Generalized Hypergeometric Functions & Meijer G -Function - 16.3 Derivatives and Contiguous Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
16.3.E5 ( z ⁒ d d z ⁑ z ) n = z n ⁒ d n d z n ⁑ z n superscript 𝑧 derivative 𝑧 𝑧 𝑛 superscript 𝑧 𝑛 derivative 𝑧 𝑛 superscript 𝑧 𝑛 {\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}=% z^{n}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}z^{n}}}
\left(z\deriv{}{z}z\right)^{n} = z^{n}\deriv[n]{}{z}z^{n}		 

(z*diff(z, z))^(n) = (z)^(n)* diff((z)^(n), [z$(n)])

(z*D[z, z])^(n) == (z)^(n)* D[(z)^(n), {z, n}]
Failure Failure
Failed [7 / 7]
Result: -.1616869430e-8-5.000000005*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 3}


Result: -5.000000005+.1616869430e-8*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 3}

... skip entries to safe data
Failed [14 / 21]
Result: Complex[-0.5000000000000001, -0.8660254037844386]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.0, -5.0]
Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
16.3.E6 z ⁒ F 1 0 ⁑ ( - ; b + 1 ; z ) + b ⁒ ( b - 1 ) ⁒ F 1 0 ⁑ ( - ; b ; z ) - b ⁒ ( b - 1 ) ⁒ F 1 0 ⁑ ( - ; b - 1 ; z ) = 0 𝑧 Gauss-hypergeometric-pFq 0 1 𝑏 1 𝑧 𝑏 𝑏 1 Gauss-hypergeometric-pFq 0 1 𝑏 𝑧 𝑏 𝑏 1 Gauss-hypergeometric-pFq 0 1 𝑏 1 𝑧 0 {\displaystyle{\displaystyle z{{}_{0}F_{1}}\left(-;b+1;z\right)+b(b-1){{}_{0}F% _{1}}\left(-;b;z\right)-b(b-1){{}_{0}F_{1}}\left(-;b-1;z\right)=0}}
z\genhyperF{0}{1}@{-}{b+1}{z}+b(b-1)\genhyperF{0}{1}@{-}{b}{z}-b(b-1)\genhyperF{0}{1}@{-}{b-1}{z} = 0		 

z*hypergeom([-], [b + 1], z)+ b*(b - 1)*hypergeom([-], [b], z)- b*(b - 1)*hypergeom([-], [b - 1], z) = 0

z*HypergeometricPFQ[{-}, {b + 1}, z]+ b*(b - 1)*HypergeometricPFQ[{-}, {b}, z]- b*(b - 1)*HypergeometricPFQ[{-}, {b - 1}, z] == 0
Error Failure - Error
16.3.E7 F 2 3 ⁑ ( a 1 + 2 , a 2 , a 3 b 1 , b 2 ; z ) ⁒ a 1 ⁒ ( a 1 + 1 ) ⁒ ( 1 - z ) + F 2 3 ⁑ ( a 1 + 1 , a 2 , a 3 b 1 , b 2 ; z ) ⁒ a 1 ⁒ ( b 1 + b 2 - 3 ⁒ a 1 - 2 + z ⁒ ( 2 ⁒ a 1 - a 2 - a 3 + 1 ) ) + F 2 3 ⁑ ( a 1 , a 2 , a 3 b 1 , b 2 ; z ) ⁒ ( ( 2 ⁒ a 1 - b 1 ) ⁒ ( 2 ⁒ a 1 - b 2 ) + a 1 - a 1 2 - z ⁒ ( a 1 - a 2 ) ⁒ ( a 1 - a 3 ) ) - F 2 3 ⁑ ( a 1 - 1 , a 2 , a 3 b 1 , b 2 ; z ) ⁒ ( a 1 - b 1 ) ⁒ ( a 1 - b 2 ) = 0 Gauss-hypergeometric-pFq 3 2 subscript π‘Ž 1 2 subscript π‘Ž 2 subscript π‘Ž 3 subscript 𝑏 1 subscript 𝑏 2 𝑧 subscript π‘Ž 1 subscript π‘Ž 1 1 1 𝑧 Gauss-hypergeometric-pFq 3 2 subscript π‘Ž 1 1 subscript π‘Ž 2 subscript π‘Ž 3 subscript 𝑏 1 subscript 𝑏 2 𝑧 subscript π‘Ž 1 subscript 𝑏 1 subscript 𝑏 2 3 subscript π‘Ž 1 2 𝑧 2 subscript π‘Ž 1 subscript π‘Ž 2 subscript π‘Ž 3 1 Gauss-hypergeometric-pFq 3 2 subscript π‘Ž 1 subscript π‘Ž 2 subscript π‘Ž 3 subscript 𝑏 1 subscript 𝑏 2 𝑧 2 subscript π‘Ž 1 subscript 𝑏 1 2 subscript π‘Ž 1 subscript 𝑏 2 subscript π‘Ž 1 superscript subscript π‘Ž 1 2 𝑧 subscript π‘Ž 1 subscript π‘Ž 2 subscript π‘Ž 1 subscript π‘Ž 3 Gauss-hypergeometric-pFq 3 2 subscript π‘Ž 1 1 subscript π‘Ž 2 subscript π‘Ž 3 subscript 𝑏 1 subscript 𝑏 2 𝑧 subscript π‘Ž 1 subscript 𝑏 1 subscript π‘Ž 1 subscript 𝑏 2 0 {\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a_{1}+2,a_{2},a_{3}\atop b_{1}% ,b_{2}};z\right)a_{1}(a_{1}+1)(1-z)+{{}_{3}F_{2}}\left({a_{1}+1,a_{2},a_{3}% \atop b_{1},b_{2}};z\right)a_{1}\left(b_{1}+b_{2}-3a_{1}-2+z(2a_{1}-a_{2}-a_{3% }+1)\right)+{{}_{3}F_{2}}\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};z\right)% \left((2a_{1}-b_{1})(2a_{1}-b_{2})+a_{1}-a_{1}^{2}-z(a_{1}-a_{2})(a_{1}-a_{3})% \right)-{{}_{3}F_{2}}\left({a_{1}-1,a_{2},a_{3}\atop b_{1},b_{2}};z\right)(a_{% 1}-b_{1})(a_{1}-b_{2})=0}}
\genhyperF{3}{2}@@{a_{1}+2,a_{2},a_{3}}{b_{1},b_{2}}{z}a_{1}(a_{1}+1)(1-z)+\genhyperF{3}{2}@@{a_{1}+1,a_{2},a_{3}}{b_{1},b_{2}}{z}a_{1}\left(b_{1}+b_{2}-3a_{1}-2+z(2a_{1}-a_{2}-a_{3}+1)\right)+\genhyperF{3}{2}@@{a_{1},a_{2},a_{3}}{b_{1},b_{2}}{z}\left((2a_{1}-b_{1})(2a_{1}-b_{2})+a_{1}-a_{1}^{2}-z(a_{1}-a_{2})(a_{1}-a_{3})\right)-\genhyperF{3}{2}@@{a_{1}-1,a_{2},a_{3}}{b_{1},b_{2}}{z}(a_{1}-b_{1})(a_{1}-b_{2}) = 0		 

hypergeom([a[1]+ 2 , a[2], a[3]], [b[1], b[2]], z)*a[1]*(a[1]+ 1)*(1 - z)+ hypergeom([a[1]+ 1 , a[2], a[3]], [b[1], b[2]], z)*a[1]*(b[1]+ b[2]- 3*a[1]- 2 + z*(2*a[1]- a[2]- a[3]+ 1))+ hypergeom([a[1], a[2], a[3]], [b[1], b[2]], z)*((2*a[1]- b[1])*(2*a[1]- b[2])+ a[1]- (a[1])^(2)- z*(a[1]- a[2])*(a[1]- a[3]))- hypergeom([a[1]- 1 , a[2], a[3]], [b[1], b[2]], z)*(a[1]- b[1])*(a[1]- b[2]) = 0

HypergeometricPFQ[{Subscript[a, 1]+ 2 , Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z]*Subscript[a, 1]*(Subscript[a, 1]+ 1)*(1 - z)+ HypergeometricPFQ[{Subscript[a, 1]+ 1 , Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z]*Subscript[a, 1]*(Subscript[b, 1]+ Subscript[b, 2]- 3*Subscript[a, 1]- 2 + z*(2*Subscript[a, 1]- Subscript[a, 2]- Subscript[a, 3]+ 1))+ HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z]*((2*Subscript[a, 1]- Subscript[b, 1])*(2*Subscript[a, 1]- Subscript[b, 2])+ Subscript[a, 1]- (Subscript[a, 1])^(2)- z*(Subscript[a, 1]- Subscript[a, 2])*(Subscript[a, 1]- Subscript[a, 3]))- HypergeometricPFQ[{Subscript[a, 1]- 1 , Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z]*(Subscript[a, 1]- Subscript[b, 1])*(Subscript[a, 1]- Subscript[b, 2]) == 0
Failure Failure Skipped - Because timed out
Failed [300 / 300]
Result: Plus[Complex[1.7372028395654344, 0.5250871122698257], Times[Complex[-0.5000000000000001, -0.8660254037844386], a]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[4.427028162877593, -11.419461015230842], Times[Complex[-0.5000000000000001, -0.8660254037844386], a]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[b, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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