Pseudogamma function

In mathematics, a pseudogamma function is a function that interpolates the factorial. The gamma function is the most famous solution to the problem of extending the notion of the factorial beyond the positive integers only. However, it is clearly not the only solution, as, for any set of points, an infinite number of curves can be drawn through those points. Such a curve, namely one which interpolates the factorial but is not equal to the gamma function, is known as a pseudogamma function.^[1] The two most famous pseudogamma functions are Hadamard's gamma function:

${\displaystyle H(x)={\frac {\psi \left(1-{\frac {x}{2}}\right)-\psi \left({\frac {1}{2}}-{\frac {x}{2}}\right)}{2\Gamma (1-x)}}={\frac {\Phi \left(-1,1,-x\right)}{\Gamma (-x)}}}$

where ${\displaystyle \Phi }$ is the Lerch zeta function. We also have the Luschny factorial:^[2]

${\displaystyle \Gamma (x+1)\left(1-{\frac {\sin \left(\pi x\right)}{\pi x}}\left({\frac {x}{2}}\left(\psi \left({\frac {x+1}{2}}\right)-\psi \left({\frac {x}{2}}\right)\right)-{\frac {1}{2}}\right)\right)}$

where Γ(x) denotes the classical gamma function

and ψ(x) denotes the digamma function. Other related pseudo gamma functions are also known.^[3]

References

1. Davis, Philip J. (1959). "Leonard Euler's Integral". The American Mathematical Monthly. 66 (10): 862–865. doi:10.1080/00029890.1959.11989422.
2. Luschny. "Is the Gamma function mis-defined? Or: Hadamard versus Euler - Who found the better Gamma function?".
3. Klimek, Matthew D. (2023). "A new entire factorial function". Ramanujan Journal. 61 (3): 757–762. arXiv:2107.11330. doi:10.1007/s11139-023-00708-2. MR 4599649.