Lesson plan
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. 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 \Phi is the Lerch zeta function, and the Luschny factorial: \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 denotes the classical gamma function and denotes the digamma function. Other related pseudogamma functions are also known. 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.
Article summary
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. 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 \Phi is the Lerch zeta function, and the Luschny factorial: \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 denotes the classical gamma function and denotes the digamma function. Other related pseudogamma functions are also known. 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.
Key claims
No claims recorded.
Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
No sources recorded.
Key claims and bibliography are generated from the same tracked claims and sources as the article page.