The graph of Q x looks like an approximation to the indicator function of the positive real numbers. We will discuss four factorial functions. Davis writes in his exposition [1, p. It cumulated in a characterization of the Gamma function by logarithmic convexity which is now known as the Bohr-Mollerup-Artin theorem. The desired condition was found in notions of convexity. What is not so well known is the fact, that there are other functions which also solve the interpolation problem for n! This identity holds for the Euler factorial exactly but for the H-Gamma function and the L-factorial function only approximately. The Lambda function is displayed in figure 14 as the upper envelope.
The function g x has the simple expansion. There is another factorial function, proposed by Peter Luschny in October , which is also continuous at all real numbers and which we will compare to Hadamard’s Gamma function. But this is not true. Thus what I had to do was to solve the equation. Then we can prove the following identity, which is a kind of super-reflection formula. This identity holds for the Euler factorial exactly but for the H-Gamma function and the L-factorial function only approximately. What remains an open question is:
Perhaps the next generation will also. P x [red] and 1 – P x [green].
If we srie Euler’s reflection theorem for z! Figure 9 shows the behavior of H x and L x with regard to logarithmic convexity. We will discuss four factorial functions the Euler factorial function n!
Hadamard’s gamma function
This identity holds for the Euler factorial exactly but for the H-Gamma function and the L-factorial function only approximately. Exp LnFactorial y ; We will show that these alternative factorial functions posses qualities which are missing from the the conventional factorial function but might be desirable in some context.
Backed up by Cantor’s set theory and an emerging theory of topology, the new function theory looked not so much at equations and identities as at the fundamental geometrical properties.
But this time things were easy to work out. However, Jacques Hadamard stated one exemplar in which we reconsidered. On the other hand, the behavior of the L-factorial meets our expectations. The problem of interpolating the values of the factorial function n! We show how to derive our ‘working’ definition. What remains an open question is: This identity shows a kind of duality between the factorial function z!
This is far from being true. From the results and methods of  we conclude that each meromorphic solution of this functional equation is a transcendental differential function over the differential field of complex rational functions. The following plot compares the logarithm of both functions:.
Figure 12 displays how wide they are off the unit.
Hadamard’s gamma function – Wikipedia
A positive function f on a, b is logarithmically convex if log f is convex on yadamard, b. According to our setup the next important question is: Thus what I had to do was to solve the equation.
Of course our factorial functions obey also a full functional equation, not only an approximate one. The Gamma function of Euler often is thought as the only function which interpolates the factorial numbers n!
I think the list of postulates which will ultimately define the ‘true’ factorial function will include:. The interpolating function is commonly known as the Gamma function. Starting from the idea of interpolating n! A particular nice one was given by Jacques Hadamard in The approximate functional equation.
My answer to Roland was: Those which are continuous and those which are not. What makes this function unique is captured by the Bohr-Mollerup-Artin theorem: I am not aware that these definitions appeared somewhere else prior to this date. But this is not true. Luschny’s factorial function L x. Davis closed his exposition  with the remark: For the L-factorial we make two observations.
This identity relates the Euler factorial z! A month later the story continued.
Is the Gamma-function misdefined?
Thus we can recycle the oscillating hsdamard P x of L x and find the following reflection theorem for L z. The ‘oscillating factor’ of the L-factorial.
What is not so well known is the fact, that there are haadmard functions which also solve the interpolation problem for n! Here is the second part, which is centered around the super reflection product.
Lambda z has a Maclaurin series expansion in even powers which can be found in the appendix. In figure 19 the green points indicate some of the small differences.