Gamma-free definition of binomial coefficients

Question

In an answer to T. Amdeberhan's recent question, Noam D. Elkies gives, as a by-product, an elementary computation of the classical integral $\int_{-\infty}^{+\infty}{\sin x\over x}dx=\pi,$ and suggests a trigonometric definition of binomial coefficients ${n\choose x}$ for non-negative integer $n$ and real or complex $x$ , as: ${n\choose x}:=\frac{n!}{\pi} \, \frac{\sin \pi x}{(n-x)(n-1-x) \cdots (1-x) x}\ ,$ which is consistent with the definition via Gamma function, $\frac{n!}{\Gamma(1+x)\Gamma(1+n-x)}$ , using the functional equation and the symmetry relation $\Gamma(x) \Gamma(1-x)=\pi/\sin\pi x$ . (We may even hide the sine function and $\pi$ , using the infinite product for $\sin(\pi x)$ , which gives a plain infinite product ${n\choose x}=\prod_{k=1}^n \Big(1+{x\over k}\Big)\prod_{k>n} \Big(1-{x^2\over k^2}\Big),$ but this formula seems somehow less handy to work with).

I'd like to elaborate this idea, mainly for didactic purposes, but not only, and see what can be deduced from the trigonometric definition without resorting to the Gamma function.

It is clear that it actually reduces to the classical definition for $x\in\mathbb{Z}$ , taking a limit as $x\to k$ .
The symmetry property ${\big({n\atop x}\big)}={n\choose n-x \ }$ and the recursive relations ${\big({n\atop x}\big)}={n\over x}{n-1\choose x-1 \ }$ and ${n-1\choose x}+{n-1\choose x-1\ }={n\choose x}$ , just follow from elementary identities.
Many identities seem to have a continuous counterpart, like Amdeberhan's $\int_{\mathbb{R}}{n\choose x}a^xb^{n-x}dx=\sum_{k\in\mathbb{Z}} {n\choose k}a^kb^{n-k}=(a+b)^n.$

While these properties motivate and justify as quite natural the choice of the above definition for the binomials ${n\choose x \ }$ for real $x$ , it remains the want of a necessity for this choice, that is, a uniqueness result analogous to the Artin-Bohr-Mollerup characterization of the Gamma function.

So here is my question:

Is there a Gamma-free characterization of the extended binomials $\textstyle{n\choose x}$ ? In other words, what property, together with one or more of the above relations, would lead to the above trigonometric definition?

(Beta function) $^{-1}$ is something like a binomial, does it count? — Fedor Petrov, Nov 19 '16 at 16:41
Yes, but what I had in mind is, for instance: suppose one didn't know $\Gamma(x)$ , and notices that $[(n+1)\int_0^1t^x(1-t)^{n-x}dt]^{-1}$ and the ${n\choose x }$ defined above satisfy the same functional relations (it's a quick integration by parts for the former and an elementary algebraic identity for the latter). What qualitative property do they share, such that one can conclude that they coincide for all $n$ and $x$ ? In the context of Gamma function, the answer would be: $\Gamma(x+y)B(x,y)$ and $\Gamma(x)\Gamma(y)$ are both logarithmically convex wrto $x$ and $y$ . — Pietro Majer, Nov 19 '16 at 17:14
And only the September silence lasts (ooops, it's already November).
https://www.youtube.com/watch?v=RpXlALmz6c4 — Włodzimierz Holsztyński, Nov 21 '16 at 04:22

score 9 · Accepted Answer · answered Nov 19 '16 at 19:52

Yes, there is a Bohr-Mollerup (or Artin-Bohr-Mollerup?) criterion for $n \choose x$ . It's probably known, but seems easier to write up anew than to find in the literature.

Proposition: The function $n\choose x$ of an integer $n \geq 0$ and a real $x$ is characterized by the initial value ${0 \choose 0} = 1$ and the recurrences $(x+1) {n \choose x+1} = (n-x) {n \choose x}, \quad\ {n \choose x} + {n \choose x+1} = {n+1 \choose x+1}$ (which imply that $n \choose x$ agrees with the usual values when $x \in \bf Z$ ), together with the condition that $n \choose x$ is a logarithmically convex function of $x \in (0,n)$ for each $n$ .

(We do not need to impose the symmetry condition ${n \choose x} = {n \choose n-x}$ , but it follows automatically from uniqueness.)

Proof sketch: We can show that the trigonometric $n \choose x$ satisfies this convexity condition using the product formula for the sine: each factor $(x+k)(x-n-k)$ (with $k=0,1,2,\ldots$ ) is logarithmically convex (downwards) on $0 \leq x \leq n$ , so the same is true of their product.

The converse can be proved a la Bohr-Mollerup. Any function of $n$ and $x$ satisfying the recurrences is of the form $f(x) {n \choose x}$ for some $1$ -periodic function $f$ . But for large $n$ and $x$ near $n/2$ the logarithm of $n \choose x$ , though convex, is nearly flat; e.g. $\log {2m \choose m-1} - 2 \log {2m \choose m} + \log {2m \choose m+1} = O(1/m).$ Thus if $f$ were nonconstant then for large $n$ the fluctuations would make $f(x) {n \choose x}$ nonconvex. Hence $f$ is constant, and then the initial value ${0 \choose 0} = 1$ makes $f(x)=1$ identically. $\Box$

The trigonometric definition of $n \choose x$ doesn't make sense if $n$ is not an integer. I think you're right that arbitrary $n$ still work if we use the $\Gamma$ definition (and initialize with $n\choose 0$ ), though then it's no longer as elementary a function. — Noam D. Elkies, Nov 19 '16 at 22:02

Włodzimierz Holsztyński · Answer 2 · 2016-11-20T19:09:53.790

Let $\ \mu\$ be a measure in $\ \mathbb C.\$ Then define $\ \mathbb P_\mu(z)$ :

$\mathbb P_\mu(z)\ :=\ e^{ \int_\mu {\log(z-t)\,dt} }$

We may conveniently define also the respective Newton coefficient:

$\binom z{\mu,n}\ :=\ \frac{\mathbb P_\mu(z)}{\mathbb P_\mu(n)}$

when $\mathbb P_\mu(n) \ne 0\$ and is well-defined (for arbitrary $\ n\in\mathbb C$ ) -- this coefficient is more like a generalized polynomial.

EXAMPLE Let $\ \mu\$ be an atomic measure where $\ \mu(\{k\}) := 1\$ for every $\ k=0\ \ldots\ n\!-\!1\$ (this time $\ n\$ is a natural number), and there are no other atoms. Then $\ \binom z{\mu,n}\$ is equal to a respective basic integer polynomial or Newton polynomial (coefficient) $\ \binom zn.$

This my definition is just a draft of a note.

EDIT Let me add another definition along similar lines (a measure). My note is a draft since--for instance--I am not discussing here the question of the set of arguments of the introduced functions; also, for the second definition below, one should also discuss the question of the measure staying in some sense away from $\ 0\in\mathbb C\$ (e.g. let $\ \mu(B(0; r)) = 0\$ for certain $\ r>0\$ (but much less can and should be assumed).

Let $\ \mu\$ be a measure in $\ \mathbb C.\$ Then define $\ \mathbf p_\mu(z)$ :

$\mathbf p_\mu(z)\ :=\ e^{ \int_\mu {\log(\frac zt-1)\,dt} }$

We may conveniently define also the respective Newton coefficient:

$\binom z{\mu;n}\ :=\ \frac{\mathbf p_\mu(z)}{\mathbf p_\mu(n)}$

when $\mathbf p_\mu(n) \ne 0\$ and well-defined (for arbitrary $\ n\in\mathbb C$ ).

WARNING This time the Newton coefficient notation features ";" instead of ",".

Gamma-free definition of binomial coefficients

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