Confluent Hypergeometric Functions - 13.24 Series

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
13.24.E1 M κ , μ ⁡ ( z ) = Γ ⁡ ( κ + μ ) ⁢ 2 2 ⁢ κ + 2 ⁢ μ ⁢ z 1 2 - κ ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ ( 2 ⁢ κ + 2 ⁢ μ ) s ⁢ ( 2 ⁢ κ ) s ( 1 + 2 ⁢ μ ) s ⁢ s ! ⁢ ( κ + μ + s ) ⁢ I κ + μ + s ⁡ ( 1 2 ⁢ z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 Euler-Gamma 𝜅 𝜇 superscript 2 2 𝜅 2 𝜇 superscript 𝑧 1 2 𝜅 superscript subscript 𝑠 0 superscript 1 𝑠 Pochhammer 2 𝜅 2 𝜇 𝑠 Pochhammer 2 𝜅 𝑠 Pochhammer 1 2 𝜇 𝑠 𝑠 𝜅 𝜇 𝑠 modified-Bessel-first-kind 𝜅 𝜇 𝑠 1 2 𝑧 {\displaystyle{\displaystyle M_{\kappa,\mu}\left(z\right)=\Gamma\left(\kappa+% \mu\right)2^{2\kappa+2\mu}z^{\frac{1}{2}-\kappa}\*\sum_{s=0}^{\infty}(-1)^{s}% \frac{{\left(2\kappa+2\mu\right)_{s}}{\left(2\kappa\right)_{s}}}{{\left(1+2\mu% \right)_{s}}s!}\*\left(\kappa+\mu+s\right)I_{\kappa+\mu+s}\left(\tfrac{1}{2}z% \right)}}
\WhittakerconfhyperM{\kappa}{\mu}@{z} = \EulerGamma@{\kappa+\mu}2^{2\kappa+2\mu}z^{\frac{1}{2}-\kappa}\*\sum_{s=0}^{\infty}(-1)^{s}\frac{\Pochhammersym{2\kappa+2\mu}{s}\Pochhammersym{2\kappa}{s}}{\Pochhammersym{1+2\mu}{s}s!}\*\left(\kappa+\mu+s\right)\modBesselI{\kappa+\mu+s}@{\tfrac{1}{2}z}		 ℜ ⁡ ( κ + μ ) > 0 , ℜ ⁡ ( ( κ + μ + s ) + k + 1 ) > 0 formulae-sequence 𝜅 𝜇 0 𝜅 𝜇 𝑠 𝑘 1 0 {\displaystyle{\displaystyle\Re(\kappa+\mu)>0,\Re((\kappa+\mu+s)+k+1)>0}} 
WhittakerM(kappa, mu, z) = GAMMA(kappa + mu)*(2)^(2*kappa + 2*mu)* (z)^((1)/(2)- kappa)* sum((- 1)^(s)*(pochhammer(2*kappa + 2*mu, s)*pochhammer(2*kappa, s))/(pochhammer(1 + 2*mu, s)*factorial(s))*(kappa + mu + s)*BesselI(kappa + mu + s, (1)/(2)*z), s = 0..infinity)

WhittakerM[\[Kappa], \[Mu], z] == Gamma[\[Kappa]+ \[Mu]]*(2)^(2*\[Kappa]+ 2*\[Mu])* (z)^(Divide[1,2]- \[Kappa])* Sum[(- 1)^(s)*Divide[Pochhammer[2*\[Kappa]+ 2*\[Mu], s]*Pochhammer[2*\[Kappa], s],Pochhammer[1 + 2*\[Mu], s]*(s)!]*(\[Kappa]+ \[Mu]+ s)*BesselI[\[Kappa]+ \[Mu]+ s, Divide[1,2]*z], {s, 0, Infinity}, GenerateConditions->None]
Failure Failure Manual Skip! Skipped - Because timed out
13.24.E2 1 Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ M κ , μ ⁡ ( z ) = 2 2 ⁢ μ ⁢ z μ + 1 2 ⁢ ∑ s = 0 ∞ p s ( μ ) ⁢ ( z ) ⁢ ( 2 ⁢ κ ⁢ z ) - 2 ⁢ μ - s ⁢ J 2 ⁢ μ + s ⁡ ( 2 ⁢ κ ⁢ z ) 1 Euler-Gamma 1 2 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 superscript 2 2 𝜇 superscript 𝑧 𝜇 1 2 superscript subscript 𝑠 0 superscript subscript 𝑝 𝑠 𝜇 𝑧 superscript 2 𝜅 𝑧 2 𝜇 𝑠 Bessel-J 2 𝜇 𝑠 2 𝜅 𝑧 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}% \left(z\right)=2^{2\mu}z^{\mu+\frac{1}{2}}\sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)% \left(2\sqrt{\kappa z}\right)^{-2\mu-s}J_{2\mu+s}\left(2\sqrt{\kappa z}\right)}}
\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{z} = 2^{2\mu}z^{\mu+\frac{1}{2}}\sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)\left(2\sqrt{\kappa z}\right)^{-2\mu-s}\BesselJ{2\mu+s}@{2\sqrt{\kappa z}}		 ℜ ⁡ ( ( 2 ⁢ μ + s ) + k + 1 ) > 0 , ℜ ⁡ ( 1 + 2 ⁢ μ ) > 0 formulae-sequence 2 𝜇 𝑠 𝑘 1 0 1 2 𝜇 0 {\displaystyle{\displaystyle\Re((2\mu+s)+k+1)>0,\Re(1+2\mu)>0}} 
(1)/(GAMMA(1 + 2*mu))*WhittakerM(kappa, mu, z) = (2)^(2*mu)* (z)^(mu +(1)/(2))* sum((p[s])^(mu)(z)*(2*sqrt(kappa*z))^(- 2*mu - s)* BesselJ(2*mu + s, 2*sqrt(kappa*z)), s = 0..infinity)

Divide[1,Gamma[1 + 2*\[Mu]]]*WhittakerM[\[Kappa], \[Mu], z] == (2)^(2*\[Mu])* (z)^(\[Mu]+Divide[1,2])* Sum[(Subscript[p, s])^(\[Mu])[z]*(2*Sqrt[\[Kappa]*z])^(- 2*\[Mu]- s)* BesselJ[2*\[Mu]+ s, 2*Sqrt[\[Kappa]*z]], {s, 0, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
13.24.E3 exp ⁡ ( - 1 2 ⁢ z ⁢ ( coth ⁡ t - 1 t ) ) ⁢ ( t sinh ⁡ t ) 1 - 2 ⁢ μ = ∑ s = 0 ∞ p s ( μ ) ⁢ ( z ) ⁢ ( - t z ) s 1 2 𝑧 hyperbolic-cotangent 𝑡 1 𝑡 superscript 𝑡 𝑡 1 2 𝜇 superscript subscript 𝑠 0 superscript subscript 𝑝 𝑠 𝜇 𝑧 superscript 𝑡 𝑧 𝑠 {\displaystyle{\displaystyle\exp\left(-\tfrac{1}{2}z\left(\coth t-\frac{1}{t}% \right)\right)\left(\frac{t}{\sinh t}\right)^{1-2\mu}=\sum_{s=0}^{\infty}p_{s}% ^{(\mu)}(z)\left(-\frac{t}{z}\right)^{s}}}
\exp@{-\tfrac{1}{2}z\left(\coth@@{t}-\frac{1}{t}\right)}\left(\frac{t}{\sinh@@{t}}\right)^{1-2\mu} = \sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)\left(-\frac{t}{z}\right)^{s}		 

exp(-(1)/(2)*z*(coth(t)-(1)/(t)))*((t)/(sinh(t)))^(1 - 2*mu) = sum((p[s])^(mu)(z)*(-(t)/(z))^(s), s = 0..infinity)

Exp[-Divide[1,2]*z*(Coth[t]-Divide[1,t])]*(Divide[t,Sinh[t]])^(1 - 2*\[Mu]) == Sum[(Subscript[p, s])^(\[Mu])[z]*(-Divide[t,z])^(s), {s, 0, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out
Failed [300 / 300]
Result: Plus[Complex[1.4000146541353637, 0.6933735030866136], Times[-1.0, NSum[Times[Power[Complex[1.299038105676658, -0.7499999999999999], s], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], p]
Test Values: {s, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, s], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[1.4000146541353637, 0.6933735030866136], Times[-1.0, NSum[Times[Power[Complex[1.299038105676658, -0.7499999999999999], s], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], p]
Test Values: {s, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, s], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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