Algebraic and Analytic Methods - 1.12 Continued Fractions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
1.12#Ex1 A k = b k ⁒ A k - 1 + a k ⁒ A k - 2 subscript 𝐴 π‘˜ subscript 𝑏 π‘˜ subscript 𝐴 π‘˜ 1 subscript π‘Ž π‘˜ subscript 𝐴 π‘˜ 2 {\displaystyle{\displaystyle A_{k}=b_{k}A_{k-1}+a_{k}A_{k-2}}}
A_{k} = b_{k}A_{k-1}+a_{k}A_{k-2}		 

A[k] = b[k]*A[k - 1]+ a[k]*A[k - 2]
 
Subscript[A, k] == Subscript[b, k]*Subscript[A, k - 1]+ Subscript[a, k]*Subscript[A, k - 2]
 Skipped - no semantic math Skipped - no semantic math - -
1.12#Ex2 B k = b k ⁒ B k - 1 + a k ⁒ B k - 2 subscript 𝐡 π‘˜ subscript 𝑏 π‘˜ subscript 𝐡 π‘˜ 1 subscript π‘Ž π‘˜ subscript 𝐡 π‘˜ 2 {\displaystyle{\displaystyle B_{k}=b_{k}B_{k-1}+a_{k}B_{k-2}}}
B_{k} = b_{k}B_{k-1}+a_{k}B_{k-2}		 

B[k] = b[k]*B[k - 1]+ a[k]*B[k - 2]
 
Subscript[B, k] == Subscript[b, k]*Subscript[B, k - 1]+ Subscript[a, k]*Subscript[B, k - 2]
 Skipped - no semantic math Skipped - no semantic math - -
1.12#Ex3 A - 1 = 1 subscript 𝐴 1 1 {\displaystyle{\displaystyle A_{-1}=1}}
A_{-1} = 1		 

A[- 1] = 1
 
Subscript[A, - 1] == 1
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1.12#Ex4 A 0 = b 0 subscript 𝐴 0 subscript 𝑏 0 {\displaystyle{\displaystyle A_{0}=b_{0}}}
A_{0} = b_{0}		 

A[0] = b[0]
 
Subscript[A, 0] == Subscript[b, 0]
 Skipped - no semantic math Skipped - no semantic math - -
1.12#Ex5 B - 1 = 0 subscript 𝐡 1 0 {\displaystyle{\displaystyle B_{-1}=0}}
B_{-1} = 0		 

B[- 1] = 0
 
Subscript[B, - 1] == 0
 Skipped - no semantic math Skipped - no semantic math - -
1.12#Ex6 B 0 = 1 subscript 𝐡 0 1 {\displaystyle{\displaystyle B_{0}=1}}
B_{0} = 1		 

B[0] = 1
 
Subscript[B, 0] == 1
 Skipped - no semantic math Skipped - no semantic math - -
1.12.E7 A n ⁒ B n - 1 - B n ⁒ A n - 1 = ( - 1 ) n - 1 ⁒ ∏ k = 1 n a k subscript 𝐴 𝑛 subscript 𝐡 𝑛 1 subscript 𝐡 𝑛 subscript 𝐴 𝑛 1 superscript 1 𝑛 1 subscript superscript product 𝑛 π‘˜ 1 subscript π‘Ž π‘˜ {\displaystyle{\displaystyle A_{n}B_{n-1}-B_{n}A_{n-1}=(-1)^{n-1}\prod^{n}_{k=% 1}a_{k}}}
A_{n}B_{n-1}-B_{n}A_{n-1} = (-1)^{n-1}\prod^{n}_{k=1}a_{k}		 

A[n]*B[n - 1]- B[n]*A[n - 1] = (- 1)^(n - 1)* product(a[k], k = 1..n)
 
Subscript[A, n]*Subscript[B, n - 1]- Subscript[B, n]*Subscript[A, n - 1] == (- 1)^(n - 1)* Product[Subscript[a, k], {k, 1, n}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
1.12.E8 C n - C n - 1 = ( - 1 ) n - 1 ⁒ ∏ k = 1 n a k B n - 1 ⁒ B n subscript 𝐢 𝑛 subscript 𝐢 𝑛 1 superscript 1 𝑛 1 subscript superscript product 𝑛 π‘˜ 1 subscript π‘Ž π‘˜ subscript 𝐡 𝑛 1 subscript 𝐡 𝑛 {\displaystyle{\displaystyle C_{n}-C_{n-1}=\frac{(-1)^{n-1}\prod^{n}_{k=1}a_{k% }}{B_{n-1}B_{n}}}}
C_{n}-C_{n-1} = \frac{(-1)^{n-1}\prod^{n}_{k=1}a_{k}}{B_{n-1}B_{n}}		 

C[n]- C[n - 1] = ((- 1)^(n - 1)* product(a[k], k = 1..n))/(B[n - 1]*B[n])
 
Subscript[C, n]- Subscript[C, n - 1] == Divide[(- 1)^(n - 1)* Product[Subscript[a, k], {k, 1, n}, GenerateConditions->None],Subscript[B, n - 1]*Subscript[B, n]]
 Skipped - no semantic math Skipped - no semantic math - -
1.12.E10 a n = A n - 1 ⁒ B n - A n ⁒ B n - 1 A n - 1 ⁒ B n - 2 - A n - 2 ⁒ B n - 1 subscript π‘Ž 𝑛 subscript 𝐴 𝑛 1 subscript 𝐡 𝑛 subscript 𝐴 𝑛 subscript 𝐡 𝑛 1 subscript 𝐴 𝑛 1 subscript 𝐡 𝑛 2 subscript 𝐴 𝑛 2 subscript 𝐡 𝑛 1 {\displaystyle{\displaystyle a_{n}=\frac{A_{n-1}B_{n}-A_{n}B_{n-1}}{A_{n-1}B_{% n-2}-A_{n-2}B_{n-1}}}}
a_{n} = \frac{A_{n-1}B_{n}-A_{n}B_{n-1}}{A_{n-1}B_{n-2}-A_{n-2}B_{n-1}}		 

a[n] = (A[n - 1]*B[n]- A[n]*B[n - 1])/(A[n - 1]*B[n - 2]- A[n - 2]*B[n - 1])
 
Subscript[a, n] == Divide[Subscript[A, n - 1]*Subscript[B, n]- Subscript[A, n]*Subscript[B, n - 1],Subscript[A, n - 1]*Subscript[B, n - 2]- Subscript[A, n - 2]*Subscript[B, n - 1]]
 Skipped - no semantic math Skipped - no semantic math - -
1.12.E11 a n = B n B n - 2 ⁒ C n - 1 - C n C n - 1 - C n - 2 subscript π‘Ž 𝑛 subscript 𝐡 𝑛 subscript 𝐡 𝑛 2 subscript 𝐢 𝑛 1 subscript 𝐢 𝑛 subscript 𝐢 𝑛 1 subscript 𝐢 𝑛 2 {\displaystyle{\displaystyle a_{n}=\frac{B_{n}}{B_{n-2}}\frac{C_{n-1}-C_{n}}{C% _{n-1}-C_{n-2}}}}
a_{n} = \frac{B_{n}}{B_{n-2}}\frac{C_{n-1}-C_{n}}{C_{n-1}-C_{n-2}}		 

a[n] = (B[n])/(B[n - 2])*(C[n - 1]- C[n])/(C[n - 1]- C[n - 2])
 
Subscript[a, n] == Divide[Subscript[B, n],Subscript[B, n - 2]]*Divide[Subscript[C, n - 1]- Subscript[C, n],Subscript[C, n - 1]- Subscript[C, n - 2]]
 Skipped - no semantic math Skipped - no semantic math - -
1.12.E12 b n = A n ⁒ B n - 2 - A n - 2 ⁒ B n A n - 1 ⁒ B n - 2 - A n - 2 ⁒ B n - 1 subscript 𝑏 𝑛 subscript 𝐴 𝑛 subscript 𝐡 𝑛 2 subscript 𝐴 𝑛 2 subscript 𝐡 𝑛 subscript 𝐴 𝑛 1 subscript 𝐡 𝑛 2 subscript 𝐴 𝑛 2 subscript 𝐡 𝑛 1 {\displaystyle{\displaystyle b_{n}=\frac{A_{n}B_{n-2}-A_{n-2}B_{n}}{A_{n-1}B_{% n-2}-A_{n-2}B_{n-1}}}}
b_{n} = \frac{A_{n}B_{n-2}-A_{n-2}B_{n}}{A_{n-1}B_{n-2}-A_{n-2}B_{n-1}}		 

b[n] = (A[n]*B[n - 2]- A[n - 2]*B[n])/(A[n - 1]*B[n - 2]- A[n - 2]*B[n - 1])
 
Subscript[b, n] == Divide[Subscript[A, n]*Subscript[B, n - 2]- Subscript[A, n - 2]*Subscript[B, n],Subscript[A, n - 1]*Subscript[B, n - 2]- Subscript[A, n - 2]*Subscript[B, n - 1]]
 Skipped - no semantic math Skipped - no semantic math - -
1.12.E13 b n = B n B n - 1 ⁒ C n - C n - 2 C n - 1 - C n - 2 subscript 𝑏 𝑛 subscript 𝐡 𝑛 subscript 𝐡 𝑛 1 subscript 𝐢 𝑛 subscript 𝐢 𝑛 2 subscript 𝐢 𝑛 1 subscript 𝐢 𝑛 2 {\displaystyle{\displaystyle b_{n}=\frac{B_{n}}{B_{n-1}}\frac{C_{n}-C_{n-2}}{C% _{n-1}-C_{n-2}}}}
b_{n} = \frac{B_{n}}{B_{n-1}}\frac{C_{n}-C_{n-2}}{C_{n-1}-C_{n-2}}		 

b[n] = (B[n])/(B[n - 1])*(C[n]- C[n - 2])/(C[n - 1]- C[n - 2])
 
Subscript[b, n] == Divide[Subscript[B, n],Subscript[B, n - 1]]*Divide[Subscript[C, n]- Subscript[C, n - 2],Subscript[C, n - 1]- Subscript[C, n - 2]]
 Skipped - no semantic math Skipped - no semantic math - -
1.12#Ex7 b 0 = A 0 subscript 𝑏 0 subscript 𝐴 0 {\displaystyle{\displaystyle b_{0}=A_{0}}}
b_{0} = A_{0}		 

b[0] = A[0]
 
Subscript[b, 0] == Subscript[A, 0]
 Skipped - no semantic math Skipped - no semantic math - -
1.12#Ex8 b 1 = B 1 subscript 𝑏 1 subscript 𝐡 1 {\displaystyle{\displaystyle b_{1}=B_{1}}}
b_{1} = B_{1}		 

b[1] = B[1]
 
Subscript[b, 1] == Subscript[B, 1]
 Skipped - no semantic math Skipped - no semantic math - -
1.12#Ex9 a 1 = A 1 - A 0 ⁒ B 1 subscript π‘Ž 1 subscript 𝐴 1 subscript 𝐴 0 subscript 𝐡 1 {\displaystyle{\displaystyle a_{1}=A_{1}-A_{0}B_{1}}}
a_{1} = A_{1}-A_{0}B_{1}		 

a[1] = A[1]- A[0]*B[1]
 
Subscript[a, 1] == Subscript[A, 1]- Subscript[A, 0]*Subscript[B, 1]
 Skipped - no semantic math Skipped - no semantic math - -
1.12#Ex10 C n ⁒ ( w ) = A n + A n - 1 ⁒ w B n + B n - 1 ⁒ w subscript 𝐢 𝑛 𝑀 subscript 𝐴 𝑛 subscript 𝐴 𝑛 1 𝑀 subscript 𝐡 𝑛 subscript 𝐡 𝑛 1 𝑀 {\displaystyle{\displaystyle C_{n}(w)=\frac{A_{n}+A_{n-1}w}{B_{n}+B_{n-1}w}}}
C_{n}(w) = \frac{A_{n}+A_{n-1}w}{B_{n}+B_{n-1}w}		 

C[n](w) = (A[n]+ A[n - 1]*w)/(B[n]+ B[n - 1]*w)
 
Subscript[C, n][w] == Divide[Subscript[A, n]+ Subscript[A, n - 1]*w,Subscript[B, n]+ Subscript[B, n - 1]*w]
 Skipped - no semantic math Skipped - no semantic math - -
1.12#Ex13 C 0 β‰  ∞ subscript 𝐢 0 {\displaystyle{\displaystyle C_{0}\not=\infty}}
C_{0}\not = \infty		 

C[0] <> infinity
 
Subscript[C, 0] \[NotEqual]*Infinity
 Skipped - no semantic math Skipped - no semantic math - -
1.12#Ex14 C n β‰  C n - 1 subscript 𝐢 𝑛 subscript 𝐢 𝑛 1 {\displaystyle{\displaystyle C_{n}\not=C_{n-1}}}
C_{n}\not = C_{n-1}		 

C[n] <> C[n - 1]
 
Subscript[C, n] \[NotEqual]*Subscript[C, n - 1]
 Skipped - no semantic math Skipped - no semantic math - -
1.12.E25 | b n | β‰₯ | a n | + 1 subscript 𝑏 𝑛 subscript π‘Ž 𝑛 1 {\displaystyle{\displaystyle|b_{n}|\geq|a_{n}|+1}}
|b_{n}| \geq |a_{n}|+1		 

abs(b[n]) >= abs(a[n])+ 1
 
Abs[Subscript[b, n]] >= Abs[Subscript[a, n]]+ 1
 Skipped - no semantic math Skipped - no semantic math - -
1.12.E26 - 1 2 ⁒ Ο€ + Ξ΄ < ph ⁑ b n 1 2 πœ‹ 𝛿 phase subscript 𝑏 𝑛 {\displaystyle{\displaystyle-\tfrac{1}{2}\pi+\delta<\operatorname{ph}b_{n}}}
-\tfrac{1}{2}\pi+\delta < \phase@@{b_{n}}		 

-(1)/(2)*Pi + delta < argument(b[n])

-Divide[1,2]*Pi + \[Delta] < Arg[Subscript[b, n]]
Failure Failure
Failed [27 / 300]
Result: -.70796327e-1 < -1.047197551
Test Values: {delta = 1.5, b[n] = 1/2-1/2*I*3^(1/2), n = 1, n = 3}


Result: -.70796327e-1 < -1.047197551
Test Values: {delta = 1.5, b[n] = 1/2-1/2*I*3^(1/2), n = 2, n = 3}


Result: -.70796327e-1 < -1.047197551
Test Values: {delta = 1.5, b[n] = 1/2-1/2*I*3^(1/2), n = 3, n = 3}


Result: -.70796327e-1 < -2.617993878
Test Values: {delta = 1.5, b[n] = -1/2*3^(1/2)-1/2*I, n = 1, n = 3}

... skip entries to safe data
Failed [49 / 100]
Result: Less[Complex[-0.7047709230104579, 0.49999999999999994], 0.5235987755982988]
Test Values: {Rule[n, 3], Rule[Ξ΄, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Less[Complex[-0.7047709230104579, 0.49999999999999994], 2.0943951023931953]
Test Values: {Rule[n, 3], Rule[Ξ΄, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.12.E26 ph ⁑ b n < 1 2 ⁒ Ο€ - Ξ΄ phase subscript 𝑏 𝑛 1 2 πœ‹ 𝛿 {\displaystyle{\displaystyle\operatorname{ph}b_{n}<\tfrac{1}{2}\pi-\delta}}
\phase@@{b_{n}} < \tfrac{1}{2}\pi-\delta		 

argument(b[n]) < (1)/(2)*Pi - delta

Arg[Subscript[b, n]] < Divide[1,2]*Pi - \[Delta]
Failure Failure
Failed [72 / 300]
Result: 3.141592654 < 3.070796327
Test Values: {delta = -1.5, b[n] = -1.5, n = 1, n = 3}


Result: 3.141592654 < 3.070796327
Test Values: {delta = -1.5, b[n] = -1.5, n = 2, n = 3}


Result: 3.141592654 < 3.070796327
Test Values: {delta = -1.5, b[n] = -1.5, n = 3, n = 3}


Result: 3.141592654 < 3.070796327
Test Values: {delta = -1.5, b[n] = -.5, n = 1, n = 3}

... skip entries to safe data
Failed [64 / 100]
Result: Less[0.5235987755982988, Complex[0.7047709230104579, -0.49999999999999994]]
Test Values: {Rule[n, 3], Rule[Ξ΄, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Less[2.0943951023931953, Complex[0.7047709230104579, -0.49999999999999994]]
Test Values: {Rule[n, 3], Rule[Ξ΄, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.12.E27 - 1 2 ⁒ Ο€ + Ξ΄ < ph ⁑ C n 1 2 πœ‹ 𝛿 phase subscript 𝐢 𝑛 {\displaystyle{\displaystyle-\tfrac{1}{2}\pi+\delta<\operatorname{ph}C_{n}}}
-\tfrac{1}{2}\pi+\delta < \phase@@{C_{n}}		 

-(1)/(2)*Pi + delta < argument(C[n])

-Divide[1,2]*Pi + \[Delta] < Arg[Subscript[C, n]]
Failure Failure
Failed [27 / 300]
Result: -.70796327e-1 < -1.047197551
Test Values: {delta = 1.5, C[n] = 1/2-1/2*I*3^(1/2), n = 1, n = 3}


Result: -.70796327e-1 < -1.047197551
Test Values: {delta = 1.5, C[n] = 1/2-1/2*I*3^(1/2), n = 2, n = 3}


Result: -.70796327e-1 < -1.047197551
Test Values: {delta = 1.5, C[n] = 1/2-1/2*I*3^(1/2), n = 3, n = 3}


Result: -.70796327e-1 < -2.617993878
Test Values: {delta = 1.5, C[n] = -1/2*3^(1/2)-1/2*I, n = 1, n = 3}

... skip entries to safe data
Failed [49 / 100]
Result: Less[Complex[-0.7047709230104579, 0.49999999999999994], 0.5235987755982988]
Test Values: {Rule[n, 3], Rule[Ξ΄, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Less[Complex[-0.7047709230104579, 0.49999999999999994], 2.0943951023931953]
Test Values: {Rule[n, 3], Rule[Ξ΄, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.12.E27 ph ⁑ C n < 1 2 ⁒ Ο€ - Ξ΄ phase subscript 𝐢 𝑛 1 2 πœ‹ 𝛿 {\displaystyle{\displaystyle\operatorname{ph}C_{n}<\tfrac{1}{2}\pi-\delta}}
\phase@@{C_{n}} < \tfrac{1}{2}\pi-\delta		 

argument(C[n]) < (1)/(2)*Pi - delta

Arg[Subscript[C, n]] < Divide[1,2]*Pi - \[Delta]
Failure Failure
Failed [72 / 300]
Result: 3.141592654 < 3.070796327
Test Values: {delta = -1.5, C[n] = -1.5, n = 1, n = 3}


Result: 3.141592654 < 3.070796327
Test Values: {delta = -1.5, C[n] = -1.5, n = 2, n = 3}


Result: 3.141592654 < 3.070796327
Test Values: {delta = -1.5, C[n] = -1.5, n = 3, n = 3}


Result: 3.141592654 < 3.070796327
Test Values: {delta = -1.5, C[n] = -.5, n = 1, n = 3}

... skip entries to safe data
Failed [64 / 100]
Result: Less[0.5235987755982988, Complex[0.7047709230104579, -0.49999999999999994]]
Test Values: {Rule[n, 3], Rule[Ξ΄, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Less[2.0943951023931953, Complex[0.7047709230104579, -0.49999999999999994]]
Test Values: {Rule[n, 3], Rule[Ξ΄, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[C, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.12.E28 βˆ‘ n = 1 ∞ | b n | = ∞ subscript superscript 𝑛 1 subscript 𝑏 𝑛 {\displaystyle{\displaystyle\sum^{\infty}_{n=1}|b_{n}|=\infty}}
\sum^{\infty}_{n=1}|b_{n}| = \infty		 

sum(abs(b[n]), n = 1..infinity) = infinity
 
Sum[Abs[Subscript[b, n]], {n, 1, Infinity}, GenerateConditions->None] == Infinity
 Skipped - no semantic math Skipped - no semantic math - -
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