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As I understood you start from the completeness relation $$\sum_{\ell=0}^\infty \frac{2\ell + 1}{2} P_\ell(x)P_\ell(y) = \delta(x-y)$$ and use that $$P_n(0) = \begin{cases} \frac{(-1)^{m}}{4^m} \tbinom{2m}{m} = \frac{(-1)^{m}}{2^{2m}} \frac{(2m)!}{\left(m!\right)^2} & \text{for} \quad n = 2m \\ 0 & \text{for} \quad n = 2m+1 \,.... 8 I think you're misinterpreting Mathematica output. First of all, notice that the first output you get from Mathematica is a message that precludes the subsequent numerical results being sensible (or at the very least must make you very cautious of them): Sum::div: Sum does not converge. If you omit the N[] wrapper, you'll get the actual sum simplified to ... 5 We need some numerical model for comparison, so this is one of them based on FEM. First we make sufficient mesh for this problem: Needs["NDSolveFEM"];Needs["MeshTools"]; L = .90; l = 1.80; w = 0.0003; bh = 17.394; bc = 22.151; ph = 8.6; pc = 13.93; pa = 10; n = 10; thi = 460; tci = 300; Ta = 380; region = Rectangle[{0, 0}, {L, l}]; ... 5 Your T function is on the left-hand side dependent on {x,y,z} but on the right-hand side not a y in the MathML code. You got confused by the name of the functions in special states of the solutions process and forget to use them consequently. The solution of the Subscript[C,1], Subscript[C,2] depends in length on the given parameters but all are not set in ... 3 I'm not sure if the definitions you posted are different from the ones used in your plot, or you have lingering definitions that are interfering, but the plot I get from your code is slightly different than the one you show. Z1D[\[Beta]_, \[Xi]_, \[Theta]_] = (1/\[Xi] (2/\[Beta]^2 - 1) + ((\[Theta] Exp[-\[Beta]])/2 - 1/2) + Sum[((-\[Beta])^n \[... 2 As a starting point, I suggest to look at this post https://math.stackexchange.com/questions/1108246/double-sum-and-zeta-function We can at least verify numerically that the sum is equal to$$S(3)=\frac1{\Gamma(3)}\int_0^{\infty}\!\! t^2 \big(\theta_3(0,e^{-t})^2-1\big)\, \mathrm{d}t With $\Gamma(3)=2$, we have NIntegrate[t^2(EllipticTheta[3,0,Exp[-t]]^2-1)...