A series in terms of the Jacobi theta fuctions

Question

On Wikipedia, one can find an expression of the series $$ \sum_{n=1}^\infty \frac{n^pq^n}{1-q^n} $$ in terms of the Jacobi theta functions for $p=3,5,7$. I'm looking for an expression of this series for $p=1$.

Answer 1

I found a formula, thanks to this StackExchange answer.

Denote by $E_2(q)$ the above series. Let $\tau$ such that $q = \exp(i\pi\tau)$. Let $$ e_2(q) = \frac{6}{\pi} E(\lambda(\tau)) j_3^2 - j_3^4 - \theta_4(0, \tau) $$ where $E$ is the complete elliptic function of the second kind, $\lambda$ is the modular lambda function, and $j_3 = \theta_3(0,\tau)$.

Then $E_2(q^2) = e_2(q)$.

Answer 2

You question is about the Eisenstein series $\,E_2.\,$ You asked

I'm looking for an expression of this series

The doubt is about what "an expression" means here. It is already known that $$ E_2(q) = 1 + \frac{24\, q}{\phi(q)} \frac{d \phi(q)}{dq} \qquad \text{ where }\qquad \phi(q) = \prod_{n=1}^\infty (1-q^n). $$

An expression using elliptic integrals and theta functions alone based on a Mathematica SE answer is $$ E_2(q) = (6/\pi)E(m)\,t_3^2 - t_3^4 - t_4^4 \quad \text{ where }\\ t_3 = \theta_3(0,q),\;t_4 = \theta_4(0,q),\; m = 1-(t_4/t_3)^4. $$ My Mathematica function and code to check this expression is

EisensteinE2[q_] := Module[{t3, t4},
    t3 = EllipticTheta[3, 0, q]; t4 = EllipticTheta[4, 0, q];
    6/\[Pi] EllipticE[1 - (t4/t3)^4] t3^2 - t3^4 - t4^4];
With[{M = 10}, 
    (1 - Sum[24 DivisorSigma[1, k] q^(2 k), {k, 1, M}]) - 
    Series[EisensteinE2[q], {q, 0, 2*M}]]
(* O[q]^21 *)

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NOTE carefully the q^(2 k) in the Sum[]. This is because the nome $\,q = e^{\pi i \tau}\,$ is standard for Jacobi theta functions $\,\theta_k(z,q)\,$ while $\,q = e^{2\pi i \tau}\,$ is standard for the Dedekind $\,\eta(\tau)\,$ function and Eisenstein series $\,E_{2k}(\tau).$