pseudogamma function

{{short description|Function that interpolates the factorial}}

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.{{cite journal |last1=Davis |first1=Philip J. |date=1959 |title=Leonhard Euler's Integral |url=https://www.tandfonline.com/doi/abs/10.1080/00029890.1959.11989422 |journal=The American Mathematical Monthly |volume=66 |issue=10 |pages=862–865 |doi=10.1080/00029890.1959.11989422}} The two most famous pseudogamma functions are Hadamard's gamma function,

$$$$

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 $\backslash Phi$ is the Lerch zeta function, and the Luschny factorial:{{cite web |last1=Luschny |title=Is the Gamma function mis-defined? Or: Hadamard versus Euler - Who found the better Gamma function? |url=https://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html}}

$$$$

\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 {{math|Γ(x)}} denotes the classical gamma function and {{math|ψ(x)}} denotes the digamma function. Other related pseudogamma functions are also known.{{cite journal

| last = Klimek | first = Matthew D.

| arxiv = 2107.11330

| doi = 10.1007/s11139-023-00708-2

| issue = 3

| journal = Ramanujan Journal

| mr = 4599649

| pages = 757–762

| title = A new entire factorial function

| volume = 61

| year = 2023}}

However, by adding conditions to the function interpolating the factorial, we obtain uniqueness of this function, most often given by the Gamma function. The most common condition is the logarithmic convexity: this is the Bohr-Mollerup theorem. See also the Wielandt theorem for other conditions.