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I want to find analytical summation formulas for these 4 types series of Legendre polynomials:

Sum[1/n*LegendreP[n - 1, Cos[ϕ]] LegendreP[n, Cos[θ]], {n, 1, ∞}]
Sum[1/n*LegendreP[n + 1, Cos[ϕ]] LegendreP[n, Cos[θ]], {n, 1, ∞}]

Sum[1/(2 n + 1)*LegendreP[n - 1, Cos[ϕ]] LegendreP[n, Cos[θ]], {n, 1, ∞}]
Sum[1/(2 n + 1)*LegendreP[n + 1, Cos[ϕ]] LegendreP[n, Cos[θ]], {n, 1, ∞}]

I find some formulas in the page of http://functions.wolfram.com/Polynomials/LegendreP/23/02/, but no formula can be used directly, so I seek help here. A relation question is https://math.stackexchange.com/questions/1829769/infinite-sum-of-legendre-polynomials.

Any advice and hint are appreciated! Thank you very much.

I try the following code in Mathematica

Sum[1/n*LegendreP[n - 1, 1]*LegendreP[n, Cos[θ]], {n, 1, ∞}]

It returns

Log[Csc[θ/2]^2/(1 + Csc[θ/2])]

then I try this code

Sum[1/n*LegendreP[n - 1, Cos[ϕ]]*LegendreP[n, Cos[θ]], {n, 1, ∞}]

it returns what I input without any change and evaluation.

Is there any method to find analytical summation or rewrite them in an integral form?

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