In a recent (unfortunately over-hyped) preprint by Saha and Sinha, <a href="https://arxiv.org/abs/2401.05733">Field theory expansions of string theory amplitudes</a> (arXiv:2401.05733), they present the following series for $\pi$:
$$\pi = 4 + \sum_{n=1}^\infty {1\over n!} \biggl({1\over n+\lambda} - {4\over 2n+1}\biggr)\biggl({(2n+1)^2 \over 4(n+\lambda)} - n \biggr)_{n-1},$$
where $\lambda$ is an arbitrary complex number and the Pochhammer symbol $(x)_n := x(x+1)\cdots(x+n-1)$.

Setting aside the unfortunate press coverage, as well as the question of the significance of this formula for $\pi$, what I'm wondering is whether it is new. The authors are physicists, and searching the literature for this type of thing is not always easy, so I figured that MathOverflow would be a natural place to ask.

A related, and perhaps easier, question is whether there are other known series for $\pi$ that involve a complex parameter $\lambda$ in the summand, but where the sum of the series is independent of the value of the parameter.

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<b>EDIT:</b> Aninda Sinha <a href="https://math.stackexchange.com/q/4876995">asked essentially the same question</a> on math.SE on March 8, 2024. See also the <a href="https://www.youtube.com/watch?v=2lvTjEZ-bbw">Numberphile interview</a> of Sinha and Saha.