Confluent Hypergeometric Functions - 13.18 Relations to Other Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
13.18.E1 M 0 , 1 2 ⁡ ( 2 ⁢ z ) = 2 ⁢ sinh ⁡ z Whittaker-confluent-hypergeometric-M 0 1 2 2 𝑧 2 𝑧 {\displaystyle{\displaystyle M_{0,\frac{1}{2}}\left(2z\right)=2\sinh z}}
\WhittakerconfhyperM{0}{\frac{1}{2}}@{2z} = 2\sinh@@{z}		 

WhittakerM(0, (1)/(2), 2*z) = 2*sinh(z)

WhittakerM[0, Divide[1,2], 2*z] == 2*Sinh[z]
Successful Successful - Successful [Tested: 7]
13.18.E2 M κ , κ - 1 2 ⁡ ( z ) = W κ , κ - 1 2 ⁡ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜅 1 2 𝑧 Whittaker-confluent-hypergeometric-W 𝜅 𝜅 1 2 𝑧 {\displaystyle{\displaystyle M_{\kappa,\kappa-\frac{1}{2}}\left(z\right)=W_{% \kappa,\kappa-\frac{1}{2}}\left(z\right)}}
\WhittakerconfhyperM{\kappa}{\kappa-\frac{1}{2}}@{z} = \WhittakerconfhyperW{\kappa}{\kappa-\frac{1}{2}}@{z}		 

WhittakerM(kappa, kappa -(1)/(2), z) = WhittakerW(kappa, kappa -(1)/(2), z)

WhittakerM[\[Kappa], \[Kappa]-Divide[1,2], z] == WhittakerW[\[Kappa], \[Kappa]-Divide[1,2], z]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 70]
13.18.E2 W κ , κ - 1 2 ⁡ ( z ) = W κ , - κ + 1 2 ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜅 1 2 𝑧 Whittaker-confluent-hypergeometric-W 𝜅 𝜅 1 2 𝑧 {\displaystyle{\displaystyle W_{\kappa,\kappa-\frac{1}{2}}\left(z\right)=W_{% \kappa,-\kappa+\frac{1}{2}}\left(z\right)}}
\WhittakerconfhyperW{\kappa}{\kappa-\frac{1}{2}}@{z} = \WhittakerconfhyperW{\kappa}{-\kappa+\frac{1}{2}}@{z}		 

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

WhittakerW[\[Kappa], \[Kappa]-Divide[1,2], z] == WhittakerW[\[Kappa], - \[Kappa]+Divide[1,2], z]
Failure Successful Successful [Tested: 70] Successful [Tested: 70]
13.18.E2 W κ , - κ + 1 2 ⁡ ( z ) = e - 1 2 ⁢ z ⁢ z κ Whittaker-confluent-hypergeometric-W 𝜅 𝜅 1 2 𝑧 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜅 {\displaystyle{\displaystyle W_{\kappa,-\kappa+\frac{1}{2}}\left(z\right)=e^{-% \frac{1}{2}z}z^{\kappa}}}
\WhittakerconfhyperW{\kappa}{-\kappa+\frac{1}{2}}@{z} = e^{-\frac{1}{2}z}z^{\kappa}		 

WhittakerW(kappa, - kappa +(1)/(2), z) = exp(-(1)/(2)*z)*(z)^(kappa)

WhittakerW[\[Kappa], - \[Kappa]+Divide[1,2], z] == Exp[-Divide[1,2]*z]*(z)^\[Kappa]
Failure Successful Successful [Tested: 70] Successful [Tested: 70]
13.18.E3 M κ , - κ - 1 2 ⁡ ( z ) = e 1 2 ⁢ z ⁢ z - κ Whittaker-confluent-hypergeometric-M 𝜅 𝜅 1 2 𝑧 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜅 {\displaystyle{\displaystyle M_{\kappa,-\kappa-\frac{1}{2}}\left(z\right)=e^{% \frac{1}{2}z}z^{-\kappa}}}
\WhittakerconfhyperM{\kappa}{-\kappa-\frac{1}{2}}@{z} = e^{\frac{1}{2}z}z^{-\kappa}		 

WhittakerM(kappa, - kappa -(1)/(2), z) = exp((1)/(2)*z)*(z)^(- kappa)

WhittakerM[\[Kappa], - \[Kappa]-Divide[1,2], z] == Exp[Divide[1,2]*z]*(z)^(- \[Kappa])
Successful Successful -
Failed [20 / 70]
Result: Complex[-0.012581208495203278, -0.029801099144953658]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, 1.5]}


Result: Complex[-0.32783156414330006, -0.2917810845255237]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, 0.5]}

... skip entries to safe data
13.18.E4 M μ - 1 2 , μ ⁡ ( z ) = 2 ⁢ μ ⁢ e 1 2 ⁢ z ⁢ z 1 2 - μ ⁢ γ ⁡ ( 2 ⁢ μ , z ) Whittaker-confluent-hypergeometric-M 𝜇 1 2 𝜇 𝑧 2 𝜇 superscript 𝑒 1 2 𝑧 superscript 𝑧 1 2 𝜇 incomplete-gamma 2 𝜇 𝑧 {\displaystyle{\displaystyle M_{\mu-\frac{1}{2},\mu}\left(z\right)=2\mu e^{% \frac{1}{2}z}z^{\frac{1}{2}-\mu}\gamma\left(2\mu,z\right)}}
\WhittakerconfhyperM{\mu-\frac{1}{2}}{\mu}@{z} = 2\mu e^{\frac{1}{2}z}z^{\frac{1}{2}-\mu}\incgamma@{2\mu}{z}		 ℜ ⁡ ( 2 ⁢ μ ) > 0 2 𝜇 0 {\displaystyle{\displaystyle\Re(2\mu)>0}} 
WhittakerM(mu -(1)/(2), mu, z) = 2*mu*exp((1)/(2)*z)*(z)^((1)/(2)- mu)* GAMMA(2*mu)-GAMMA(2*mu, z)

WhittakerM[\[Mu]-Divide[1,2], \[Mu], z] == 2*\[Mu]*Exp[Divide[1,2]*z]*(z)^(Divide[1,2]- \[Mu])* Gamma[2*\[Mu], 0, z]
Failure Successful
Failed [35 / 35]
Result: -.5507089801-1.429327526*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -2.178955063-1.073512810*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Successful [Tested: 35]
13.18.E5 W μ - 1 2 , μ ⁡ ( z ) = e 1 2 ⁢ z ⁢ z 1 2 - μ ⁢ Γ ⁡ ( 2 ⁢ μ , z ) Whittaker-confluent-hypergeometric-W 𝜇 1 2 𝜇 𝑧 superscript 𝑒 1 2 𝑧 superscript 𝑧 1 2 𝜇 incomplete-Gamma 2 𝜇 𝑧 {\displaystyle{\displaystyle W_{\mu-\frac{1}{2},\mu}\left(z\right)=e^{\frac{1}% {2}z}z^{\frac{1}{2}-\mu}\Gamma\left(2\mu,z\right)}}
\WhittakerconfhyperW{\mu-\frac{1}{2}}{\mu}@{z} = e^{\frac{1}{2}z}z^{\frac{1}{2}-\mu}\incGamma@{2\mu}{z}		 

WhittakerW(mu -(1)/(2), mu, z) = exp((1)/(2)*z)*(z)^((1)/(2)- mu)* GAMMA(2*mu, z)

WhittakerW[\[Mu]-Divide[1,2], \[Mu], z] == Exp[Divide[1,2]*z]*(z)^(Divide[1,2]- \[Mu])* Gamma[2*\[Mu], z]
Successful Successful - Successful [Tested: 70]
13.18.E6 M - 1 4 , 1 4 ⁡ ( z 2 ) = 1 2 ⁢ e 1 2 ⁢ z 2 ⁢ π ⁢ z ⁢ erf ⁡ ( z ) Whittaker-confluent-hypergeometric-M 1 4 1 4 superscript 𝑧 2 1 2 superscript 𝑒 1 2 superscript 𝑧 2 𝜋 𝑧 error-function 𝑧 {\displaystyle{\displaystyle M_{-\frac{1}{4},\frac{1}{4}}\left(z^{2}\right)=% \tfrac{1}{2}e^{\frac{1}{2}z^{2}}\sqrt{\pi z}\operatorname{erf}\left(z\right)}}
\WhittakerconfhyperM{-\frac{1}{4}}{\frac{1}{4}}@{z^{2}} = \tfrac{1}{2}e^{\frac{1}{2}z^{2}}\sqrt{\pi z}\erf@{z}		 

WhittakerM(-(1)/(4), (1)/(4), (z)^(2)) = (1)/(2)*exp((1)/(2)*(z)^(2))*sqrt(Pi*z)*erf(z)

WhittakerM[-Divide[1,4], Divide[1,4], (z)^(2)] == Divide[1,2]*Exp[Divide[1,2]*(z)^(2)]*Sqrt[Pi*z]*Erf[z]
Failure Failure
Failed [2 / 7]
Result: .7978557562-.9869289445*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}


Result: 1.482664004+.2744150982*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

Failed [2 / 7]
Result: Complex[0.7978557563768727, -0.986928944338508]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[1.4826640039189691, 0.2744150979001404]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

13.18.E7 W - 1 4 , + 1 4 ⁡ ( z 2 ) = e 1 2 ⁢ z 2 ⁢ π ⁢ z ⁢ erfc ⁡ ( z ) Whittaker-confluent-hypergeometric-W 1 4 1 4 superscript 𝑧 2 superscript 𝑒 1 2 superscript 𝑧 2 𝜋 𝑧 complementary-error-function 𝑧 {\displaystyle{\displaystyle W_{-\frac{1}{4},+\frac{1}{4}}\left(z^{2}\right)=e% ^{\frac{1}{2}z^{2}}\sqrt{\pi z}\operatorname{erfc}\left(z\right)}}
\WhittakerconfhyperW{-\frac{1}{4}}{+\frac{1}{4}}@{z^{2}} = e^{\frac{1}{2}z^{2}}\sqrt{\pi z}\erfc@{z}		 

WhittakerW(-(1)/(4), +(1)/(4), (z)^(2)) = exp((1)/(2)*(z)^(2))*sqrt(Pi*z)*erfc(z)

WhittakerW[-Divide[1,4], +Divide[1,4], (z)^(2)] == Exp[Divide[1,2]*(z)^(2)]*Sqrt[Pi*z]*Erfc[z]
Failure Failure
Failed [2 / 7]
Result: -1.928317415+.502368653e-1*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}


Result: -2.674168572+2.656547698*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

Failed [2 / 7]
Result: Complex[-1.9283174154667808, 0.050236864945780724]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-2.6741685713500765, 2.656547698651725]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

13.18.E7 W - 1 4 , - 1 4 ⁡ ( z 2 ) = e 1 2 ⁢ z 2 ⁢ π ⁢ z ⁢ erfc ⁡ ( z ) Whittaker-confluent-hypergeometric-W 1 4 1 4 superscript 𝑧 2 superscript 𝑒 1 2 superscript 𝑧 2 𝜋 𝑧 complementary-error-function 𝑧 {\displaystyle{\displaystyle W_{-\frac{1}{4},-\frac{1}{4}}\left(z^{2}\right)=e% ^{\frac{1}{2}z^{2}}\sqrt{\pi z}\operatorname{erfc}\left(z\right)}}
\WhittakerconfhyperW{-\frac{1}{4}}{-\frac{1}{4}}@{z^{2}} = e^{\frac{1}{2}z^{2}}\sqrt{\pi z}\erfc@{z}		 

WhittakerW(-(1)/(4), -(1)/(4), (z)^(2)) = exp((1)/(2)*(z)^(2))*sqrt(Pi*z)*erfc(z)

WhittakerW[-Divide[1,4], -Divide[1,4], (z)^(2)] == Exp[Divide[1,2]*(z)^(2)]*Sqrt[Pi*z]*Erfc[z]
Failure Failure
Failed [2 / 7]
Result: -1.928317415+.502368653e-1*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}


Result: -2.674168572+2.656547698*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

Failed [2 / 7]
Result: Complex[-1.928317415466781, 0.05023686494578061]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-2.674168571350077, 2.6565476986517247]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

13.18.E8 M 0 , ν ⁡ ( 2 ⁢ z ) = 2 2 ⁢ ν + 1 2 ⁢ Γ ⁡ ( 1 + ν ) ⁢ z ⁢ I ν ⁡ ( z ) Whittaker-confluent-hypergeometric-M 0 𝜈 2 𝑧 superscript 2 2 𝜈 1 2 Euler-Gamma 1 𝜈 𝑧 modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle M_{0,\nu}\left(2z\right)=2^{2\nu+\frac{1}{2}}% \Gamma\left(1+\nu\right)\sqrt{z}I_{\nu}\left(z\right)}}
\WhittakerconfhyperM{0}{\nu}@{2z} = 2^{2\nu+\frac{1}{2}}\EulerGamma@{1+\nu}\sqrt{z}\modBesselI{\nu}@{z}		 ℜ ⁡ ( 1 + ν ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 1 𝜈 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(1+\nu)>0,\Re(\nu+k+1)>0}} 
WhittakerM(0, nu, 2*z) = (2)^(2*nu +(1)/(2))* GAMMA(1 + nu)*sqrt(z)*BesselI(nu, z)

WhittakerM[0, \[Nu], 2*z] == (2)^(2*\[Nu]+Divide[1,2])* Gamma[1 + \[Nu]]*Sqrt[z]*BesselI[\[Nu], z]
Successful Successful -
Failed [7 / 56]
Result: Complex[-0.8586367168171446, -0.6707313588072118]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -0.5]}


Result: Complex[0.33759646322286985, -0.8589803343001376]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, -0.5]}

... skip entries to safe data
13.18.E9 W 0 , ν ⁡ ( 2 ⁢ z ) = 2 ⁢ z / π ⁢ K ν ⁡ ( z ) Whittaker-confluent-hypergeometric-W 0 𝜈 2 𝑧 2 𝑧 𝜋 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle W_{0,\nu}\left(2z\right)=\sqrt{\ifrac{2z}{\pi}}K_% {\nu}\left(z\right)}}
\WhittakerconfhyperW{0}{\nu}@{2z} = \sqrt{\ifrac{2z}{\pi}}\modBesselK{\nu}@{z}		 

WhittakerW(0, nu, 2*z) = sqrt((2*z)/(Pi))*BesselK(nu, z)

WhittakerW[0, \[Nu], 2*z] == Sqrt[Divide[2*z,Pi]]*BesselK[\[Nu], z]
Successful Successful - Successful [Tested: 70]
13.18.E10 W 0 , 1 3 ⁡ ( 4 3 ⁢ z 3 2 ) = 2 ⁢ π ⁢ z 1 4 ⁢ Ai ⁡ ( z ) Whittaker-confluent-hypergeometric-W 0 1 3 4 3 superscript 𝑧 3 2 2 𝜋 superscript 𝑧 1 4 Airy-Ai 𝑧 {\displaystyle{\displaystyle W_{0,\frac{1}{3}}\left(\tfrac{4}{3}z^{\frac{3}{2}% }\right)=2\sqrt{\pi}z^{\frac{1}{4}}\mathrm{Ai}\left(z\right)}}
\WhittakerconfhyperW{0}{\frac{1}{3}}@{\tfrac{4}{3}z^{\frac{3}{2}}} = 2\sqrt{\pi}z^{\frac{1}{4}}\AiryAi@{z}		 

WhittakerW(0, (1)/(3), (4)/(3)*(z)^((3)/(2))) = 2*sqrt(Pi)*(z)^((1)/(4))* AiryAi(z)

WhittakerW[0, Divide[1,3], Divide[4,3]*(z)^(Divide[3,2])] == 2*Sqrt[Pi]*(z)^(Divide[1,4])* AiryAi[z]
Failure Failure
Failed [1 / 7]
Result: -.246840478+.5335590044*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

Failed [1 / 7]
Result: Complex[-0.24684047859323988, 0.533559004293784]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

13.18.E12 M - 1 2 ⁢ a , - 1 4 ⁡ ( 1 2 ⁢ z 2 ) = 2 1 2 ⁢ a - 1 ⁢ Γ ⁡ ( 1 2 ⁢ a + 3 4 ) ⁢ z / π ⁢ ( U ⁡ ( a , z ) + U ⁡ ( a , - z ) ) Whittaker-confluent-hypergeometric-M 1 2 𝑎 1 4 1 2 superscript 𝑧 2 superscript 2 1 2 𝑎 1 Euler-Gamma 1 2 𝑎 3 4 𝑧 𝜋 parabolic-U 𝑎 𝑧 parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle M_{-\frac{1}{2}a,-\frac{1}{4}}\left(\tfrac{1}{2}z% ^{2}\right)=2^{\frac{1}{2}a-1}\Gamma\left(\tfrac{1}{2}a+\tfrac{3}{4}\right)% \sqrt{\ifrac{z}{\pi}}\*\left(U\left(a,z\right)+U\left(a,-z\right)\right)}}
\WhittakerconfhyperM{-\frac{1}{2}a}{-\frac{1}{4}}@{\tfrac{1}{2}z^{2}} = 2^{\frac{1}{2}a-1}\EulerGamma@{\tfrac{1}{2}a+\tfrac{3}{4}}\sqrt{\ifrac{z}{\pi}}\*\left(\paraU@{a}{z}+\paraU@{a}{-z}\right)		 ℜ ⁡ ( 1 2 ⁢ a + 3 4 ) > 0 1 2 𝑎 3 4 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}a+\tfrac{3}{4})>0}} 
WhittakerM(-(1)/(2)*a, -(1)/(4), (1)/(2)*(z)^(2)) = (2)^((1)/(2)*a - 1)* GAMMA((1)/(2)*a +(3)/(4))*sqrt((z)/(Pi))*(CylinderU(a, z)+ CylinderU(a, - z))

WhittakerM[-Divide[1,2]*a, -Divide[1,4], Divide[1,2]*(z)^(2)] == (2)^(Divide[1,2]*a - 1)* Gamma[Divide[1,2]*a +Divide[3,4]]*Sqrt[Divide[z,Pi]]*(ParabolicCylinderD[- 1/2 -(a), z]+ ParabolicCylinderD[- 1/2 -(a), - z])
Failure Failure
Failed [8 / 28]
Result: -.4546011384-.8349579092*I
Test Values: {a = 3/2, z = -1/2+1/2*I*3^(1/2)}


Result: .58169427e-2+1.789104086*I
Test Values: {a = 3/2, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [8 / 28]
Result: Complex[-0.454601138107828, -0.8349579095614801]
Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[0.005816942543956816, 1.7891040854776739]
Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
13.18.E13 M - 1 2 ⁢ a , 1 4 ⁡ ( 1 2 ⁢ z 2 ) = 2 1 2 ⁢ a - 2 ⁢ Γ ⁡ ( 1 2 ⁢ a + 1 4 ) ⁢ z / π ⁢ ( U ⁡ ( a , - z ) - U ⁡ ( a , z ) ) Whittaker-confluent-hypergeometric-M 1 2 𝑎 1 4 1 2 superscript 𝑧 2 superscript 2 1 2 𝑎 2 Euler-Gamma 1 2 𝑎 1 4 𝑧 𝜋 parabolic-U 𝑎 𝑧 parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle M_{-\frac{1}{2}a,\frac{1}{4}}\left(\tfrac{1}{2}z^% {2}\right)=2^{\frac{1}{2}a-2}\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{4}\right)% \sqrt{\ifrac{z}{\pi}}\*\left(U\left(a,-z\right)-U\left(a,z\right)\right)}}
\WhittakerconfhyperM{-\frac{1}{2}a}{\frac{1}{4}}@{\tfrac{1}{2}z^{2}} = 2^{\frac{1}{2}a-2}\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{4}}\sqrt{\ifrac{z}{\pi}}\*\left(\paraU@{a}{-z}-\paraU@{a}{z}\right)		 ℜ ⁡ ( 1 2 ⁢ a + 1 4 ) > 0 1 2 𝑎 1 4 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}a+\tfrac{1}{4})>0}} 
WhittakerM(-(1)/(2)*a, (1)/(4), (1)/(2)*(z)^(2)) = (2)^((1)/(2)*a - 2)* GAMMA((1)/(2)*a +(1)/(4))*sqrt((z)/(Pi))*(CylinderU(a, - z)- CylinderU(a, z))

WhittakerM[-Divide[1,2]*a, Divide[1,4], Divide[1,2]*(z)^(2)] == (2)^(Divide[1,2]*a - 2)* Gamma[Divide[1,2]*a +Divide[1,4]]*Sqrt[Divide[z,Pi]]*(ParabolicCylinderD[- 1/2 -(a), - z]- ParabolicCylinderD[- 1/2 -(a), z])
Failure Failure
Failed [6 / 21]
Result: .3997621251-.6252084121*I
Test Values: {a = 3/2, z = -1/2+1/2*I*3^(1/2)}


Result: .9306149059+.2046923958*I
Test Values: {a = 3/2, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [6 / 21]
Result: Complex[0.3997621252402044, -0.6252084117529283]
Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[0.9306149056064967, 0.20469239560568858]
Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
13.18.E14 M 1 4 + n , - 1 4 ⁡ ( z 2 ) = ( - 1 ) n ⁢ n ! ( 2 ⁢ n ) ! ⁢ e - 1 2 ⁢ z 2 ⁢ z ⁢ H 2 ⁢ n ⁡ ( z ) Whittaker-confluent-hypergeometric-M 1 4 𝑛 1 4 superscript 𝑧 2 superscript 1 𝑛 𝑛 2 𝑛 superscript 𝑒 1 2 superscript 𝑧 2 𝑧 Hermite-polynomial-H 2 𝑛 𝑧 {\displaystyle{\displaystyle M_{\frac{1}{4}+n,-\frac{1}{4}}\left(z^{2}\right)=% (-1)^{n}\frac{n!}{(2n)!}e^{-\frac{1}{2}z^{2}}\sqrt{z}H_{2n}\left(z\right)}}
\WhittakerconfhyperM{\frac{1}{4}+n}{-\frac{1}{4}}@{z^{2}} = (-1)^{n}\frac{n!}{(2n)!}e^{-\frac{1}{2}z^{2}}\sqrt{z}\HermitepolyH{2n}@{z}		 

WhittakerM((1)/(4)+ n, -(1)/(4), (z)^(2)) = (- 1)^(n)*(factorial(n))/(factorial(2*n))*exp(-(1)/(2)*(z)^(2))*sqrt(z)*HermiteH(2*n, z)

WhittakerM[Divide[1,4]+ n, -Divide[1,4], (z)^(2)] == (- 1)^(n)*Divide[(n)!,(2*n)!]*Exp[-Divide[1,2]*(z)^(2)]*Sqrt[z]*HermiteH[2*n, z]
Failure Failure
Failed [6 / 21]
Result: 4.741276300-.776142297*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1}


Result: 9.155588595+2.115036937*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}

... skip entries to safe data
Failed [6 / 21]
Result: Complex[4.741276296912009, -0.7761422976118018]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[9.15558858680754, 2.115036935310196]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
13.18.E15 M 3 4 + n , 1 4 ⁡ ( z 2 ) = ( - 1 ) n ⁢ n ! ( 2 ⁢ n + 1 ) ! ⁢ e - 1 2 ⁢ z 2 ⁢ z 2 ⁢ H 2 ⁢ n + 1 ⁡ ( z ) Whittaker-confluent-hypergeometric-M 3 4 𝑛 1 4 superscript 𝑧 2 superscript 1 𝑛 𝑛 2 𝑛 1 superscript 𝑒 1 2 superscript 𝑧 2 𝑧 2 Hermite-polynomial-H 2 𝑛 1 𝑧 {\displaystyle{\displaystyle M_{\frac{3}{4}+n,\frac{1}{4}}\left(z^{2}\right)=(% -1)^{n}\frac{n!}{(2n+1)!}\frac{e^{-\frac{1}{2}z^{2}}\sqrt{z}}{2}H_{2n+1}\left(% z\right)}}
\WhittakerconfhyperM{\frac{3}{4}+n}{\frac{1}{4}}@{z^{2}} = (-1)^{n}\frac{n!}{(2n+1)!}\frac{e^{-\frac{1}{2}z^{2}}\sqrt{z}}{2}\HermitepolyH{2n+1}@{z}		 

WhittakerM((3)/(4)+ n, (1)/(4), (z)^(2)) = (- 1)^(n)*(factorial(n))/(factorial(2*n + 1))*(exp(-(1)/(2)*(z)^(2))*sqrt(z))/(2)*HermiteH(2*n + 1, z)

WhittakerM[Divide[3,4]+ n, Divide[1,4], (z)^(2)] == (- 1)^(n)*Divide[(n)!,(2*n + 1)!]*Divide[Exp[-Divide[1,2]*(z)^(2)]*Sqrt[z],2]*HermiteH[2*n + 1, z]
Failure Failure
Failed [6 / 21]
Result: 2.634248102+.148339259*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1}


Result: 3.481689250+1.400565410*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}

... skip entries to safe data
Failed [6 / 21]
Result: Complex[2.6342480998741933, 0.14833925882834587]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[3.4816892469231746, 1.4005654089276338]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
13.18.E16 W 1 4 + 1 2 ⁢ n , 1 4 ⁡ ( z 2 ) = 2 - n ⁢ e - 1 2 ⁢ z 2 ⁢ z ⁢ H n ⁡ ( z ) Whittaker-confluent-hypergeometric-W 1 4 1 2 𝑛 1 4 superscript 𝑧 2 superscript 2 𝑛 superscript 𝑒 1 2 superscript 𝑧 2 𝑧 Hermite-polynomial-H 𝑛 𝑧 {\displaystyle{\displaystyle W_{\frac{1}{4}+\frac{1}{2}n,\frac{1}{4}}\left(z^{% 2}\right)=2^{-n}e^{-\frac{1}{2}z^{2}}\sqrt{z}H_{n}\left(z\right)}}
\WhittakerconfhyperW{\frac{1}{4}+\frac{1}{2}n}{\frac{1}{4}}@{z^{2}} = 2^{-n}e^{-\frac{1}{2}z^{2}}\sqrt{z}\HermitepolyH{n}@{z}		 

WhittakerW((1)/(4)+(1)/(2)*n, (1)/(4), (z)^(2)) = (2)^(- n)* exp(-(1)/(2)*(z)^(2))*sqrt(z)*HermiteH(n, z)

WhittakerW[Divide[1,4]+Divide[1,2]*n, Divide[1,4], (z)^(2)] == (2)^(- n)* Exp[-Divide[1,2]*(z)^(2)]*Sqrt[z]*HermiteH[n, z]
Failure Failure
Failed [6 / 21]
Result: 1.704303716-.6267307130*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1}


Result: -2.370638149+.3880711488*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}

... skip entries to safe data
Failed [6 / 21]
Result: Complex[1.7043037156649337, -0.6267307126437623]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-2.370638148456005, 0.388071148805901]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
13.18.E17 W 1 2 ⁢ α + 1 2 + n , 1 2 ⁢ α ⁡ ( z ) = ( - 1 ) n ⁢ ( α + 1 ) n ⁢ M 1 2 ⁢ α + 1 2 + n , 1 2 ⁢ α ⁡ ( z ) Whittaker-confluent-hypergeometric-W 1 2 𝛼 1 2 𝑛 1 2 𝛼 𝑧 superscript 1 𝑛 Pochhammer 𝛼 1 𝑛 Whittaker-confluent-hypergeometric-M 1 2 𝛼 1 2 𝑛 1 2 𝛼 𝑧 {\displaystyle{\displaystyle W_{\frac{1}{2}\alpha+\frac{1}{2}+n,\frac{1}{2}% \alpha}\left(z\right)=(-1)^{n}{\left(\alpha+1\right)_{n}}M_{\frac{1}{2}\alpha+% \frac{1}{2}+n,\frac{1}{2}\alpha}\left(z\right)}}
\WhittakerconfhyperW{\frac{1}{2}\alpha+\frac{1}{2}+n}{\frac{1}{2}\alpha}@{z} = (-1)^{n}\Pochhammersym{\alpha+1}{n}\WhittakerconfhyperM{\frac{1}{2}\alpha+\frac{1}{2}+n}{\frac{1}{2}\alpha}@{z}		 

WhittakerW((1)/(2)*alpha +(1)/(2)+ n, (1)/(2)*alpha, z) = (- 1)^(n)* pochhammer(alpha + 1, n)*WhittakerM((1)/(2)*alpha +(1)/(2)+ n, (1)/(2)*alpha, z)

WhittakerW[Divide[1,2]*\[Alpha]+Divide[1,2]+ n, Divide[1,2]*\[Alpha], z] == (- 1)^(n)* Pochhammer[\[Alpha]+ 1, n]*WhittakerM[Divide[1,2]*\[Alpha]+Divide[1,2]+ n, Divide[1,2]*\[Alpha], z]
Failure Failure Successful [Tested: 63] Successful [Tested: 63]
13.18.E17 ( - 1 ) n ⁢ ( α + 1 ) n ⁢ M 1 2 ⁢ α + 1 2 + n , 1 2 ⁢ α ⁡ ( z ) = ( - 1 ) n ⁢ n ! ⁢ e - 1 2 ⁢ z ⁢ z 1 2 ⁢ α + 1 2 ⁢ L n ( α ) ⁡ ( z ) superscript 1 𝑛 Pochhammer 𝛼 1 𝑛 Whittaker-confluent-hypergeometric-M 1 2 𝛼 1 2 𝑛 1 2 𝛼 𝑧 superscript 1 𝑛 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 1 2 𝛼 1 2 Laguerre-polynomial-L 𝛼 𝑛 𝑧 {\displaystyle{\displaystyle(-1)^{n}{\left(\alpha+1\right)_{n}}M_{\frac{1}{2}% \alpha+\frac{1}{2}+n,\frac{1}{2}\alpha}\left(z\right)=(-1)^{n}n!e^{-\frac{1}{2% }z}z^{\frac{1}{2}\alpha+\frac{1}{2}}L^{(\alpha)}_{n}\left(z\right)}}
(-1)^{n}\Pochhammersym{\alpha+1}{n}\WhittakerconfhyperM{\frac{1}{2}\alpha+\frac{1}{2}+n}{\frac{1}{2}\alpha}@{z} = (-1)^{n}n!e^{-\frac{1}{2}z}z^{\frac{1}{2}\alpha+\frac{1}{2}}\LaguerrepolyL[\alpha]{n}@{z}		 

(- 1)^(n)* pochhammer(alpha + 1, n)*WhittakerM((1)/(2)*alpha +(1)/(2)+ n, (1)/(2)*alpha, z) = (- 1)^(n)* factorial(n)*exp(-(1)/(2)*z)*(z)^((1)/(2)*alpha +(1)/(2))* LaguerreL(n, alpha, z)

(- 1)^(n)* Pochhammer[\[Alpha]+ 1, n]*WhittakerM[Divide[1,2]*\[Alpha]+Divide[1,2]+ n, Divide[1,2]*\[Alpha], z] == (- 1)^(n)* (n)!*Exp[-Divide[1,2]*z]*(z)^(Divide[1,2]*\[Alpha]+Divide[1,2])* LaguerreL[n, \[Alpha], z]
Missing Macro Error Successful Skip - symbolical successful subtest Successful [Tested: 63]
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