In a recent (unfortunately over-hyped) preprint by Saha and Sinha, Field theory expansions of string theory amplitudes (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.

In a recent (unfortunately over-hyped) preprint by Saha and Sinha, Field theory expansions of string theory amplitudes (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.