How Bernoulli Did It:

Transcription

1 How Beroulli Did It: Sums of Powers, Geeratig Fuctios, ad Beroulli Numbers Hieu D. Nguye (Joit wor with Abdul Hasse) Rowa Uiversity Rowa Math Semiar October, 26 MAA Colum: Ed Sadifer s How Euler Did it:

2 Beroulli Challege Prize Problem: Fid the sum of the th powers of the first atural umbers, i.e Rules: Studets oly. No techology allowed. Hit: The aswer is a 32-digit umber. (Beroulli did it i less tha half of a quarter of a hour ) 2

3 Jacob (Jacques) Beroulli The MacTutor History of Math Archive 3

4 The MacTutor History of Math Archive Leohard Euler

5 Sums of Powers = 5() = =? Joha Faulhaber, Academia Algebrae (63) ( + ) = = = 2 N (Pythagoreas) ( + )(2+ ) = = = M (Archimedes) ( + ) 2 = = = 2 N (Nichomacus) = = M N (Haytham) 6 5

6 = = (28N 672N + 222N Faulhaber s Formulas 4688N N 7422N N 85 N ) / 45 = M( b + b N b N ) (eve powers) 2r r 2 r = N ( c + c N c N ) (odd powers greater tha ) 2r + 2 r 2 r 6

7 Jacob Beroulli, Ars Cojectadi (73) 2 = = = = = = = = = = = =? p p p p + = = = = =

8 p = :, 2 2 p = 2:,, p = 3:,, p = 4:,, = = = p = 5:,, = ,,, 3 4 = = ,, = =, p p( p )( p 2) p( p )( p 2)( p 3)( p 4),,, A, B, C,... p A=, B=, C =,

9 Beroulli s formula p p( p )( p 2) = + + A + B p p p+ p p p 3 p( p )( p 2)( p 3)( p 4) p 3 + C + (Origial) Beroulli umbers Moder formula A=, B=, C =, p p! p+ p! p p! p p! p 2 = B + B + B2 + B3 +...!( p+ )!! p! 2!( p )! 3!( p 2)! p = ( ) δ p p! B!( p+ )! p+ B =, B =, B2 =, B3 =, B4 =, B5 =,

10 Figurate Numbers "It appears to me that if oe wats to mae progress i mathematics oe should study the masters, ad ot the pupils. --- Niels Abel (82-829) Passage from Ars Cojectadi (p. 95) We will observe i passig that, may [scholars] egaged i the cotemplatio of figurate umbers (amog them Faulhaber ad Remmeli of Ulm, Wallis, Mercator, i his Logarithmotechia ad others) but I do ot ow of oe who gave a geeral ad scietific proof of this property. Wallis i his Arithmetica Ifiitorum ivestigated by meas of iductio the ratios that the series of squares, cubes, ad other powers of atural umbers have to the series of terms each equal to the greatest term. This he put i the foudatio of his method.

11 His ext step was to establish 76 properties of trigoal, pyramidal, ad other figurate umbers, but it would have bee better ad more fittig to the ature of the subject if the process would have bee reversed ad he would have first give a discussio of figurate umbers, demostrated i a geeral ad accurate way, ad oly the have proceeded with the ivestigatio of the sums of powers. Eve disregardig the fact that the method of iductio is ot sufficietly scietific ad, moreover, requires special wor for every ew series; it is a method of commo judgmet that the simpler ad more primitive thigs should precede others. Such are the figurate umbers as related to the powers, sice they are formed by additio, while the others are formed by multiplicatio; Traslated from the Lati by Professor Jeuthiel Gisburg, Yeshiva College, New Yor City.

12 ( = ) Beroulli s Triagle (Pascal s) of Figurate Numbers (Biomial Coefficiets) A B C D E... Figurate Idetities A+ A A = B B+ B B = C C+ C C = D... A B C... D =, =, ( )( 2) =, 2 ( )( 2)( 3) =, 23 2

13 Beroulli s proof of his formula based o figurate idetities: Case p = : Assume the figurate idetity 2 = = B B2... B C = holds: The 2 ( ) ( ) = = = ( + ) = ( ) = + = Note: Observe that Beroulli s proof resembles the moder oe usig the priciple of mathematical iductio eve though he himself viewed iductio as ot sufficietly scietific. 3

14 Case p = 2: 3 2 ( )( 2) ( )( 2) = = The sice ( )( 2) 3 = = = Assume the figurate idetity C + C C = D holds: ad , = = 4

15 it follows that = = Hece, 3 2 Case p = 3: 2 = = = Assume the figurate idetity D + D D = E holds: 5

16 ( )( 2)( 3) ( )( 2)( 3) = = 6 The sice ad ( )( 2)( 3) = = 2 +, = + +, = 6

17 it follows that = Hece, = =

18 Proof for arbitrary p? Ufortuately Beroulli does ot provide oe, but the recipe for a iductive proof is evidet i his demostratio for p =, 2, 3. Beroulli s formula for p = : ( ) =

19 Leoard Euler (755) e t Geeratig Fuctios t = t t t t t t t t = ! 6 2! 3 4! 42 6! t t t t = B + B + B + B + B +! 2! 4! 6! Beroulli umbers e t t t = B! = B =, B =, B2 =, B3 =, B4 =, B5 =,

20 Equate series coefficiets of t: m t t t t t = ( e ) B = B! m= m!! m+ t B + = B = t ( m ) m m!! + = + = = =!( + )! Recursive formula: = B if > =!( + )! if = = : B = B = : + B = B = 2 2 B B B = 2: + + = B2 = 2

21 Beroulli Polyomials xt te t = B ( x) t e! = Equate series coefficiets of t: xt t t te = ( e ) B ( x)! m= m= m+ ( xt) t t = B ( x)! m!! + xt B ( x) + = t ( m )! + = +!( + )! = = 2

22 Recursive formula = : B ( x) = = B ( x) x =!( + )!! B ( x) = : + B( x) = x B( x) = x ( ) ( ) 2( ) 2 2 B x B x B x x = 2: + + = B ( x) = x x = = : B3 ( x) x x x Observe that B () = B 22

23 Iterval [-2,2] Iterval [,] B ( x) B ( x) B ( x) 2 B ( x) 3 23

24 Some Properties of Beroulli Numbers ad Polyomials ( ) (2 π ) ζ (2 ) = = 2 2(2 )! B B ( x) = ( ) B ( x) 2 if = B = if > r = ( ) r + r= r B p = Bp Bp p B( x) = B x ( ) () What about derivative ad itegral properties? 24

25 Derivative property Appell Sequeces B'( x) = d ( x / 2) = = B( x) dx d 2 B2'( x) = ( x x+ ) = 2x = 2 x = 2 B( x) dx 6 2 d B3 '( x) = ( x x + x) = 3x 3x+ = 3 x x+ = 3 B2 ( x) dx Itegral property: 2 B ( x) dx= x dx x x = = B2 ( x) dx= x x + x dx= B 2( x) dx = x x + dx = 25

26 Theorem: (Appell?) The followig two defiitios of Beroulli polyomials are equivalet: I. Geeratig fuctio: xt te t = B ( x) t e! = II. Appell sequece with zero mea: (i) B ( x ) = (ii) B + '( x) = ( + ) B ( x) (iii) B ( x) dx if = = if > 26

27 Explicit costructio of Beroulli polyomials as a Appell sequece: B '( x) = B ( x) B ( x) = B ( x) dx= dx= x+ C B ( x) dx = ( x + C ) dx = x + C x = 2 2 B ( x) = B x+ B B ( x) = B x + 2Bx+ B Geeral formula + C = C = = B B ( x) = B x + 3Bx + 3B x+ B ( ) 2 B( x) = Bx + Bx + B2x B 2 27

28 Proof of Theorem ( ) xt d te d t B ( x) dx e = dx t =! xt te ' t t B ( x) t = e =! t t =! ( )! + + ' B( x) B+ ( x) = = + B '( ) + x = B ( x) + B x dx B x dx B x dx! B B = =!!( )! +!( + )! ( ) = = if = = (Recursive formula) if > ( ) 28

29 Hypergeometric Beroulli Polyomials Recursive formula = 2 xt te /2 t = B (2, x) t e t! = B (2, ) x x m = =, (3) m =!(3)! 34 (3 + m ) m > = : B (2, x) = = : B (2, x) = x 3 2 = 2: (2, ) = B2 x x x = 3: (2, ) = B3 x x x x 29

30 We ca defie hypergeometric Beroulli umbers as 2 B (2) = B (2,) Iterval [-2,2] Iterval [,] B ( x) B ( x) B ( x) 2 B ( x) 3 3

31 Weighted meas: ( xb ) (2, x) ( ) (2, ) = ( )( /3) = ( + 4 /3 /3) xb xdx x x dx x x dx 3 2 = x /3+ 2 x /3 x/3 = 3

32 Theorem: The followig two defiitios of hypergeometric Beroulli polyomials are equivalet: I. Geeratig fuctio: II. Appell sequece with zero first momet: (i) B ( x ) = 2 xt te /2 t = B (2, x) t e t! (ii) B '( x) ( ) B ( x) + = + /2 if = (iii) ( xb ) ( ) xdx= if > = 32

33 Aswer to Prize Problem: = ( ) = = ( ) + + = ( ) = ( ) 66 = 9,49,924,24,424,243,424,24,924,242,5 33

34 "With the help of this table it too me less tha half of a quarter of a hour to fid that the teth powers of the first umbers beig added together will yield the sum From this it will become clear how useless was the wor of Ismael Bullialdus spet o the compilatio of his volumious Arithmetica Ifiitorum i which he did othig more tha compute with immese labor the sums of the first six powers, which is oly a part of what we have accomplished i the space of a sigle page." 34

35 Refereces [] Appell, P. E. "Sur ue classe de polyomes." Aales d'école Normal Superieur, Ser. 2 9, 9-44, 882. [2] K. Dilcher, Beroulli umbers ad cofluet hypergeometric fuctios, Number Theory for the Milleium, I (Urbaa, IL, 2), , A K Peters, Natic, MA, 22. [3] L. Euler, Istitutioes calculi differetialis, St. Petersburg, 755, reprited i Opera Omia Series I vol. Eglish traslatio of Part I by Joh Blato, Spriger, NY, 2. [4] A. Hasse ad H. D. Nguye, Hypergeometric Zeta Fuctios, Preprit, 25. Available at the Mathematics ArXiv: [5] F. T. Howard, Numbers Geerated by the Reciprocal of e^x--x, Math. Comput. 3 (977) No. 38, [6] D. J. Pegelley, The bridge betwee the cotiuous ad the discrete via origial sources, i Study the Masters: The Abel-Fauvel Coferece, Kristiasad, 22 (eds. Otto Bee et al), Natioal Ceter for Mathematics Educatio, Uiversity of Gotheburg, Swede. [7] E. Sadifer, How Euler Did It: Beroulli Numbers, September 25, MAA Colum 35