Unit 42 A Math Fleet on the Pascal s Triangle and Determinant Patterns of Reciprocals of Factorials Sea

Transcription

1 Unit A Math Fleet on the Pascal s Triangle and Determinant Patterns of Reciprocals of Factorials Sea Heavier-than-air flying machines are impossible. Lord Kelvin, president of Royal Society (89) Pascal s triangle (PT) and determinant patterns of reciprocals of factorials (DPRF) are D mathematical objects that cast a long shadow, see [- 8]. Here is shown the connection of PT with structured sums of powers of integers and with Pi squared. Continue to exhaust the megapotential of PT: ** PT and sums of powers of integers with ± oscillating Patterns, f.e. m + m - m - m + m +6 m -7 m -8 m +9 m + m - m - m ± ** PT and determinant patterns of reciprocals of factorials ** Find relations of PT with e, other mathematical constants,,, etc. ** PT and integer sequences, e.g. Bernoulli and Euler numbers ** PT and Taylor series of elementary functions ** PT and continued fractions ** PT and the sequence of primes ** PT and the ternary tree of Pythagorean triples

2 References D.E. Knuth. The Art of Computer Programming. Vol. Addison-Wesley 997, ISBN p S.Wolfram. A New Kind of Science. Wolfram Media, ISBN

3 Math Ship One Pascal s triangle and the sums of powers of integers n i= i m=6 (-n) (-n) (-n) (-n) (-n) 6 (-n) (-n) (-) (-) (-) (-) (-) 6 (-) (-) 7 (6+) by (6+) determinants

4 n i= (-) i n-i m=6 n mod n n n n n n mod (6+) by (6+) determinants

5 Math Ship Two Pascal s triangle, the sums of powers of integers, and Pi squared n i= i m=6 (-n) (-n) (-n) (-n) (-n) 6 (-n) (-n) (-) (-) (-) (-) (-) 6 (-) (-) 7 (6+) by (6+) determinants

6 p p D(k = 7, n)= p 6 n i= n i= n (-) i n-i k (-) i n-i i= i= p p 6 p D(k, n) = D(k,) 7 7 by 7 determinant n n-i (-) i = i = p ; p = n(n+) k + D(k, ) lim k = D(k +, )

7 Recurrence relation k k i k D(k, n) = ( p) + ( ) D(k i, n) i= k i k k ( ) D(k i, n)= ( p) i= k i D(, n)=, D(, n)= i k Relation between PT D(k, ) and determinant patterns of reciprocals of factorials D(k = 7, )= by 7 determinant

8 eb( k = 7) = !! 6! 8!!!!!! 6! 8!!!!!!! 7! 9!!!!!! 7! 9!!!!! 7! 9!!!! 7!!!!!!

9 be(k = 7)= 77 7! 8!! 9 87! 6!!!!!! 7! 9!!!!!! 6! 8!!!!!! 6! 8!!!!! 6! 8!!!! 6!!!!!!

10 7 7 7! 8 8!! ! 6!!!!! 7! 9!!!!!!! 7! 9!!!!!! 7! 9!! b(k = 7)=!!! 7! 9!!!! 7!!!!!!

11 k = 9 m= = k ( ) ( ) ( ) ( ) ( )

12 Relation between D(k,) and patterns of determinants of reciprocals of factorials D(k, ) eb(k)= (k)! be(k)= eb(k)= b(k)= k k k k D(k, ) eb(k)= k k (k)! k ( ) (k ) = D(k, ) k (k)= D(k, ) ( ) D(k, ) (k)! k B k = ( ) Bernoulli number [9] k

13 Aus Symmetriegruenden fuehren wir noch e(k) ein. e(k = 7) = !! 6! 8!!!!!! 6! 8!!!!!!! 6! 8!!!!!! 6! 8!!!!! 6! 8!!!! 6!!!!!!

14 e(k = 7) = !! 6! 8!!!! 7 9!! 6! 8!!!! 7 9!!! 6! 8!!! 7 9!!! 6! 8!! 7!!! 6! 8!!!! 6!!!!!!

15 Relation between e(k) and Euler numbers [] k E = ( ) (k)! e(k) k # e( k = 7)= !! 6! 8!!!! 7 9!! 6! 8!!!! 7 9!!! 6! 8!!! 7 9!!! 6! 8!! 7!!! 6! 8!!!! 6!!!!!! e # (k)=(k+)be(k+) =(-) k- (k)!(k+)

16 How about ## e (k = 7)= 7!! 6! 8!!!! 7 9!! 6! 8!!!! 7 9!!! 6! 8!!! 7 9!!! 6! 8!! 7!!! 6! 8!!!! 6!!!!!!???

17 ## e (k = 7, a)= a a (6 a)!! 7 9 a a a a a a a!! 6! 8!!!! 7 9 a a a a a a!!! 6! 8!!! 7 9 a a a a a!!! 6! 8!! 7 a a a a!!! 6! 8! a a a!!! 6! a a!!! a!!

18 ** e (k,a) lim k ** e (k+,a) a A. 8.. = = A

19 be(k) lim[f(k) = ] = k e(k) Convergence behavior of f(k) be(k) k f(k)= e(k)

20 = ,,, Difference quotient f(k) f(k +) df = limdf(k)= = 9 k f(k +) f(k + ) Linear convergence acceleration f(k) = [(df)f(k +) f(k)] df f(k)= [(df) f(k + ) (df )[(df) ] df(df +)f(k +)+ f(k)] k f(k)

21 More k f(k)

22 e(k) be(k) lim = lim = k e(k +) k be(k +) eb(k) b(k) lim = lim = k eb(k +) k b(k +) Convergence behavior of e(k+)/e(k) and be(k+)/be(k) (/) k f(k)=e(k+)/e(k) g(k)= be(k+)/be(k)

23 Difference quotients f(k) f(k +) df = lim[df(k)= ] = 9 k f(k +) f(k + ) g(k) g(k +) dg= lim[dg(k)= ] = 9 k g(k +) g(k + ) f(k) f(k +) dfg= lim[dfg(k)= ] = k g(k +) g(k) Huygens convergence acceleration z(k) = [(dfg)f(k)+ g(k)] dfg+ e.g. z() = (, , ) =,

24 Math Ship Three Pascal s triangle, the sums of powers of integers, and Pi squared n i= i m=6 (-n) (-n) (-n) (-n) (-n) 6 (-n) (-n) (-) (-) (-) (-) (-) 6 (-) (-) 7 (6+) by (6+) determinants

25 n p p (k = 7, n)= p i= n n i i= k+ i = p 6 6 p 6 p by 7 determinant n p i = i = ; p = n(n+) i= i= (k, n) (k,) k + (k, ) lim k = (k +, )

26 6 Math Ship Four Pascal s triangle and Pi squared Pasca l's triangle 6 d r(k = 7)=

27 Recurrence relation k i k + d r(k)= ( ) d r(k i) i= k + i k i k + ( ) d r(k i)= i= k + i d r()=, d r()= k + d r(k) lim k = d r(k +) Convergence behavior

28 k + d r(k) k f(k) = d r(k +) =

29 Difference quotient f(k) f(k +) df = limdf(k)= = 9 k f(k +) f(k + ) Linear convergence acceleration df f(k) = f(k +) f(k) df df e.g. 9 f(6) = f(7) f(6) = = Tripel accelaration f(k) f(k +) f(k +) f(k + ) k f(k)= f(k) f(k +)+ f(k + )

30 Exercise Integration of 6 d r(k = 7)=

31 & Recurrence relation k i k + d r(k)= ( ) d r(k i) i= k + i k i k + ( ) d r(k i)= i= k + i d r()=, d r()= & k + d r(k) lim k = d r(k +) & Difference quotient f(k) f(k +) df = limdf(k)= = 9 k f(k +) f(k + ) & Linear convergence acceleration df f(k) = f(k +) f(k) df df

32 Factorization of dr(k) dr()= dr()= dr()=7 dr()==* dr()=7=*69 dr(6)=87=7**7 dr(7)=9969=7*67 dr(8)=8869= *7*867 dr(9)=969=***8*67 dr()=987 = *89**9*68 dr()=98 = *7*** 9797 dr()= = *7*89*679 dr()= = 7***7*99*69 dr()= = ****7*988 dr()= = 7*7*68*67*697 dr(6)= = 7***7*99*69 dr(7)=

33 = *7*9*67777 dr(8)= =9*776*87*9966 dr(9)= = *7**668*76699* dr()= = *7*7**97*68** 68679*97* dr()= = **7***87*6** 887*78799*

34 d b(k = 7)= Factorization of db(k) db()= db()== db()=6= * db()=6= ** db()=68= **69 db(6)=76= 6 * * *7 db(7)=7= 6 * *7*67 db(8)=7988= 7 * * *867 db(9)= = 7 * **7 *8*67 db()= = 9 * * *7* **9 db()=7898 = 9 * * ***9797 db()= = * 7 * *7 ** *679

35 db()= = * * *7 ***99*69 db()= = * * *7**7*988 db()= = * 6 * *7 * **7*68*697 db(6)= = * 7 * *7 ***7 *68697 db(7)= = * 7 * *7**7* db(8)= = 7 8 * *7 ** *7*9 * 9966 db(9)= = 7 * 8 * *7 **7*9* 76699* db()= = 8 * 8 * *7 * **

36 7*9* db()= = 8 * * *7 * **7* 9*9*889*9799*79887 db()= = * * 9 *7 6 * * *7 *9 *9**7* **7*68679* db()= = * * *7 6 * * *7 *9* *9*7* **87*78799*

37 db()= = * * 9 *7 6 * * *7 *9 * *9* *7***87*8* db(6)= = * * *7 6 * * *7 *9 * *9** 7***7*67*99*9* * db(7)=

38 = * * 9 *7 7 * * *7*9 * *9** 7***7*7879* 9*7966* * db(8)= = 6 * * *7 * * *7 *9 * *9** 7***7*87**67* * db(6)= =

39 6 * 8 * *7 9 * * *7 *9 * *9 * * 7* **7**9*6 *86 67* db()= = 96 * 8 * *7 6 * 8 * 7 *7 *9 * *9 * *7 ** *7 **9*6*67 *7* 7*79*8*89*97*

40 db(9)= = 87 * * *7 * 8 * 8 *7 *9 * *9 * *7 * * *7**9*6 *67*7* 7*79*8*89* * * * The prime factorization structures of dr(k) and db(k) are significantly different!

41 db(k) contains many small primaries (small noise ), more precisely the mostly complete sequence of primaries from to about k, the order of the prime factor is about k, of about int(k/), of about int(k/) and so on to (the larger k, the smaller is the about ), and a small number of increasingly large primes ( factor bombs ), comparable with monster waves in special simple continued fractions (see Unit ). Recurrence relation????????????????? k + 6 d b(k) lim k = d b(k +) The quotient of two immediately consecutive row sums of the determinants d r(k) and column sums d b(k) converges to = (+ ) for k to.

42 Math Ship Five Modified Pascal s triangle and Pi squared Modified Pascal's triangle 9 d r(k = 7)=

43 Factorization of dr(k) dr(6)= = 6 *69*7*68697 dr(7)= = 6 *7*9*67777 dr(8)= = 8 *776*87*9966 dr(9)= = 7 *7**668*76699* d b(k = 7)=

44 Factorization of db(k) db(6)= = 8 * *7 * **7 *9**9** db(7)= = 8 * *7 * *7**9** The prime factorization structures of dr(k) and db(k) are significantly different! db(k) contains many small primaries (small noise ) and a small number of large primes ( factor bombs ), comparable with monster waves in special simple continued fractions (see Unit ). k + d r(k) lim k = d r(k +) k + d b(k) lim k = d b(k +)

45 The quotient of two immediately consecutive row sums of the determinants d r(k) and column sums d b(k) converges to = (+ ) for k to. 7 Math Ship Six Modified Pascal s triangle and Pi squared Modified Pascal's triangle

46 d r(k = 7)= d b(k = 7)= k + d r(k) lim k = d r(k +) k + 6 d b(k) lim k = d b(k +)

47 The quotient of two immediately consecutive row sums of the determinants d r(k) and column sums d b(k) converges to = (+ ) for k to. Math Ship Seven Modified Pascal s triangle and Pi squared Modified Pascal's triangle

48 d r(k = 7)= d b(k = 7)= Factorization of db(k) db()= db()=6=* db()==**7 db()=7= ***

49 db()=876= * **69 db(6)=7= * **7**7 db(7)=68776= * **7*7*67 k + d r(k) lim k = d r(k +) 8 k + d b(k) lim k = d b(k +) The quotient of two immediately consecutive row sums of the determinants d r(k) and column sums d b(k) converges to = (+ ) for k to.

50 Math Ship Eight Modified Pascal s triangle, the sum of powers of integers, and Pi squared Modified Pascal's triangle

51 ?(m= 6, n) (-n) (-n) (-n) (-n) 9 7 (-n) (-n) (-n) (-) (-) (-) (-) 9 7 (-) (-) (-) (6+) by (6+) determinants

52 p p 7 (k = 7, n)= p 9 n i= n i= n i= i i k i = = (k, n) (k,) p p 6 p by 7 determinant p(n+) ; p = n(n+) 6 k + (k, ) lim k = (k +, )

53 p p G(k = 7, n)= p 9 p 6 6 p 6 p 8 6? (k, n) =?? (k,) 7 by 7 determinant k + G(k, ) lim k = G(k +, )

54 Math Ship with error Pascal s triangle and Pi squared Pasca l's triangle 6 9

55 Pascal's triangle with error

56 6 d r(k = 7)= d b(k = 7)= k + d r(k) lim k = d r(k +) k + 6 d b(k) lim k = d b(k +)

57 The quotient of two immediately consecutive row sums of the determinants d r(k) and column sums d b(k) converges to = (+ ) for k to. Math Ship with error Pascal s triangle and Pi squared Pasca l's triangle 6 9

58 Pascal's triangle with error 6 d r(k = 7)=

59 d b(k = 7)= k + d r(k) lim k = d r(k +) k + 6 d b(k) lim k = d b(k +) d b(k) converges to The quotient of two immediately consecutive row sums of the determinants d r(k) and column sums = (+ ) for k to.

60 Math Ship Eighteen Pascal s triangle, the sums of powers of integers, Pi squared, and more n i= i 6 (-n) (-n) (-n) (-n) (-n) 6 (-n) (-n) (-) (-) (-) (-) (-) 6 (-) (-) 7 (6+) by (6+) determinant

61 n mod n n n n n n 6 (-) n-i i m= 6 = n i= mod (6+) by (6+) determinant = (nmod) (m) d(n m) (m) d(n= m)

62 + + (m= 6) = = by 6 determinants

63 n n n + n n (m) d(n=, m)= ( ) 6 by 6 determinant Reduction of (m= k) (m= k)= m

64 Reduction of (k ) (k )=!! 9! 8! 6!!!!!! 8! 6!!! k ( ) (k )!!!!! 7!!! 6!!!!! k by k determinant

65 k (k ) Recurrence relation of (k ) k + i (i )+ (k )= i ( ) k i i i= Connection between (m= k ) and the Bernoulli numbers B k k k ( ) (k )= B k k

66 k + (k ) lim k = (k +) 8 Reduction of d(n,m=k) d(n, m= k / k = 7)= n n n n n 6 n 7 n 8 n 9 n n n n n n by determinant

67 p p p k k = ( ) p 6 p 6 p 7 p 7 by 7 determinant P=n(n+) 7 n i= n Reduction of ( ) i i= n i m n i m d(n=, m) ( ) i = d(n, m) for m ( ) m= k n= j k + (k ) lim k = (k +) 8

68 tanx= () () () (7) 7 (9) 9 x + x + x + x + x +...!!! 7! 9! x x!!! 7! 9!!!! 6! 8!!!! 6! = x 7 x!!! 9 x!!!

69 (x) (x) (x)!! 6! 8!!!!! 7! 9!!!! 7! = 7 (x)!!! 9 (x)!!!

70 tanhx= () () () (7) 7 (9) 9 x + x + x + x + x +...!!! 7! 9! x x x 7 x 9 x!!! 7! 9!!!! 6! 8!!!! 6! =!!!!!!

71 tan x = () () (7) 6 (9) 8 () x + x + x + x + x +...!! 6! 8!! 7 9!! 6! 8!! 7 x!!! 6! 8! x!!! 6! = 6 x!!! 8 x x!!!

72 But!!! 7! 9! 7 x!!! 6! 8! x!!! 6! tanx = cosx x!!! 7 x 9 x!! 6 8!!! 7!! 7 = x+ x + x + x +...

73 x x!! 6! 8!!!!! 6! 8!!!! 6! = 6 cosx x!!! 8 x x!! 6 8 = + x + x + x + x +...! 6 8!!! 6! 8!

74 7 9!! 6! 8!! 7 x!!! 6! 8! x!!! 6! =!!! 6 cos x x 8 x x!! 6 8 = + x + x + x + x +...! !!! 6! 8!

75 coshx x x 6 x 8 x x!! 6! 8!!!!! 6! 8!!!! 6! =!!!!! 6 8 = x + x x + x +...! 6 8!!! 6! 8!

76 coshx = cos x!! 6! 8!!! 6! 8! x!! 6! x!! 6 x! 8 x

77 = x x 7!! 6! 8! 7!! 6! 8!!! 6!!!! 6 x 8 x cosx cosh x = cosix cosh ix x x 6 x 8 x!! 6! 8!!! 6! 8!!! 6!!!!

78 (coshx) cosh x x sinh x = (cos x) cos x xsin x = x x 7!! 6! 8! 7!! 6! 8!!! 6!!!! 6 x 8 x

79 (cosh) coshx xsinhx x coshx = (cosx) cosx xsinx x cos x x x 6 x 8 x 7!! 6! 8! 7!! 6! 8!!! 6!!!!

80 (cosh) coshx xsinhx x coshx = (cosx) cosx xsinx x sinx 7!! 6! 8! 7 x!! 6! 8! x!! 6! x!! 6 x! 8 x

81 (cosh) coshx xsinhx x sinhx = (cosx) cosx xsinx x sinx x x x 8 x 7!! 6! 8! 7!! 6! 8!!! 6! 6 x!!!

82 x sinx x +x!! 7! 9!!!!! 7! 9!!!! 7! = 6 x!!! 8 +x x!! 6 8 = + x + x + x + x +...!

83 Connection between (m= k ) and the Bernoulli numbers B k k k ( ) (k )= B k k (-) (-) (-) (-) (-) 6 = (7! ) (-) (-) 7 (6+) by (6+) determinant

84 mod (6+) by (6+) determinant = ( ) 6 In number theory, 69 is a "marker" (similar to the radioactive markers in biology): whenever it appears in a computation, one can be sure that Bernoulli numbers are involved.

85 Math Ship Hundred Thirty Seven The general sandbox ** PT die rote und die blaue Partei ** Determinant patterns of reciprocals of Factorials the Bernoulli and the Euler Wing DPRF Vermutung!! 6!!!!!! 7! 6!!!!! r(k) =!!!!!!!!!!!! k by k determinant

86 r(i+) = fuer i =,,,, r(k) sequence,,,,,,,,,, ,,,, ,,,, ,, Bi r(i)= (i)! r(i) lim = ( ) i r(i+ )

87 PT Vermutung s(k)= k by k determinant s(i) = fuer i =,,,, s(k) sequence <,, -,, 6,, -7,, 796,, -79,, 686,, -977,, ,, ,, > s(i-) = i ( i -)B i /i s(i-)/r(i) = i ( i -)(i-)!

88 The Bernoulli wing!! 6!!!!!! 7! 6!!!!! r(k)=!!!!!!!!!!!! k by k determinant

89 7 7 7! 8 8!! ! 6!!!!! b(k = 7)=!! 9! 7!!!!!!! 9! 7!!!! 7! 9!!!! 7! 9!!!! 7!!!!!!!

90 D(k = 7, )= by 7 determinant

91 r(i) = ( ) i i b(i) i ( ) = D(i, ) i ( )(i)! i i i s(i ) = ( ) D(i, )

92 !!!!! 6!!!!!!!!! 6!!!!!!!!!!!!!!!!!!

93 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ±x e = ± x + x ± x x ±... = ± x x ± x x ±... x x +x!!!!!!!! x!!! +x!!!

94 x!!!! x!! x! x x!

95 !!!!!!!!! 6! 6! 7!!!!!!!!!!!!!!!!!!! 6!!!!!! 7! 6!!!!!

96 Bessel functions of the first kind / i J (x) = J (x) ( ) x / I (x) = I (x) i= i!(i+) i+!!!!!! 7 x!! x x!!!! x !!!!!!!!!!!!!!!!

97 i ( ) x J (x) i!(i+)! i= i!i! i= i J (x) ( ) x 7 i+ i!!!!!!!!!! x!!!!!!!!!! x!!!!!!!! x!!!!!! x!!!! 9 x!!

98 x I (x) i= i!(i+)! i I (x) x i= i!i! x 7 i+!!!!!!!!!! x!!!!!!!!!! x!!!!!!!! x!!!!!! x!!!! 9!!

99 scf (einfache Kettenbruch) Darstellung fuer J () = J () [;,,,,,,,,,,6,,7,,8,,9,,,,,,,,,,,,,...] und I () I () [;,,,,,6,7,8,9,,,,,,,6, 7,8,9,,,,,,,6,7,8,...]

100 Zum Vergleich i ( ) i+ x sin x i= (i+)! tan x i cos x ( ) i x i!i! x x x 7 x i=!!! 7! 9!!!! 6! 8!!!! 6!!!! 9 x!!!

101 tanh x x x sinh x i+ i= cosh x i x 7 x i= 9 x x (i+)! x (i)!!!! 7! 9!!!! 6! 8!!!! 6!!!!!!! scf (einfache Kettenbruch) Darstellung fuer

102 tan = = [,,,,,,,7,,9,,,,,,,,7,, 9,,,,,,,,7,,9, ] tanh = = [,,,, 7, 9,,,, 7, 9,,,, 7, 9,,,, 7, 9,,,,7, 9,,,, 7, ]

103 i= i!i! i J (x) ( ) x 6 8 i x!!!!!!!!!! x!!!!!!!!!! x!!!!!!!! x!!!!!! x!!!! x!!

104 Zum Vergleich i cos x ( ) x (i)! x x x i= 6 x 8 x i!! 6! 8!!!!! 6! 8!!!! 6!!!! x!!!

105 J () = J () [;,,,,,,,,,,6,,7,,8,,9,,,,,,,,,,,,,...] und I () I () [;,,,,,6,7,8,9,,,,,,,6, 7,8,9,,,,,,,6,7,8,...] tan = =[;,,,,,,7,,9,,,,,,,,7,, 9,,,,,,,,7,,9, ] tanh = =[;,,, 7, 9,,,, 7, 9,,,, 7, 9,,,, 7, 9,,,,7, 9,,,, 7, ]

106 Und dazu J () = J () [;,,,,,,6,,8,,,...] und I () I () [;,,6,8,,...] tan. = = [;,,,,8,,,,6,,,, ] tanh. = = [;,6,,,8,,6,, ]