Closed form for partition function with sinh

Theorem

Define $$\left( \left( x \right) \right)= \left\{ \begin{array}{ll} x - \lfloor x \rfloor - \dfrac{1}{2} &, \quad x \not\in\mathbb{Z} \\ 0 , \quad x \in \mathbb{Z}, \end{array} \right.$$ a sawtooth function, define $$s(h,k)=\displaystyle\sum_{j=1}^{k-1} \dfrac{j}{k} \left( \left( \dfrac{hj}{k} \right) \right),$$ and define $$A_k(n)=\displaystyle\sum_{0<h<k, (h,k)=1} \exp \left( \pi i s(h,k) - \dfrac{2 \pi i h n}{k} \right).$$ Then, taking $p(0)=1$, the following formula holds: $$p(n)=\dfrac{1}{\pi \sqrt{2}} \displaystyle\sum_{k=1}^{\infty} \sqrt{k}A_k(n) \dfrac{\mathrm{d}}{\mathrm{d}n} \left[ \dfrac{\sinh \left( \dfrac{\pi}{k} \sqrt{\dfrac{2}{3}} \sqrt{n-\dfrac{1}{24}} \right)}{\sqrt{n-\dfrac{1}{24}}} \right],$$ where $p$ denotes the partition function, $\pi$ denotes pi, and $\sinh$ denotes the hyperbolic sine.

References

• 1964: {{ #if: |{{{2}}}|Milton Abramowitz}}{{#if: Irene A. Stegun|{{#if: |, {{ #if: |{{{2}}}|Irene A. Stegun}}{{#if: |, [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]{{#if: |, [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]{{#if: |, [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]] and [[Mathematician:{{{author6}}}|{{ #if: |{{{2}}}|{{{author6}}}}}]]| and [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]]}}| and [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]}}| and [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]}}| and {{ #if: |{{{2}}}|Irene A. Stegun}}}}|}}: [[Book:Milton Abramowitz/Handbook of mathematical functions{{#if: |/Volume {{{volume}}}|}}{{#if: |/{{{edpage}}}}}|Handbook of mathematical functions{{#if: |: Volume {{{volume}}}|}}{{#if: |: {{{eddisplay}}}|{{#if: | ({{{ed}}} ed.)}}}}]]{{#if: | (translated by [[Mathematician:{{{translated}}}|{{ #if: |{{{2}}}|{{{translated}}}}}]])}}{{#if: |, {{{publisher}}}|}}{{#if: |, ISBN {{{isbn}}}|}}{{#if: Generating function for partition function | ... (previous)|}}{{#if: Pure recurrence relation for partition function | ... (next)|}}{{#if: |: Entry: {{#if: |[[{{{entryref}}}|{{{entry}}}]]|{{{entry}}}}}|}}: $24.2.1 \mathrm{I}.C.$