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Question: Converting a function expressed by the hypergeom function to the Legendre function of the first and the second kind

Question:Converting a function expressed by the hypergeom function to the Legendre function of the first and the second kind

Najibeh Borhani 50 Maple
hypergeom legendrep
March 03 2022
2

Could you help me to convert the following maple solution expressed by the hypergeom function to the LegendreP and Q function?

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diff(T[3](t), t, t)+3*(diff(a(t), t))*(diff(T[3](t), t))/a(t)+(2*(diff(a(t), t, t))/a(t)+6*(diff(a(t), t))^2/a(t)^2+(-Omega^2+6)/a(t)^2)*T[3](t)

diff(diff(T[3](t), t), t)+3*(diff(a(t), t))*(diff(T[3](t), t))/a(t)+(2*(diff(diff(a(t), t), t))/a(t)+6*(diff(a(t), t))^2/a(t)^2+(-Omega^2+6)/a(t)^2)*T[3](t)

 (1)
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"a(t) :=Zeta*(1-(1-t/(Zeta^()))^(2))^(1/(2)) "

proc (t) options operator, arrow, function_assign; Zeta*(1-(1-t/Zeta)^2)^(1/2) end proc

 (2)
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ODE2 := diff(T[3](t), t, t)+3*(diff(a(t), t))*(diff(T[3](t), t))/a(t)+(2*(diff(a(t), t, t))/a(t)+6*(diff(a(t), t))^2/a(t)^2+(-Omega^2+6)/a(t)^2)*T[3](t)

diff(diff(T[3](t), t), t)+3*(1-t/Zeta)*(diff(T[3](t), t))/((1-(1-t/Zeta)^2)*Zeta)+(2*(-(1-t/Zeta)^2/((1-(1-t/Zeta)^2)^(3/2)*Zeta)-1/((1-(1-t/Zeta)^2)^(1/2)*Zeta))/(Zeta*(1-(1-t/Zeta)^2)^(1/2))+6*(1-t/Zeta)^2/((1-(1-t/Zeta)^2)^2*Zeta^2)+(-Omega^2+6)/(Zeta^2*(1-(1-t/Zeta)^2)))*T[3](t)

 (3)
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generalsol := dsolve(ODE2, T[3](t))

T[3](t) = _C1*hypergeom([1/2+(-Omega^2+1)^(1/2), 1/2-(-Omega^2+1)^(1/2)], [1-((1/2)*I)*15^(1/2)], (1/2)*t/Zeta)*t^(-((1/4)*I)*15^(1/2)-1/4)*(2*Zeta-t)^(((1/4)*I)*15^(1/2)-1/4)+_C2*(-(-2*Zeta+t)*t)^(((1/4)*I)*15^(1/2)-1/4)*hypergeom([((1/2)*I)*15^(1/2)+1/2+(-Omega^2+1)^(1/2), ((1/2)*I)*15^(1/2)+1/2-(-Omega^2+1)^(1/2)], [1+((1/2)*I)*15^(1/2)], (1/2)*t/Zeta)

 (4)
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convert(_C1*hypergeom([1/2+sqrt(-Omega^2+1), 1/2-sqrt(-Omega^2+1)], [1-I*sqrt(15)*(1/2)], t/(2*Zeta))*t^(-I*sqrt(15)*(1/4)-1/4)*(2*Zeta-t)^(I*sqrt(15)*(1/4)-1/4), LegendreP)

_C1*GAMMA(1-((1/2)*I)*15^(1/2))*(-t/Zeta)^(((1/4)*I)*15^(1/2))*LegendreP(-1/2+(-Omega^2+1)^(1/2), ((1/2)*I)*15^(1/2), 1-t/Zeta)*t^(-((1/4)*I)*15^(1/2)-1/4)*(2*Zeta-t)^(((1/4)*I)*15^(1/2)-1/4)/((2*Zeta-t)/Zeta)^(((1/4)*I)*15^(1/2))

 (5)
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convert(_C2*(-t*(t-2*Zeta))^(I*sqrt(15)*(1/4)-1/4)*hypergeom([I*sqrt(15)*(1/2)+1/2+sqrt(-Omega^2+1), I*sqrt(15)*(1/2)+1/2-sqrt(-Omega^2+1)], [1+I*sqrt(15)*(1/2)], t/(2*Zeta)), LegendreQ)

_C2*(-t*(-2*Zeta+t))^(((1/4)*I)*15^(1/2)-1/4)*hypergeom([((1/2)*I)*15^(1/2)+1/2+(-Omega^2+1)^(1/2), ((1/2)*I)*15^(1/2)+1/2-(-Omega^2+1)^(1/2)], [1+((1/2)*I)*15^(1/2)], (1/2)*t/Zeta)

 (6)
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NULL

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