1.12

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
1.12#Ex1	${\displaystyle{\displaystyle A_{k}=b_{k}A_{k-1}+a_{k}A_{k-2}}}$ 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	${\displaystyle{\displaystyle B_{k}=b_{k}B_{k-1}+a_{k}B_{k-2}}}$ 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	${\displaystyle{\displaystyle A_{-1}=1}}$ A_{-1} = 1		A[- 1] = 1	Subscript[A, - 1] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
1.12#Ex4	${\displaystyle{\displaystyle A_{0}=b_{0}}}$ 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	${\displaystyle{\displaystyle B_{-1}=0}}$ B_{-1} = 0		B[- 1] = 0	Subscript[B, - 1] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
1.12#Ex6	${\displaystyle{\displaystyle B_{0}=1}}$ B_{0} = 1		B[0] = 1	Subscript[B, 0] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
1.12.E7	${\displaystyle{\displaystyle A_{n}B_{n-1}-B_{n}A_{n-1}=(-1)^{n-1}\prod^{n}_{k=1}a_{k}}}$ 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	${\displaystyle{\displaystyle C_{n}-C_{n-1}=\frac{(-1)^{n-1}\prod^{n}_{k=1}a_{k}}{B_{n-1}B_{n}}}}$ 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	${\displaystyle{\displaystyle a_{n}=\frac{A_{n-1}B_{n}-A_{n}B_{n-1}}{A_{n-1}B_{n-2}-A_{n-2}B_{n-1}}}}$ 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	${\displaystyle{\displaystyle a_{n}=\frac{B_{n}}{B_{n-2}}\frac{C_{n-1}-C_{n}}{C_{n-1}-C_{n-2}}}}$ 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	${\displaystyle{\displaystyle b_{n}=\frac{A_{n}B_{n-2}-A_{n-2}B_{n}}{A_{n-1}B_{n-2}-A_{n-2}B_{n-1}}}}$ 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	${\displaystyle{\displaystyle b_{n}=\frac{B_{n}}{B_{n-1}}\frac{C_{n}-C_{n-2}}{C_{n-1}-C_{n-2}}}}$ 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	${\displaystyle{\displaystyle b_{0}=A_{0}}}$ 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	${\displaystyle{\displaystyle b_{1}=B_{1}}}$ 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	${\displaystyle{\displaystyle a_{1}=A_{1}-A_{0}B_{1}}}$ 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	${\displaystyle{\displaystyle C_{n}(w)=\frac{A_{n}+A_{n-1}w}{B_{n}+B_{n-1}w}}}$ 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	${\displaystyle{\displaystyle C_{0}\not=\infty}}$ C_{0}\not = \infty		C[0] <> infinity	Subscript[C, 0] \[NotEqual]*Infinity	Skipped - no semantic math	Skipped - no semantic math	-	-
1.12#Ex14	${\displaystyle{\displaystyle C_{n}\not=C_{n-1}}}$ 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	${\displaystyle{\displaystyle|b_{n}|\geq|a_{n}|+1}}$ |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	${\displaystyle{\displaystyle-\tfrac{1}{2}\pi+\delta<\operatorname{ph}b_{n}}}$ -\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[δ, 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[δ, 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	${\displaystyle{\displaystyle\operatorname{ph}b_{n}<\tfrac{1}{2}\pi-\delta}}$ \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[δ, 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[δ, 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	${\displaystyle{\displaystyle-\tfrac{1}{2}\pi+\delta<\operatorname{ph}C_{n}}}$ -\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[δ, 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[δ, 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	${\displaystyle{\displaystyle\operatorname{ph}C_{n}<\tfrac{1}{2}\pi-\delta}}$ \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[δ, 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[δ, 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	${\displaystyle{\displaystyle\sum^{\infty}_{n=1}|b_{n}|=\infty}}$ \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	-	-