x) (ii) '( x) ( x) ( x) ( x) (iii) ( x) if if Exaple : ( x) ( x) ( x) ( x C) x C x x Cx C C 4

15 Euler-Maclauri Suatio (EMS) q ( ) ( ) f ( ) f ( x) [ f ( ) f ()] f ( ) f () ( )! f (q )! q (q ) ( ), [, ] Exaple : f ( x) x f ( ) q x f ( x) ( ) ( ) q (q ) f ( ) f () f ( ) ( )! (q )! [ f ( ) f ()] EMS: ( ) 5

16 Repeated Itegratio by Parts f ( x) Special Case ( = ) u f ( x), dv ( x) du f '( x), v ( x) ( x) (4) 4( ) ( ) ( x) f '( x) [ () f () () f ()] ( x) f ''( x)!! 3 ( x) f ''( x) [ f () f ()] [ () f '() () f '()]! 3! 3 ( x) f '''( x) 3! [ f () f ()] [ f '() f '()] [ 3 () f ''() 3 () f ''()]! 3! 4 ( x) f '''( x) 4! 4!... ( x) f ( x) [ ( x) f ( x)] ( x) f '( x) x f x 6

17 f f f f f f! 3! 3 [ () ()] [ '() '()] [ ''() ''()] [ (4) 5( x) f ( x) 4() '''() 4() '''()] 5( ) (5) ( ) f f x f x 4! 5! 5!... Assue f a polyoial of degree p: f x f f! p ( ) ( ) ( ) [ () ( ) ()] f x f f f f! p ( ) ( ) ( ) [ () ()] [ () ()] q ( ) ( ) f ( ) f ( x) [ f ( ) f ()] f ( ) f () ( )! f (q )! q (q ) ( ), [, ] 7

18 Exaple 3: f ( x) f ( x) x x ( > ) [ f() f()] [ ] ( )! ( )! f () f () x x ( )! ( )! ( ) ( ) EMS: ( )!! ( )! x x if (or ) ( )! otherwise ( )! f x f f f f! p ( ) ( ) ( ) [ () ()] [ () ()]!!( )! (Recall forula: ( x) x ) 8

19 Exaple 3: f ( x) ( x) f ( x) ( x) ( > ) if if [ f () f ()] [ () ()] if eve if odd Derivatives of eroulli polyoials! '( x) ( x) ( x) ( )! ( )! ( x) ( x) ( )! 9

20 f () f () () () ( ) ( ) ( ) ( )! [ ( ) () ( ) ()] ( ( ))!! ( )! if odd if eve For > :! if (or ) otherwise! if otherwise f f! ( ) ( ) [ () ()] (!)!

21 EMS: f x f f f f! p ( ) ( ) ( ) [ () ()] [ () ()] (Trivial idetity!) =

22 Exaple 4: f( x) ( x) ( x) ( > ) f ( x) ( x) ( x) ( x) ( x) ( x) [ f () f ()] [ ()] ( d f ) ( x) [( x) ( x)]?

23 Leibiz s Product Rule: f ( x) g( x) h( x) f '( x) g '( x) h( x) g( x) h'( x) f ''( x) [ g ''( x) h'( x) g '( x) h'( x)] [ g '( x) h'( x) g( x) h''( x)] g ''( x) h'( x) g '( x) h'( x) g( x) h''( x) f '''( x) g '''( x) h( x) 3 g ''( x) h'( x) 3 g '( x) h ''( x) g( x) h '''( x) ( ) ( ) ( ) ( ) '( )... ( ) ( ) ( ) ( ) ( ) ( ) f x g x h x g x h x g x h x g ( ) x h ( ) x ( ) ( ) 3

24 f ( x) ( x) ( x) d f ( x) [ x] ( x) ( ) ( ) f ( ( ) ( ) x) ( x) ( x) (!! x) ( x) ( x) ( )! ( )! () f () ( ) ( )!! ( ) () () ( )! ( )!! ( ) () ( )!!! ( ) ( )! ( )! where if if 4

25 f ( ) ( ) () f ()!!! [ ( ) ] ( )! ( )! ( )! EMS: ( ) f x f f f f! p ( ) ( ) ( ) [ () ()] [ () ()] ( ) ( ) 5

26 ( ) ( ) ( ) ( ) ( ) Assue : ( ) ( ) ( ) (recover Euler s forula) 6

27 Hypergeoetric eroulli Polyoials (N = ) (, x), (, x) x, 3 3 (, x) x x, 3(, x) x x x Appell sequece with zero first oet: (i) (, x) (ii) '(, x) (, x) (iii) ( x) (, x) / if if We defie hypergeoetric eroulli ubers by () (,) (), (), (), 3 ()

28 Exaple 5: f ( x) (, x) ( > ) f ( x) (, x) () / if () if Proof: ( x) (, x) / if if ( x) (, x) (,) (, x) (, x) / if if / if if [ f () f ()] [ (,) (,)] () () ( > ) 8

29 Proof: [ (,) (,)] [ (,) (,)] (,) x (, ) (,) () () ( ) ( ) ( ) ( ) f () f () (,) (,)! (,) (,) ( )!! ( ) (, ) ( )! x! (,) ( )!!! () ( )! ( )! 9

30 !! () () ()!! ( )! ( )! ( ) ( ) f f!! ( )!!( )! () EMS: () () () () () f x f f f f! p ( ) ( ) ( ) [ () ()] [ () ()] () () () () 3

31 Theore: () () Efficiet forula for (): () () () if eve j j j j () if odd j j j j Iterestig Proble: Fid a forula for () i ters of oly eroulli ubers. 3

32 Other choices for f(x): Further Exploratio f ( x) ( x) ( x) ( x) (Cubic recurrece forula?) x x f ( x) E ( x) (Euler polyoials) f ( x) H ( x) ( ) e x x / / d ( e ) (Herite polyoials) 3

33 Geeralized Euler-Maclauri Suatio (GEMS) Appell Sequece: { A ( x)} (i) A ( x) (ii) A '(, x) A (, x) Repeated Itegratio by Parts: b a b f ( x) A ( x) f ( x) a b [ A ( x) f ( x)] A ( x) f '( x) b a a [ A ( b) f ( b) A ( a) f ( a)]! b a A ( x) f ''( x) A ( x) f '( x)! b a 33

34 b a f ( x) [ A ( b) f ( b) A ( a) f ( a)] [ A ( b) f '( b) A ( a) f '( a)]! A3 ( x) f ''( x) 3! 3! b a b a A ( x) f '''( x) 3 GEMS: b a ( ) f x A b f b A a f a p ( ) ( ) ( ) [ ( ) ( ) ( ) ( )]! ( ) p! b a A x f x ( p) p ( ) ( ) Choices for A (x): A ( x) (, x) A( x) E ( x) A( x) H( x) 34

35 Refereces [] L. Euler, [E] Istitutioes calculi differetialis (73), Part II, Chapter 5, Traslatio by David Pegelley, available at The Euler Archive: [] A. Hasse ad H. Nguye, Hypergeoetric eroulli Polyoials ad Appell Sequeces, to appear i Iter. J. Nuber Theory. 35