# What do Gegenbauer polynomials have to do with the birthday problem?

I was reading up on Gegenbauer polynomials for a blog post about the kissing number problem, which I also asked a separate question about later.

On the Wolfram MathWorld plot summary for Gegenbauer polynomials, toward the bottom, there is a list of "See Also" topics. This list includes the birthday problem.

I don't see any connection between the two topics. Visiting the plot summary for the birthday problem doesn't yield anything about Gegenbauer polynomials that I can see. (As Dietrich Burde notes in the comments, however, it does link to hypergeometric functions, which in turn link to Gegenbauer polynomials.) Am I missing something, or is this perhaps a mistake?

• It just says: see also Birthday problem etc. This does not mean there is a direct close relationship. In some books on algebra it also says: algebra - see also Chinese literature, or see the Alhambra in Spain. Then the connection is just that also wallpaper groups belong to algebra, or that there is a Chinese remainder theorem. – Dietrich Burde Feb 27 at 22:00
• @DietrichBurde: I recognize that the connection may be indirect. But I don't see what that could be, and on that basis, one could justify including any number of things in the "See Also" list. In fact, there are only eight things in the list. Out of the top eight related topics, I would not have thought one of them would be the birthday problem. (Also, the situation is not symmetric, as I noted.) – Brian Tung Feb 27 at 22:03
• I should add that I actually kind of hope there is an interesting connection I've overlooked. I just can't figure out what that might be. – Brian Tung Feb 27 at 22:03
• @DietrichBurde: Well, OK, that's interesting. (In my defense, it's still not perfectly symmetric, but I'm willing to admit I overlooked that. <g>) But I don't see what that connection is. Do you happen to know? – Brian Tung Feb 27 at 22:04
• Ahh, I see. I'd say "explained" is putting it a bit strongly. :-) It's too bad; I was hoping there would be something more interesting there. – Brian Tung Feb 27 at 22:13

The link you found is probably historical. A previous version of the Birthday Problem page gave an expression for $$Q_2(n,d)$$ - the probability that a birthday is shared by exactly $$2$$ and no more people out of a group of $$n$$ people - in terms of an Ultraspherical Polynomial. The current Gegenbauer Polynomial page says Gegenbauer Polynomials are "are proportional to (or, depending on the normalization, equal to) the ultraspherical polynomials".
I cannot judge whether the change is a correction or a simplification, though the current Birthday Problem and Gegenbauer Polynomial pages each mention hypergeometric functions of the $$\,_2F_1$$ type.