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Saha-Sinha Formula

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The Saha-Sinha formula is a parameterized series representation of pi given by

pi=4+sum_(n=1)^infty1/(n!)(1/(n+lambda)-4/(2n+1))(((2n+1)^2)/(4(n+lambda))-n)_(n-1),
(1)

where (a)_k is the Pochhammer symbol. The series is absolutely convergent for the complex number parameter lambda satisfying Re(lambda)>-1. Its sum is independent of lambda, although its rate of convergence depends on the parameter.

Saha and Sinha (2024) obtained the formula from parameterized expansions involving the beta function. Rosengren (2025) gave rigorous proofs and generalizations using partial fraction decomposition.

For Re(lambda)>0, another representation from the same family is

pi=sum_(n=0)^infty1/(n!)(1-lambda+((lambda-1/2)^2)/(lambda+n))_n(4/(2n+1)-1/(lambda+n)).
(2)

This follows by setting x_1=x_2=1/2 in Rosengren's (2025) parameterized beta function expansion. At lambda=1/2, this second representation reduces to

pi/2=sum_(n=0)^infty((1/2)_n)/(n!(2n+1)),
(3)

which is the Taylor series for the inverse sine evaluated at 1.

For each fixed positive integer n, the summand in the first representation tends to 4(-1)^n/(2n+1) as lambda->+infty. These limiting terms give the Gregory series,

pi/4=sum_(n=0)^infty((-1)^n)/(2n+1).
(4)

See also

Beta Function, Gregory Series, Pi Formulas

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References

Bischoff, M. "String Theorists Accidentally Find a New Formula for Pi." Scientific American. Aug. 29, 2024. https://www.scientificamerican.com/article/string-theorists-accidentally-find-a-new-formula-for-pi/.Chow, T. "Possible New Series for pi." MathOverflow. Jun. 26, 2024. https://mathoverflow.net/questions/473931/possible-new-series-for-pi.Rosengren, H. "String Theory Amplitudes and Partial Fractions." Ramanujan J. 67, 26, 2025. https://doi.org/10.1007/s11139-025-01080-z.Saha, A. P. and Sinha, A. "Field Theory Expansions of String Theory Amplitudes." Phys. Rev. Lett. 132, 221601, 2024. https://doi.org/10.1103/PhysRevLett.132.221601.Stadnicki, S. "Is the New Series for pi a Big (or Even Medium) Deal?" Mathematics Stack Exchange. Jun. 25, 2024. https://math.stackexchange.com/questions/4937730/is-the-new-series-for-a-big-or-even-medium-deal.

Cite this as:

Weisstein, Eric W. "Saha-Sinha Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Saha-SinhaFormula.html

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