Confluent Hypergeometric Functions - 13.15 Recurrence Relations and Derivatives

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
13.15.E1 ( κ - μ - 1 2 ) ⁢ M κ - 1 , μ ⁡ ( z ) + ( z - 2 ⁢ κ ) ⁢ M κ , μ ⁡ ( z ) + ( κ + μ + 1 2 ) ⁢ M κ + 1 , μ ⁡ ( z ) = 0 𝜅 𝜇 1 2 Whittaker-confluent-hypergeometric-M 𝜅 1 𝜇 𝑧 𝑧 2 𝜅 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 𝜅 𝜇 1 2 Whittaker-confluent-hypergeometric-M 𝜅 1 𝜇 𝑧 0 {\displaystyle{\displaystyle(\kappa-\mu-\tfrac{1}{2})M_{\kappa-1,\mu}\left(z% \right)+(z-2\kappa)M_{\kappa,\mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2})M_{% \kappa+1,\mu}\left(z\right)=0}}
(\kappa-\mu-\tfrac{1}{2})\WhittakerconfhyperM{\kappa-1}{\mu}@{z}+(z-2\kappa)\WhittakerconfhyperM{\kappa}{\mu}@{z}+(\kappa+\mu+\tfrac{1}{2})\WhittakerconfhyperM{\kappa+1}{\mu}@{z} = 0		 

(kappa - mu -(1)/(2))*WhittakerM(kappa - 1, mu, z)+(z - 2*kappa)*WhittakerM(kappa, mu, z)+(kappa + mu +(1)/(2))*WhittakerM(kappa + 1, mu, z) = 0

(\[Kappa]- \[Mu]-Divide[1,2])*WhittakerM[\[Kappa]- 1, \[Mu], z]+(z - 2*\[Kappa])*WhittakerM[\[Kappa], \[Mu], z]+(\[Kappa]+ \[Mu]+Divide[1,2])*WhittakerM[\[Kappa]+ 1, \[Mu], z] == 0
Successful Successful -
Failed [84 / 300]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}


Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -0.5]}

... skip entries to safe data
13.15.E2 2 ⁢ μ ⁢ ( 1 + 2 ⁢ μ ) ⁢ z ⁢ M κ - 1 2 , μ - 1 2 ⁡ ( z ) - ( z + 2 ⁢ μ ) ⁢ ( 1 + 2 ⁢ μ ) ⁢ M κ , μ ⁡ ( z ) + ( κ + μ + 1 2 ) ⁢ z ⁢ M κ + 1 2 , μ + 1 2 ⁡ ( z ) = 0 2 𝜇 1 2 𝜇 𝑧 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝜇 1 2 𝑧 𝑧 2 𝜇 1 2 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 𝜅 𝜇 1 2 𝑧 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝜇 1 2 𝑧 0 {\displaystyle{\displaystyle 2\mu(1+2\mu)\sqrt{z}M_{\kappa-\frac{1}{2},\mu-% \frac{1}{2}}\left(z\right)-(z+2\mu)(1+2\mu)M_{\kappa,\mu}\left(z\right)+(% \kappa+\mu+\tfrac{1}{2})\sqrt{z}M_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z% \right)=0}}
2\mu(1+2\mu)\sqrt{z}\WhittakerconfhyperM{\kappa-\frac{1}{2}}{\mu-\frac{1}{2}}@{z}-(z+2\mu)(1+2\mu)\WhittakerconfhyperM{\kappa}{\mu}@{z}+(\kappa+\mu+\tfrac{1}{2})\sqrt{z}\WhittakerconfhyperM{\kappa+\frac{1}{2}}{\mu+\frac{1}{2}}@{z} = 0		 

2*mu*(1 + 2*mu)*sqrt(z)*WhittakerM(kappa -(1)/(2), mu -(1)/(2), z)-(z + 2*mu)*(1 + 2*mu)*WhittakerM(kappa, mu, z)+(kappa + mu +(1)/(2))*sqrt(z)*WhittakerM(kappa +(1)/(2), mu +(1)/(2), z) = 0

2*\[Mu]*(1 + 2*\[Mu])*Sqrt[z]*WhittakerM[\[Kappa]-Divide[1,2], \[Mu]-Divide[1,2], z]-(z + 2*\[Mu])*(1 + 2*\[Mu])*WhittakerM[\[Kappa], \[Mu], z]+(\[Kappa]+ \[Mu]+Divide[1,2])*Sqrt[z]*WhittakerM[\[Kappa]+Divide[1,2], \[Mu]+Divide[1,2], z] == 0
Successful Failure -
Failed [81 / 300]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}


Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -0.5]}

... skip entries to safe data
13.15.E3 ( κ - μ - 1 2 ) ⁢ M κ - 1 2 , μ + 1 2 ⁡ ( z ) + ( 1 + 2 ⁢ μ ) ⁢ z ⁢ M κ , μ ⁡ ( z ) - ( κ + μ + 1 2 ) ⁢ M κ + 1 2 , μ + 1 2 ⁡ ( z ) = 0 𝜅 𝜇 1 2 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝜇 1 2 𝑧 1 2 𝜇 𝑧 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 𝜅 𝜇 1 2 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝜇 1 2 𝑧 0 {\displaystyle{\displaystyle(\kappa-\mu-\tfrac{1}{2})M_{\kappa-\frac{1}{2},\mu% +\frac{1}{2}}\left(z\right)+(1+2\mu)\sqrt{z}M_{\kappa,\mu}\left(z\right)-(% \kappa+\mu+\tfrac{1}{2})M_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0}}
(\kappa-\mu-\tfrac{1}{2})\WhittakerconfhyperM{\kappa-\frac{1}{2}}{\mu+\frac{1}{2}}@{z}+(1+2\mu)\sqrt{z}\WhittakerconfhyperM{\kappa}{\mu}@{z}-(\kappa+\mu+\tfrac{1}{2})\WhittakerconfhyperM{\kappa+\frac{1}{2}}{\mu+\frac{1}{2}}@{z} = 0		 

(kappa - mu -(1)/(2))*WhittakerM(kappa -(1)/(2), mu +(1)/(2), z)+(1 + 2*mu)*sqrt(z)*WhittakerM(kappa, mu, z)-(kappa + mu +(1)/(2))*WhittakerM(kappa +(1)/(2), mu +(1)/(2), z) = 0

(\[Kappa]- \[Mu]-Divide[1,2])*WhittakerM[\[Kappa]-Divide[1,2], \[Mu]+Divide[1,2], z]+(1 + 2*\[Mu])*Sqrt[z]*WhittakerM[\[Kappa], \[Mu], z]-(\[Kappa]+ \[Mu]+Divide[1,2])*WhittakerM[\[Kappa]+Divide[1,2], \[Mu]+Divide[1,2], z] == 0
Successful Failure -
Failed [84 / 300]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}


Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -0.5]}

... skip entries to safe data
13.15.E4 2 ⁢ μ ⁢ M κ - 1 2 , μ - 1 2 ⁡ ( z ) - 2 ⁢ μ ⁢ M κ + 1 2 , μ - 1 2 ⁡ ( z ) - z ⁢ M κ , μ ⁡ ( z ) = 0 2 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝜇 1 2 𝑧 2 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝜇 1 2 𝑧 𝑧 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 0 {\displaystyle{\displaystyle 2\mu M_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(% z\right)-2\mu M_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-\sqrt{z}M_{% \kappa,\mu}\left(z\right)=0}}
2\mu\WhittakerconfhyperM{\kappa-\frac{1}{2}}{\mu-\frac{1}{2}}@{z}-2\mu\WhittakerconfhyperM{\kappa+\frac{1}{2}}{\mu-\frac{1}{2}}@{z}-\sqrt{z}\WhittakerconfhyperM{\kappa}{\mu}@{z} = 0		 

2*mu*WhittakerM(kappa -(1)/(2), mu -(1)/(2), z)- 2*mu*WhittakerM(kappa +(1)/(2), mu -(1)/(2), z)-sqrt(z)*WhittakerM(kappa, mu, z) = 0

2*\[Mu]*WhittakerM[\[Kappa]-Divide[1,2], \[Mu]-Divide[1,2], z]- 2*\[Mu]*WhittakerM[\[Kappa]+Divide[1,2], \[Mu]-Divide[1,2], z]-Sqrt[z]*WhittakerM[\[Kappa], \[Mu], z] == 0
Successful Failure -
Failed [78 / 300]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}


Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -0.5]}

... skip entries to safe data
13.15.E5 2 ⁢ μ ⁢ ( 1 + 2 ⁢ μ ) ⁢ M κ , μ ⁡ ( z ) - 2 ⁢ μ ⁢ ( 1 + 2 ⁢ μ ) ⁢ z ⁢ M κ - 1 2 , μ - 1 2 ⁡ ( z ) - ( κ - μ - 1 2 ) ⁢ z ⁢ M κ - 1 2 , μ + 1 2 ⁡ ( z ) = 0 2 𝜇 1 2 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 2 𝜇 1 2 𝜇 𝑧 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝜇 1 2 𝑧 𝜅 𝜇 1 2 𝑧 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝜇 1 2 𝑧 0 {\displaystyle{\displaystyle 2\mu(1+2\mu)M_{\kappa,\mu}\left(z\right)-2\mu(1+2% \mu)\sqrt{z}M_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-(\kappa-\mu-% \tfrac{1}{2})\sqrt{z}M_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0}}
2\mu(1+2\mu)\WhittakerconfhyperM{\kappa}{\mu}@{z}-2\mu(1+2\mu)\sqrt{z}\WhittakerconfhyperM{\kappa-\frac{1}{2}}{\mu-\frac{1}{2}}@{z}-(\kappa-\mu-\tfrac{1}{2})\sqrt{z}\WhittakerconfhyperM{\kappa-\frac{1}{2}}{\mu+\frac{1}{2}}@{z} = 0		 

2*mu*(1 + 2*mu)*WhittakerM(kappa, mu, z)- 2*mu*(1 + 2*mu)*sqrt(z)*WhittakerM(kappa -(1)/(2), mu -(1)/(2), z)-(kappa - mu -(1)/(2))*sqrt(z)*WhittakerM(kappa -(1)/(2), mu +(1)/(2), z) = 0

2*\[Mu]*(1 + 2*\[Mu])*WhittakerM[\[Kappa], \[Mu], z]- 2*\[Mu]*(1 + 2*\[Mu])*Sqrt[z]*WhittakerM[\[Kappa]-Divide[1,2], \[Mu]-Divide[1,2], z]-(\[Kappa]- \[Mu]-Divide[1,2])*Sqrt[z]*WhittakerM[\[Kappa]-Divide[1,2], \[Mu]+Divide[1,2], z] == 0
Successful Failure -
Failed [81 / 300]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}


Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -0.5]}

... skip entries to safe data
13.15.E6 2 ⁢ μ ⁢ ( 1 + 2 ⁢ μ ) ⁢ z ⁢ M κ + 1 2 , μ - 1 2 ⁡ ( z ) + ( z - 2 ⁢ μ ) ⁢ ( 1 + 2 ⁢ μ ) ⁢ M κ , μ ⁡ ( z ) + ( κ - μ - 1 2 ) ⁢ z ⁢ M κ - 1 2 , μ + 1 2 ⁡ ( z ) = 0 2 𝜇 1 2 𝜇 𝑧 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝜇 1 2 𝑧 𝑧 2 𝜇 1 2 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 𝜅 𝜇 1 2 𝑧 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝜇 1 2 𝑧 0 {\displaystyle{\displaystyle 2\mu(1+2\mu)\sqrt{z}M_{\kappa+\frac{1}{2},\mu-% \frac{1}{2}}\left(z\right)+(z-2\mu)(1+2\mu)M_{\kappa,\mu}\left(z\right)+(% \kappa-\mu-\tfrac{1}{2})\sqrt{z}M_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z% \right)=0}}
2\mu(1+2\mu)\sqrt{z}\WhittakerconfhyperM{\kappa+\frac{1}{2}}{\mu-\frac{1}{2}}@{z}+(z-2\mu)(1+2\mu)\WhittakerconfhyperM{\kappa}{\mu}@{z}+(\kappa-\mu-\tfrac{1}{2})\sqrt{z}\WhittakerconfhyperM{\kappa-\frac{1}{2}}{\mu+\frac{1}{2}}@{z} = 0		 

2*mu*(1 + 2*mu)*sqrt(z)*WhittakerM(kappa +(1)/(2), mu -(1)/(2), z)+(z - 2*mu)*(1 + 2*mu)*WhittakerM(kappa, mu, z)+(kappa - mu -(1)/(2))*sqrt(z)*WhittakerM(kappa -(1)/(2), mu +(1)/(2), z) = 0

2*\[Mu]*(1 + 2*\[Mu])*Sqrt[z]*WhittakerM[\[Kappa]+Divide[1,2], \[Mu]-Divide[1,2], z]+(z - 2*\[Mu])*(1 + 2*\[Mu])*WhittakerM[\[Kappa], \[Mu], z]+(\[Kappa]- \[Mu]-Divide[1,2])*Sqrt[z]*WhittakerM[\[Kappa]-Divide[1,2], \[Mu]+Divide[1,2], z] == 0
Successful Failure -
Failed [81 / 300]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}


Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -0.5]}

... skip entries to safe data
13.15.E7 2 ⁢ μ ⁢ ( 1 + 2 ⁢ μ ) ⁢ z ⁢ M κ + 1 2 , μ - 1 2 ⁡ ( z ) - 2 ⁢ μ ⁢ ( 1 + 2 ⁢ μ ) ⁢ M κ , μ ⁡ ( z ) + ( κ + μ + 1 2 ) ⁢ z ⁢ M κ + 1 2 , μ + 1 2 ⁡ ( z ) = 0 2 𝜇 1 2 𝜇 𝑧 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝜇 1 2 𝑧 2 𝜇 1 2 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 𝜅 𝜇 1 2 𝑧 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝜇 1 2 𝑧 0 {\displaystyle{\displaystyle 2\mu(1+2\mu)\sqrt{z}M_{\kappa+\frac{1}{2},\mu-% \frac{1}{2}}\left(z\right)-2\mu(1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa+% \mu+\tfrac{1}{2})\sqrt{z}M_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=% 0}}
2\mu(1+2\mu)\sqrt{z}\WhittakerconfhyperM{\kappa+\frac{1}{2}}{\mu-\frac{1}{2}}@{z}-2\mu(1+2\mu)\WhittakerconfhyperM{\kappa}{\mu}@{z}+(\kappa+\mu+\tfrac{1}{2})\sqrt{z}\WhittakerconfhyperM{\kappa+\frac{1}{2}}{\mu+\frac{1}{2}}@{z} = 0		 

2*mu*(1 + 2*mu)*sqrt(z)*WhittakerM(kappa +(1)/(2), mu -(1)/(2), z)- 2*mu*(1 + 2*mu)*WhittakerM(kappa, mu, z)+(kappa + mu +(1)/(2))*sqrt(z)*WhittakerM(kappa +(1)/(2), mu +(1)/(2), z) = 0

2*\[Mu]*(1 + 2*\[Mu])*Sqrt[z]*WhittakerM[\[Kappa]+Divide[1,2], \[Mu]-Divide[1,2], z]- 2*\[Mu]*(1 + 2*\[Mu])*WhittakerM[\[Kappa], \[Mu], z]+(\[Kappa]+ \[Mu]+Divide[1,2])*Sqrt[z]*WhittakerM[\[Kappa]+Divide[1,2], \[Mu]+Divide[1,2], z] == 0
Successful Failure -
Failed [81 / 300]
Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}


Result: Indeterminate
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -0.5]}

... skip entries to safe data
13.15.E8 W κ + 1 2 , μ + 1 2 ⁡ ( z ) - z ⁢ W κ , μ ⁡ ( z ) + ( κ - μ - 1 2 ) ⁢ W κ - 1 2 , μ + 1 2 ⁡ ( z ) = 0 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝜇 1 2 𝑧 𝑧 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 𝜅 𝜇 1 2 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝜇 1 2 𝑧 0 {\displaystyle{\displaystyle W_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z% \right)-\sqrt{z}W_{\kappa,\mu}\left(z\right)+(\kappa-\mu-\tfrac{1}{2})W_{% \kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0}}
\WhittakerconfhyperW{\kappa+\frac{1}{2}}{\mu+\frac{1}{2}}@{z}-\sqrt{z}\WhittakerconfhyperW{\kappa}{\mu}@{z}+(\kappa-\mu-\tfrac{1}{2})\WhittakerconfhyperW{\kappa-\frac{1}{2}}{\mu+\frac{1}{2}}@{z} = 0		 

WhittakerW(kappa +(1)/(2), mu +(1)/(2), z)-sqrt(z)*WhittakerW(kappa, mu, z)+(kappa - mu -(1)/(2))*WhittakerW(kappa -(1)/(2), mu +(1)/(2), z) = 0

WhittakerW[\[Kappa]+Divide[1,2], \[Mu]+Divide[1,2], z]-Sqrt[z]*WhittakerW[\[Kappa], \[Mu], z]+(\[Kappa]- \[Mu]-Divide[1,2])*WhittakerW[\[Kappa]-Divide[1,2], \[Mu]+Divide[1,2], z] == 0
Successful Failure - Successful [Tested: 300]
13.15.E9 W κ + 1 2 , μ - 1 2 ⁡ ( z ) - z ⁢ W κ , μ ⁡ ( z ) + ( κ + μ - 1 2 ) ⁢ W κ - 1 2 , μ - 1 2 ⁡ ( z ) = 0 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝜇 1 2 𝑧 𝑧 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 𝜅 𝜇 1 2 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝜇 1 2 𝑧 0 {\displaystyle{\displaystyle W_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z% \right)-\sqrt{z}W_{\kappa,\mu}\left(z\right)+(\kappa+\mu-\tfrac{1}{2})W_{% \kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)=0}}
\WhittakerconfhyperW{\kappa+\frac{1}{2}}{\mu-\frac{1}{2}}@{z}-\sqrt{z}\WhittakerconfhyperW{\kappa}{\mu}@{z}+(\kappa+\mu-\tfrac{1}{2})\WhittakerconfhyperW{\kappa-\frac{1}{2}}{\mu-\frac{1}{2}}@{z} = 0		 

WhittakerW(kappa +(1)/(2), mu -(1)/(2), z)-sqrt(z)*WhittakerW(kappa, mu, z)+(kappa + mu -(1)/(2))*WhittakerW(kappa -(1)/(2), mu -(1)/(2), z) = 0

WhittakerW[\[Kappa]+Divide[1,2], \[Mu]-Divide[1,2], z]-Sqrt[z]*WhittakerW[\[Kappa], \[Mu], z]+(\[Kappa]+ \[Mu]-Divide[1,2])*WhittakerW[\[Kappa]-Divide[1,2], \[Mu]-Divide[1,2], z] == 0
Successful Failure - Successful [Tested: 300]
13.15.E10 2 ⁢ μ ⁢ W κ , μ ⁡ ( z ) - z ⁢ W κ + 1 2 , μ + 1 2 ⁡ ( z ) + z ⁢ W κ + 1 2 , μ - 1 2 ⁡ ( z ) = 0 2 𝜇 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 𝑧 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝜇 1 2 𝑧 𝑧 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝜇 1 2 𝑧 0 {\displaystyle{\displaystyle 2\mu W_{\kappa,\mu}\left(z\right)-\sqrt{z}W_{% \kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)+\sqrt{z}W_{\kappa+\frac{1}{2% },\mu-\frac{1}{2}}\left(z\right)=0}}
2\mu\WhittakerconfhyperW{\kappa}{\mu}@{z}-\sqrt{z}\WhittakerconfhyperW{\kappa+\frac{1}{2}}{\mu+\frac{1}{2}}@{z}+\sqrt{z}\WhittakerconfhyperW{\kappa+\frac{1}{2}}{\mu-\frac{1}{2}}@{z} = 0		 

2*mu*WhittakerW(kappa, mu, z)-sqrt(z)*WhittakerW(kappa +(1)/(2), mu +(1)/(2), z)+sqrt(z)*WhittakerW(kappa +(1)/(2), mu -(1)/(2), z) = 0

2*\[Mu]*WhittakerW[\[Kappa], \[Mu], z]-Sqrt[z]*WhittakerW[\[Kappa]+Divide[1,2], \[Mu]+Divide[1,2], z]+Sqrt[z]*WhittakerW[\[Kappa]+Divide[1,2], \[Mu]-Divide[1,2], z] == 0
Successful Failure - Successful [Tested: 300]
13.15.E11 W κ + 1 , μ ⁡ ( z ) + ( 2 ⁢ κ - z ) ⁢ W κ , μ ⁡ ( z ) + ( κ - μ - 1 2 ) ⁢ ( κ + μ - 1 2 ) ⁢ W κ - 1 , μ ⁡ ( z ) = 0 Whittaker-confluent-hypergeometric-W 𝜅 1 𝜇 𝑧 2 𝜅 𝑧 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 𝜅 𝜇 1 2 𝜅 𝜇 1 2 Whittaker-confluent-hypergeometric-W 𝜅 1 𝜇 𝑧 0 {\displaystyle{\displaystyle W_{\kappa+1,\mu}\left(z\right)+(2\kappa-z)W_{% \kappa,\mu}\left(z\right)+(\kappa-\mu-\tfrac{1}{2})(\kappa+\mu-\tfrac{1}{2})W_% {\kappa-1,\mu}\left(z\right)=0}}
\WhittakerconfhyperW{\kappa+1}{\mu}@{z}+(2\kappa-z)\WhittakerconfhyperW{\kappa}{\mu}@{z}+(\kappa-\mu-\tfrac{1}{2})(\kappa+\mu-\tfrac{1}{2})\WhittakerconfhyperW{\kappa-1}{\mu}@{z} = 0		 

WhittakerW(kappa + 1, mu, z)+(2*kappa - z)*WhittakerW(kappa, mu, z)+(kappa - mu -(1)/(2))*(kappa + mu -(1)/(2))*WhittakerW(kappa - 1, mu, z) = 0

WhittakerW[\[Kappa]+ 1, \[Mu], z]+(2*\[Kappa]- z)*WhittakerW[\[Kappa], \[Mu], z]+(\[Kappa]- \[Mu]-Divide[1,2])*(\[Kappa]+ \[Mu]-Divide[1,2])*WhittakerW[\[Kappa]- 1, \[Mu], z] == 0
Successful Successful - Successful [Tested: 300]
13.15.E12 ( κ - μ - 1 2 ) ⁢ z ⁢ W κ - 1 2 , μ + 1 2 ⁡ ( z ) + 2 ⁢ μ ⁢ W κ , μ ⁡ ( z ) - ( κ + μ - 1 2 ) ⁢ z ⁢ W κ - 1 2 , μ - 1 2 ⁡ ( z ) = 0 𝜅 𝜇 1 2 𝑧 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝜇 1 2 𝑧 2 𝜇 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 𝜅 𝜇 1 2 𝑧 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝜇 1 2 𝑧 0 {\displaystyle{\displaystyle(\kappa-\mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1% }{2},\mu+\frac{1}{2}}\left(z\right)+2\mu W_{\kappa,\mu}\left(z\right)-(\kappa+% \mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)=% 0}}
(\kappa-\mu-\tfrac{1}{2})\sqrt{z}\WhittakerconfhyperW{\kappa-\frac{1}{2}}{\mu+\frac{1}{2}}@{z}+2\mu\WhittakerconfhyperW{\kappa}{\mu}@{z}-(\kappa+\mu-\tfrac{1}{2})\sqrt{z}\WhittakerconfhyperW{\kappa-\frac{1}{2}}{\mu-\frac{1}{2}}@{z} = 0		 

(kappa - mu -(1)/(2))*sqrt(z)*WhittakerW(kappa -(1)/(2), mu +(1)/(2), z)+ 2*mu*WhittakerW(kappa, mu, z)-(kappa + mu -(1)/(2))*sqrt(z)*WhittakerW(kappa -(1)/(2), mu -(1)/(2), z) = 0

(\[Kappa]- \[Mu]-Divide[1,2])*Sqrt[z]*WhittakerW[\[Kappa]-Divide[1,2], \[Mu]+Divide[1,2], z]+ 2*\[Mu]*WhittakerW[\[Kappa], \[Mu], z]-(\[Kappa]+ \[Mu]-Divide[1,2])*Sqrt[z]*WhittakerW[\[Kappa]-Divide[1,2], \[Mu]-Divide[1,2], z] == 0
Successful Failure - Successful [Tested: 300]
13.15.E13 ( κ + μ - 1 2 ) ⁢ z ⁢ W κ - 1 2 , μ - 1 2 ⁡ ( z ) - ( z + 2 ⁢ μ ) ⁢ W κ , μ ⁡ ( z ) + z ⁢ W κ + 1 2 , μ + 1 2 ⁡ ( z ) = 0 𝜅 𝜇 1 2 𝑧 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝜇 1 2 𝑧 𝑧 2 𝜇 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 𝑧 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝜇 1 2 𝑧 0 {\displaystyle{\displaystyle(\kappa+\mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1% }{2},\mu-\frac{1}{2}}\left(z\right)-(z+2\mu)W_{\kappa,\mu}\left(z\right)+\sqrt% {z}W_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0}}
(\kappa+\mu-\tfrac{1}{2})\sqrt{z}\WhittakerconfhyperW{\kappa-\frac{1}{2}}{\mu-\frac{1}{2}}@{z}-(z+2\mu)\WhittakerconfhyperW{\kappa}{\mu}@{z}+\sqrt{z}\WhittakerconfhyperW{\kappa+\frac{1}{2}}{\mu+\frac{1}{2}}@{z} = 0		 

(kappa + mu -(1)/(2))*sqrt(z)*WhittakerW(kappa -(1)/(2), mu -(1)/(2), z)-(z + 2*mu)*WhittakerW(kappa, mu, z)+sqrt(z)*WhittakerW(kappa +(1)/(2), mu +(1)/(2), z) = 0

(\[Kappa]+ \[Mu]-Divide[1,2])*Sqrt[z]*WhittakerW[\[Kappa]-Divide[1,2], \[Mu]-Divide[1,2], z]-(z + 2*\[Mu])*WhittakerW[\[Kappa], \[Mu], z]+Sqrt[z]*WhittakerW[\[Kappa]+Divide[1,2], \[Mu]+Divide[1,2], z] == 0
Successful Failure - Successful [Tested: 300]
13.15.E14 ( κ - μ - 1 2 ) ⁢ z ⁢ W κ - 1 2 , μ + 1 2 ⁡ ( z ) - ( z - 2 ⁢ μ ) ⁢ W κ , μ ⁡ ( z ) + z ⁢ W κ + 1 2 , μ - 1 2 ⁡ ( z ) = 0 𝜅 𝜇 1 2 𝑧 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝜇 1 2 𝑧 𝑧 2 𝜇 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 𝑧 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝜇 1 2 𝑧 0 {\displaystyle{\displaystyle(\kappa-\mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1% }{2},\mu+\frac{1}{2}}\left(z\right)-(z-2\mu)W_{\kappa,\mu}\left(z\right)+\sqrt% {z}W_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)=0}}
(\kappa-\mu-\tfrac{1}{2})\sqrt{z}\WhittakerconfhyperW{\kappa-\frac{1}{2}}{\mu+\frac{1}{2}}@{z}-(z-2\mu)\WhittakerconfhyperW{\kappa}{\mu}@{z}+\sqrt{z}\WhittakerconfhyperW{\kappa+\frac{1}{2}}{\mu-\frac{1}{2}}@{z} = 0		 

(kappa - mu -(1)/(2))*sqrt(z)*WhittakerW(kappa -(1)/(2), mu +(1)/(2), z)-(z - 2*mu)*WhittakerW(kappa, mu, z)+sqrt(z)*WhittakerW(kappa +(1)/(2), mu -(1)/(2), z) = 0

(\[Kappa]- \[Mu]-Divide[1,2])*Sqrt[z]*WhittakerW[\[Kappa]-Divide[1,2], \[Mu]+Divide[1,2], z]-(z - 2*\[Mu])*WhittakerW[\[Kappa], \[Mu], z]+Sqrt[z]*WhittakerW[\[Kappa]+Divide[1,2], \[Mu]-Divide[1,2], z] == 0
Successful Failure - Successful [Tested: 300]
13.15.E15 d n d z n ⁡ ( e 1 2 ⁢ z ⁢ z μ - 1 2 ⁢ M κ , μ ⁡ ( z ) ) = ( - 1 ) n ⁢ ( - 2 ⁢ μ ) n ⁢ e 1 2 ⁢ z ⁢ z μ - 1 2 ⁢ ( n + 1 ) ⁢ M κ - 1 2 ⁢ n , μ - 1 2 ⁢ n ⁡ ( z ) derivative 𝑧 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 superscript 1 𝑛 Pochhammer 2 𝜇 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 𝑛 1 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝑛 𝜇 1 2 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {\frac{1}{2}z}z^{\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}{% \left(-2\mu\right)_{n}}e^{\frac{1}{2}z}z^{\mu-\frac{1}{2}(n+1)}M_{\kappa-\frac% {1}{2}n,\mu-\frac{1}{2}n}\left(z\right)}}
\deriv[n]{}{z}\left(e^{\frac{1}{2}z}z^{\mu-\frac{1}{2}}\WhittakerconfhyperM{\kappa}{\mu}@{z}\right) = (-1)^{n}\Pochhammersym{-2\mu}{n}e^{\frac{1}{2}z}z^{\mu-\frac{1}{2}(n+1)}\WhittakerconfhyperM{\kappa-\frac{1}{2}n}{\mu-\frac{1}{2}n}@{z}		 

diff(exp((1)/(2)*z)*(z)^(mu -(1)/(2))* WhittakerM(kappa, mu, z), [z$(n)]) = (- 1)^(n)* pochhammer(- 2*mu, n)*exp((1)/(2)*z)*(z)^(mu -(1)/(2)*(n + 1))* WhittakerM(kappa -(1)/(2)*n, mu -(1)/(2)*n, z)

D[Exp[Divide[1,2]*z]*(z)^(\[Mu]-Divide[1,2])* WhittakerM[\[Kappa], \[Mu], z], {z, n}] == (- 1)^(n)* Pochhammer[- 2*\[Mu], n]*Exp[Divide[1,2]*z]*(z)^(\[Mu]-Divide[1,2]*(n + 1))* WhittakerM[\[Kappa]-Divide[1,2]*n, \[Mu]-Divide[1,2]*n, z]
Failure Failure Skipped - Because timed out
Failed [210 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}


Result: DirectedInfinity[]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}

... skip entries to safe data
13.15.E16 d n d z n ⁡ ( e 1 2 ⁢ z ⁢ z - μ - 1 2 ⁢ M κ , μ ⁡ ( z ) ) = ( 1 2 + μ - κ ) n ( 1 + 2 ⁢ μ ) n ⁢ e 1 2 ⁢ z ⁢ z - μ - 1 2 ⁢ ( n + 1 ) ⁢ M κ - 1 2 ⁢ n , μ + 1 2 ⁢ n ⁡ ( z ) derivative 𝑧 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 Pochhammer 1 2 𝜇 𝜅 𝑛 Pochhammer 1 2 𝜇 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 𝑛 1 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝑛 𝜇 1 2 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {\frac{1}{2}z}z^{-\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)=\frac{{% \left(\frac{1}{2}+\mu-\kappa\right)_{n}}}{{\left(1+2\mu\right)_{n}}}e^{\frac{1% }{2}z}z^{-\mu-\frac{1}{2}(n+1)}M_{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}\left(z% \right)}}
\deriv[n]{}{z}\left(e^{\frac{1}{2}z}z^{-\mu-\frac{1}{2}}\WhittakerconfhyperM{\kappa}{\mu}@{z}\right) = \frac{\Pochhammersym{\frac{1}{2}+\mu-\kappa}{n}}{\Pochhammersym{1+2\mu}{n}}e^{\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}\WhittakerconfhyperM{\kappa-\frac{1}{2}n}{\mu+\frac{1}{2}n}@{z}		 

diff(exp((1)/(2)*z)*(z)^(- mu -(1)/(2))* WhittakerM(kappa, mu, z), [z$(n)]) = (pochhammer((1)/(2)+ mu - kappa, n))/(pochhammer(1 + 2*mu, n))*exp((1)/(2)*z)*(z)^(- mu -(1)/(2)*(n + 1))* WhittakerM(kappa -(1)/(2)*n, mu +(1)/(2)*n, z)

D[Exp[Divide[1,2]*z]*(z)^(- \[Mu]-Divide[1,2])* WhittakerM[\[Kappa], \[Mu], z], {z, n}] == Divide[Pochhammer[Divide[1,2]+ \[Mu]- \[Kappa], n],Pochhammer[1 + 2*\[Mu], n]]*Exp[Divide[1,2]*z]*(z)^(- \[Mu]-Divide[1,2]*(n + 1))* WhittakerM[\[Kappa]-Divide[1,2]*n, \[Mu]+Divide[1,2]*n, z]
Failure Failure Skipped - Because timed out
Failed [210 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}


Result: DirectedInfinity[]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}

... skip entries to safe data
13.15.E17 ( z ⁢ d d z ⁡ z ) n ⁢ ( e 1 2 ⁢ z ⁢ z - κ - 1 ⁢ M κ , μ ⁡ ( z ) ) = ( 1 2 + μ - κ ) n ⁢ e 1 2 ⁢ z ⁢ z n - κ - 1 ⁢ M κ - n , μ ⁡ ( z ) superscript 𝑧 derivative 𝑧 𝑧 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜅 1 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 Pochhammer 1 2 𝜇 𝜅 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝑛 𝜅 1 Whittaker-confluent-hypergeometric-M 𝜅 𝑛 𝜇 𝑧 {\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(e^{\frac{1}{2}z}z^{-\kappa-1}M_{\kappa,\mu}\left(z\right)\right)={\left(% \tfrac{1}{2}+\mu-\kappa\right)_{n}}e^{\frac{1}{2}z}z^{n-\kappa-1}M_{\kappa-n,% \mu}\left(z\right)}}
\left(z\deriv{}{z}z\right)^{n}\left(e^{\frac{1}{2}z}z^{-\kappa-1}\WhittakerconfhyperM{\kappa}{\mu}@{z}\right) = \Pochhammersym{\tfrac{1}{2}+\mu-\kappa}{n}e^{\frac{1}{2}z}z^{n-\kappa-1}\WhittakerconfhyperM{\kappa-n}{\mu}@{z}		 

(z*diff(z, z))^(n)*(exp((1)/(2)*z)*(z)^(- kappa - 1)* WhittakerM(kappa, mu, z)) = pochhammer((1)/(2)+ mu - kappa, n)*exp((1)/(2)*z)*(z)^(n - kappa - 1)* WhittakerM(kappa - n, mu, z)

(z*D[z, z])^(n)*(Exp[Divide[1,2]*z]*(z)^(- \[Kappa]- 1)* WhittakerM[\[Kappa], \[Mu], z]) == Pochhammer[Divide[1,2]+ \[Mu]- \[Kappa], n]*Exp[Divide[1,2]*z]*(z)^(n - \[Kappa]- 1)* WhittakerM[\[Kappa]- n, \[Mu], z]
Failure Failure
Failed [300 / 300]
Result: .3585110760+.454218427e-1*I
Test Values: {kappa = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -.1773224730-.5602797385*I
Test Values: {kappa = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.35851107533499493, 0.045421842889073805]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.1773224737195902, -0.560279739303586]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
13.15.E18 d n d z n ⁡ ( e - 1 2 ⁢ z ⁢ z μ - 1 2 ⁢ M κ , μ ⁡ ( z ) ) = ( - 1 ) n ⁢ ( - 2 ⁢ μ ) n ⁢ e - 1 2 ⁢ z ⁢ z μ - 1 2 ⁢ ( n + 1 ) ⁢ M κ + 1 2 ⁢ n , μ - 1 2 ⁢ n ⁡ ( z ) derivative 𝑧 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 superscript 1 𝑛 Pochhammer 2 𝜇 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 𝑛 1 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝑛 𝜇 1 2 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {-\frac{1}{2}z}z^{\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}% {\left(-2\mu\right)_{n}}e^{-\frac{1}{2}z}z^{\mu-\frac{1}{2}(n+1)}M_{\kappa+% \frac{1}{2}n,\mu-\frac{1}{2}n}\left(z\right)}}
\deriv[n]{}{z}\left(e^{-\frac{1}{2}z}z^{\mu-\frac{1}{2}}\WhittakerconfhyperM{\kappa}{\mu}@{z}\right) = (-1)^{n}\Pochhammersym{-2\mu}{n}e^{-\frac{1}{2}z}z^{\mu-\frac{1}{2}(n+1)}\WhittakerconfhyperM{\kappa+\frac{1}{2}n}{\mu-\frac{1}{2}n}@{z}		 

diff(exp(-(1)/(2)*z)*(z)^(mu -(1)/(2))* WhittakerM(kappa, mu, z), [z$(n)]) = (- 1)^(n)* pochhammer(- 2*mu, n)*exp(-(1)/(2)*z)*(z)^(mu -(1)/(2)*(n + 1))* WhittakerM(kappa +(1)/(2)*n, mu -(1)/(2)*n, z)

D[Exp[-Divide[1,2]*z]*(z)^(\[Mu]-Divide[1,2])* WhittakerM[\[Kappa], \[Mu], z], {z, n}] == (- 1)^(n)* Pochhammer[- 2*\[Mu], n]*Exp[-Divide[1,2]*z]*(z)^(\[Mu]-Divide[1,2]*(n + 1))* WhittakerM[\[Kappa]+Divide[1,2]*n, \[Mu]-Divide[1,2]*n, z]
Failure Failure Skipped - Because timed out
Failed [210 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}


Result: DirectedInfinity[]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}

... skip entries to safe data
13.15.E19 d n d z n ⁡ ( e - 1 2 ⁢ z ⁢ z - μ - 1 2 ⁢ M κ , μ ⁡ ( z ) ) = ( - 1 ) n ⁢ ( 1 2 + μ + κ ) n ( 1 + 2 ⁢ μ ) n ⁢ e - 1 2 ⁢ z ⁢ z - μ - 1 2 ⁢ ( n + 1 ) ⁢ M κ + 1 2 ⁢ n , μ + 1 2 ⁢ n ⁡ ( z ) derivative 𝑧 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 superscript 1 𝑛 Pochhammer 1 2 𝜇 𝜅 𝑛 Pochhammer 1 2 𝜇 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 𝑛 1 Whittaker-confluent-hypergeometric-M 𝜅 1 2 𝑛 𝜇 1 2 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {-\frac{1}{2}z}z^{-\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)=(-1)^{n% }\frac{{\left(\frac{1}{2}+\mu+\kappa\right)_{n}}}{{\left(1+2\mu\right)_{n}}}e^% {-\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}\*M_{\kappa+\frac{1}{2}n,\mu+\frac{1}{% 2}n}\left(z\right)}}
\deriv[n]{}{z}\left(e^{-\frac{1}{2}z}z^{-\mu-\frac{1}{2}}\WhittakerconfhyperM{\kappa}{\mu}@{z}\right) = (-1)^{n}\frac{\Pochhammersym{\frac{1}{2}+\mu+\kappa}{n}}{\Pochhammersym{1+2\mu}{n}}e^{-\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}\*\WhittakerconfhyperM{\kappa+\frac{1}{2}n}{\mu+\frac{1}{2}n}@{z}		 

diff(exp(-(1)/(2)*z)*(z)^(- mu -(1)/(2))* WhittakerM(kappa, mu, z), [z$(n)]) = (- 1)^(n)*(pochhammer((1)/(2)+ mu + kappa, n))/(pochhammer(1 + 2*mu, n))*exp(-(1)/(2)*z)*(z)^(- mu -(1)/(2)*(n + 1))* WhittakerM(kappa +(1)/(2)*n, mu +(1)/(2)*n, z)

D[Exp[-Divide[1,2]*z]*(z)^(- \[Mu]-Divide[1,2])* WhittakerM[\[Kappa], \[Mu], z], {z, n}] == (- 1)^(n)*Divide[Pochhammer[Divide[1,2]+ \[Mu]+ \[Kappa], n],Pochhammer[1 + 2*\[Mu], n]]*Exp[-Divide[1,2]*z]*(z)^(- \[Mu]-Divide[1,2]*(n + 1))* WhittakerM[\[Kappa]+Divide[1,2]*n, \[Mu]+Divide[1,2]*n, z]
Failure Failure Skipped - Because timed out
Failed [210 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}


Result: DirectedInfinity[]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}

... skip entries to safe data
13.15.E20 ( z ⁢ d d z ⁡ z ) n ⁢ ( e - 1 2 ⁢ z ⁢ z κ - 1 ⁢ M κ , μ ⁡ ( z ) ) = ( 1 2 + μ + κ ) n ⁢ e - 1 2 ⁢ z ⁢ z κ + n - 1 ⁢ M κ + n , μ ⁡ ( z ) superscript 𝑧 derivative 𝑧 𝑧 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜅 1 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 Pochhammer 1 2 𝜇 𝜅 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜅 𝑛 1 Whittaker-confluent-hypergeometric-M 𝜅 𝑛 𝜇 𝑧 {\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(e^{-\frac{1}{2}z}z^{\kappa-1}M_{\kappa,\mu}\left(z\right)\right)={\left(% \tfrac{1}{2}+\mu+\kappa\right)_{n}}e^{-\frac{1}{2}z}z^{\kappa+n-1}\*M_{\kappa+% n,\mu}\left(z\right)}}
\left(z\deriv{}{z}z\right)^{n}\left(e^{-\frac{1}{2}z}z^{\kappa-1}\WhittakerconfhyperM{\kappa}{\mu}@{z}\right) = \Pochhammersym{\tfrac{1}{2}+\mu+\kappa}{n}e^{-\frac{1}{2}z}z^{\kappa+n-1}\*\WhittakerconfhyperM{\kappa+n}{\mu}@{z}		 

(z*diff(z, z))^(n)*(exp(-(1)/(2)*z)*(z)^(kappa - 1)* WhittakerM(kappa, mu, z)) = pochhammer((1)/(2)+ mu + kappa, n)*exp(-(1)/(2)*z)*(z)^(kappa + n - 1)* WhittakerM(kappa + n, mu, z)

(z*D[z, z])^(n)*(Exp[-Divide[1,2]*z]*(z)^(\[Kappa]- 1)* WhittakerM[\[Kappa], \[Mu], z]) == Pochhammer[Divide[1,2]+ \[Mu]+ \[Kappa], n]*Exp[-Divide[1,2]*z]*(z)^(\[Kappa]+ n - 1)* WhittakerM[\[Kappa]+ n, \[Mu], z]
Failure Failure
Failed [300 / 300]
Result: -.221105652e-1-.2375136134*I
Test Values: {kappa = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 1}


Result: .3191037849-.7838469226*I
Test Values: {kappa = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.02211056528532032, -0.23751361332195844]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.31910378464483535, -0.7838469223028885]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
13.15.E21 d n d z n ⁡ ( e 1 2 ⁢ z ⁢ z - μ - 1 2 ⁢ W κ , μ ⁡ ( z ) ) = ( - 1 ) n ⁢ ( 1 2 + μ - κ ) n ⁢ e 1 2 ⁢ z ⁢ z - μ - 1 2 ⁢ ( n + 1 ) ⁢ W κ - 1 2 ⁢ n , μ + 1 2 ⁢ n ⁡ ( z ) derivative 𝑧 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 1 𝑛 Pochhammer 1 2 𝜇 𝜅 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 𝑛 1 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝑛 𝜇 1 2 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {\frac{1}{2}z}z^{-\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}% {\left(\tfrac{1}{2}+\mu-\kappa\right)_{n}}e^{\frac{1}{2}z}z^{-\mu-\frac{1}{2}(% n+1)}\*W_{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}\left(z\right)}}
\deriv[n]{}{z}\left(e^{\frac{1}{2}z}z^{-\mu-\frac{1}{2}}\WhittakerconfhyperW{\kappa}{\mu}@{z}\right) = (-1)^{n}\Pochhammersym{\tfrac{1}{2}+\mu-\kappa}{n}e^{\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}\*\WhittakerconfhyperW{\kappa-\frac{1}{2}n}{\mu+\frac{1}{2}n}@{z}		 

diff(exp((1)/(2)*z)*(z)^(- mu -(1)/(2))* WhittakerW(kappa, mu, z), [z$(n)]) = (- 1)^(n)* pochhammer((1)/(2)+ mu - kappa, n)*exp((1)/(2)*z)*(z)^(- mu -(1)/(2)*(n + 1))* WhittakerW(kappa -(1)/(2)*n, mu +(1)/(2)*n, z)

D[Exp[Divide[1,2]*z]*(z)^(- \[Mu]-Divide[1,2])* WhittakerW[\[Kappa], \[Mu], z], {z, n}] == (- 1)^(n)* Pochhammer[Divide[1,2]+ \[Mu]- \[Kappa], n]*Exp[Divide[1,2]*z]*(z)^(- \[Mu]-Divide[1,2]*(n + 1))* WhittakerW[\[Kappa]-Divide[1,2]*n, \[Mu]+Divide[1,2]*n, z]
Failure Failure Skipped - Because timed out
Failed [192 / 300]
Result: Plus[Complex[-2.7003415598242593, -2.135803172450526], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, -1.5]], []], Times[2, Plus[1, ], Plus[1, , Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, -1.5]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-1, 2], Times[-1, -1.5]]], WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Rational[1, 2], Power[E, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-3, 2],<syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.8050385267502765, -1.4779965316225212], Times[2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, -1.5]], []], Times[2, Plus[1, ], Plus[1, , Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, -1.5]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-1, 2], Times[-1, -1.5]]], WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Rational[1, 2], Power[E, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-3, 2], Times[-1, -1.5]]], Plus[Times[-1, WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, -1.5, WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, WhittakerW[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}

... skip entries to safe data
13.15.E22 d n d z n ⁡ ( e 1 2 ⁢ z ⁢ z μ - 1 2 ⁢ W κ , μ ⁡ ( z ) ) = ( - 1 ) n ⁢ ( 1 2 - μ - κ ) n ⁢ e 1 2 ⁢ z ⁢ z μ - 1 2 ⁢ ( n + 1 ) ⁢ W κ - 1 2 ⁢ n , μ - 1 2 ⁢ n ⁡ ( z ) derivative 𝑧 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 1 𝑛 Pochhammer 1 2 𝜇 𝜅 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 𝑛 1 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝑛 𝜇 1 2 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {\frac{1}{2}z}z^{\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}{% \left(\tfrac{1}{2}-\mu-\kappa\right)_{n}}e^{\frac{1}{2}z}z^{\mu-\frac{1}{2}(n+% 1)}\*W_{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}\left(z\right)}}
\deriv[n]{}{z}\left(e^{\frac{1}{2}z}z^{\mu-\frac{1}{2}}\WhittakerconfhyperW{\kappa}{\mu}@{z}\right) = (-1)^{n}\Pochhammersym{\tfrac{1}{2}-\mu-\kappa}{n}e^{\frac{1}{2}z}z^{\mu-\frac{1}{2}(n+1)}\*\WhittakerconfhyperW{\kappa-\frac{1}{2}n}{\mu-\frac{1}{2}n}@{z}		 

diff(exp((1)/(2)*z)*(z)^(mu -(1)/(2))* WhittakerW(kappa, mu, z), [z$(n)]) = (- 1)^(n)* pochhammer((1)/(2)- mu - kappa, n)*exp((1)/(2)*z)*(z)^(mu -(1)/(2)*(n + 1))* WhittakerW(kappa -(1)/(2)*n, mu -(1)/(2)*n, z)

D[Exp[Divide[1,2]*z]*(z)^(\[Mu]-Divide[1,2])* WhittakerW[\[Kappa], \[Mu], z], {z, n}] == (- 1)^(n)* Pochhammer[Divide[1,2]- \[Mu]- \[Kappa], n]*Exp[Divide[1,2]*z]*(z)^(\[Mu]-Divide[1,2]*(n + 1))* WhittakerW[\[Kappa]-Divide[1,2]*n, \[Mu]-Divide[1,2]*n, z]
Failure Failure Skipped - Because timed out
Failed [192 / 300]
Result: Plus[Complex[-3.1506729340368813, -11.027912097410434], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, -1.5]], []], Times[2, Plus[1, ], Plus[1, , Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, -1.5]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-1, 2], -1.5]], WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Rational[1, 2], Power[E, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-3, 2], -1.5]], P<syntaxhighlight lang=mathematica>Result: Plus[Complex[32.491056912593166, 25.892568815057246], Times[2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, -1.5]], []], Times[2, Plus[1, ], Plus[1, , Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, -1.5]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-1, 2], -1.5]], WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Rational[1, 2], Power[E, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-3, 2], -1.5]], Plus[Times[-1, WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, -1.5, WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, WhittakerW[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}

... skip entries to safe data
13.15.E23 ( z ⁢ d d z ⁡ z ) n ⁢ ( e 1 2 ⁢ z ⁢ z - κ - 1 ⁢ W κ , μ ⁡ ( z ) ) = ( 1 2 + μ - κ ) n ⁢ ( 1 2 - μ - κ ) n ⁢ e 1 2 ⁢ z ⁢ z n - κ - 1 ⁢ W κ - n , μ ⁡ ( z ) superscript 𝑧 derivative 𝑧 𝑧 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜅 1 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 Pochhammer 1 2 𝜇 𝜅 𝑛 Pochhammer 1 2 𝜇 𝜅 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝑛 𝜅 1 Whittaker-confluent-hypergeometric-W 𝜅 𝑛 𝜇 𝑧 {\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(e^{\frac{1}{2}z}z^{-\kappa-1}W_{\kappa,\mu}\left(z\right)\right)={\left(% \tfrac{1}{2}+\mu-\kappa\right)_{n}}{\left(\tfrac{1}{2}-\mu-\kappa\right)_{n}}e% ^{\frac{1}{2}z}z^{n-\kappa-1}W_{\kappa-n,\mu}\left(z\right)}}
\left(z\deriv{}{z}z\right)^{n}\left(e^{\frac{1}{2}z}z^{-\kappa-1}\WhittakerconfhyperW{\kappa}{\mu}@{z}\right) = \Pochhammersym{\tfrac{1}{2}+\mu-\kappa}{n}\Pochhammersym{\tfrac{1}{2}-\mu-\kappa}{n}e^{\frac{1}{2}z}z^{n-\kappa-1}\WhittakerconfhyperW{\kappa-n}{\mu}@{z}		 

(z*diff(z, z))^(n)*(exp((1)/(2)*z)*(z)^(- kappa - 1)* WhittakerW(kappa, mu, z)) = pochhammer((1)/(2)+ mu - kappa, n)*pochhammer((1)/(2)- mu - kappa, n)*exp((1)/(2)*z)*(z)^(n - kappa - 1)* WhittakerW(kappa - n, mu, z)

(z*D[z, z])^(n)*(Exp[Divide[1,2]*z]*(z)^(- \[Kappa]- 1)* WhittakerW[\[Kappa], \[Mu], z]) == Pochhammer[Divide[1,2]+ \[Mu]- \[Kappa], n]*Pochhammer[Divide[1,2]- \[Mu]- \[Kappa], n]*Exp[Divide[1,2]*z]*(z)^(n - \[Kappa]- 1)* WhittakerW[\[Kappa]- n, \[Mu], z]
 Failure Failure
Failed [300 / 300]
Result: 2.468472246+1.546856952*I
Test Values: {kappa = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 1}

Result: 1.885026449+1.175257266*I
Test Values: {kappa = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[2.4684722428383408, 1.546856950437671]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[1.8850264475606715, 1.175257265810332]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
13.15.E24 d n d z n ⁡ ( e - 1 2 ⁢ z ⁢ z - μ - 1 2 ⁢ W κ , μ ⁡ ( z ) ) = ( - 1 ) n ⁢ e - 1 2 ⁢ z ⁢ z - μ - 1 2 ⁢ ( n + 1 ) ⁢ W κ + 1 2 ⁢ n , μ + 1 2 ⁢ n ⁡ ( z ) derivative 𝑧 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 1 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 𝑛 1 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝑛 𝜇 1 2 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {-\frac{1}{2}z}z^{-\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)=(-1)^{n% }e^{-\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}W_{\kappa+\frac{1}{2}n,\mu+\frac{1}% {2}n}\left(z\right)}}
\deriv[n]{}{z}\left(e^{-\frac{1}{2}z}z^{-\mu-\frac{1}{2}}\WhittakerconfhyperW{\kappa}{\mu}@{z}\right) = (-1)^{n}e^{-\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}\WhittakerconfhyperW{\kappa+\frac{1}{2}n}{\mu+\frac{1}{2}n}@{z}		 

diff(exp(-(1)/(2)*z)*(z)^(- mu -(1)/(2))* WhittakerW(kappa, mu, z), [z$(n)]) = (- 1)^(n)* exp(-(1)/(2)*z)*(z)^(- mu -(1)/(2)*(n + 1))* WhittakerW(kappa +(1)/(2)*n, mu +(1)/(2)*n, z)
 
D[Exp[-Divide[1,2]*z]*(z)^(- \[Mu]-Divide[1,2])* WhittakerW[\[Kappa], \[Mu], z], {z, n}] == (- 1)^(n)* Exp[-Divide[1,2]*z]*(z)^(- \[Mu]-Divide[1,2]*(n + 1))* WhittakerW[\[Kappa]+Divide[1,2]*n, \[Mu]+Divide[1,2]*n, z]
 Failure Failure Skipped - Because timed out
Failed [192 / 300]
Result: Plus[Complex[0.5001431347806349, -0.3406797899835502], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, -1.5]], []], Times[2, Plus[1, ], Plus[1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[2, -1.5]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-1, 2], Times[-1, -1.5]]], WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Rational[-1, 2], Power[E, Times[Rational[-1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-3, 2], Times[-1, <syntaxhighlight lang=mathematica>Result: Plus[Complex[0.332118444019996, 0.20129597063218943], Times[2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, -1.5]], []], Times[2, Plus[1, ], Plus[1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[2, -1.5]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-1, 2], Times[-1, -1.5]]], WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Rational[-1, 2], Power[E, Times[Rational[-1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-3, 2], Times[-1, -1.5]]], Plus[WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, -1.5, WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, WhittakerW[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}

... skip entries to safe data
13.15.E25 d n d z n ⁡ ( e - 1 2 ⁢ z ⁢ z μ - 1 2 ⁢ W κ , μ ⁡ ( z ) ) = ( - 1 ) n ⁢ e - 1 2 ⁢ z ⁢ z μ - 1 2 ⁢ ( n + 1 ) ⁢ W κ + 1 2 ⁢ n , μ - 1 2 ⁢ n ⁡ ( z ) derivative 𝑧 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 1 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜇 1 2 𝑛 1 Whittaker-confluent-hypergeometric-W 𝜅 1 2 𝑛 𝜇 1 2 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {-\frac{1}{2}z}z^{\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}% e^{-\frac{1}{2}z}z^{\mu-\frac{1}{2}(n+1)}W_{\kappa+\frac{1}{2}n,\mu-\frac{1}{2% }n}\left(z\right)}}
\deriv[n]{}{z}\left(e^{-\frac{1}{2}z}z^{\mu-\frac{1}{2}}\WhittakerconfhyperW{\kappa}{\mu}@{z}\right) = (-1)^{n}e^{-\frac{1}{2}z}z^{\mu-\frac{1}{2}(n+1)}\WhittakerconfhyperW{\kappa+\frac{1}{2}n}{\mu-\frac{1}{2}n}@{z}		 

diff(exp(-(1)/(2)*z)*(z)^(mu -(1)/(2))* WhittakerW(kappa, mu, z), [z$(n)]) = (- 1)^(n)* exp(-(1)/(2)*z)*(z)^(mu -(1)/(2)*(n + 1))* WhittakerW(kappa +(1)/(2)*n, mu -(1)/(2)*n, z)
 
D[Exp[-Divide[1,2]*z]*(z)^(\[Mu]-Divide[1,2])* WhittakerW[\[Kappa], \[Mu], z], {z, n}] == (- 1)^(n)* Exp[-Divide[1,2]*z]*(z)^(\[Mu]-Divide[1,2]*(n + 1))* WhittakerW[\[Kappa]+Divide[1,2]*n, \[Mu]-Divide[1,2]*n, z]
 Failure Failure Skipped - Because timed out
Failed [192 / 300]
Result: Plus[Complex[-3.483681927072143, -5.36298237509452], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, -1.5]], []], Times[2, Plus[1, ], Plus[1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[-2, -1.5]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-1, 2], -1.5]], WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Rational[1, 2], Power[E, Times[Rational[-1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-3, 2], -1.5]], Plus[Times[-1,<syntaxhighlight lang=mathematica>Result: Plus[Complex[24.085306751162083, 11.80402713986923], Times[2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, -1.5]], []], Times[2, Plus[1, ], Plus[1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[-2, -1.5]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-1, 2], -1.5]], WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Rational[1, 2], Power[E, Times[Rational[-1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[Rational[-3, 2], -1.5]], Plus[Times[-1, WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, -1.5, WhittakerW[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, WhittakerW[Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5]}

... skip entries to safe data
13.15.E26 ( z ⁢ d d z ⁡ z ) n ⁢ ( e - 1 2 ⁢ z ⁢ z κ - 1 ⁢ W κ , μ ⁡ ( z ) ) = ( - 1 ) n ⁢ e - 1 2 ⁢ z ⁢ z κ + n - 1 ⁢ W κ + n , μ ⁡ ( z ) superscript 𝑧 derivative 𝑧 𝑧 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜅 1 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 1 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜅 𝑛 1 Whittaker-confluent-hypergeometric-W 𝜅 𝑛 𝜇 𝑧 {\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(e^{-\frac{1}{2}z}z^{\kappa-1}W_{\kappa,\mu}\left(z\right)\right)=(-1)^{n% }e^{-\frac{1}{2}z}z^{\kappa+n-1}W_{\kappa+n,\mu}\left(z\right)}}
\left(z\deriv{}{z}z\right)^{n}\left(e^{-\frac{1}{2}z}z^{\kappa-1}\WhittakerconfhyperW{\kappa}{\mu}@{z}\right) = (-1)^{n}e^{-\frac{1}{2}z}z^{\kappa+n-1}\WhittakerconfhyperW{\kappa+n}{\mu}@{z}		 

(z*diff(z, z))^(n)*(exp(-(1)/(2)*z)*(z)^(kappa - 1)* WhittakerW(kappa, mu, z)) = (- 1)^(n)* exp(-(1)/(2)*z)*(z)^(kappa + n - 1)* WhittakerW(kappa + n, mu, z)
 
(z*D[z, z])^(n)*(Exp[-Divide[1,2]*z]*(z)^(\[Kappa]- 1)* WhittakerW[\[Kappa], \[Mu], z]) == (- 1)^(n)* Exp[-Divide[1,2]*z]*(z)^(\[Kappa]+ n - 1)* WhittakerW[\[Kappa]+ n, \[Mu], z]
 Failure Failure
Failed [300 / 300]
Result: .2623016537+.1488103823*I
Test Values: {kappa = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 1}

Result: .1952811915+.4851862634*I
Test Values: {kappa = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.26230165366126323, 0.1488103820981603]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.1952811914323972, 0.4851862632402242]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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