Hi,

I do not know any calculator that calculates the binomial coefficients C(n,k) when n is not an integer. Of course k must be an integer >=0, but n can be any real or complex number. To be precise

C(z,k) = z(z-1)(z-2)...(z-k+1)/k!

is the coefficient of x^k in the binomial expansion of (1+x)^z (for |x|<1, of course).

For the sake of motivation, one may ask what what are the second, fourth, sixth, eight-order, etc. asymptotic expansions of the Lorentz factor

1/sqrt(1-v^2/c^2) = 1 + ...

from relativity. The coefficients of (v/c)^2, (v/c)^4, (v/c)^6, (v/c)^8, etc., are simply

-C(-1/2,1) = 0.5

C(-1/2,2) = 0.375

-C(-1/2,3) = 0.3125

C(-1/2,4) = 0.2734375

As far as I could check neither the 34S nor even the HP50G implement C(n,k) other than for integers with n>=k>=0 (returning Domain Errors otherwise), and yet no essential change in the algorithm should be needed to extend the domain (if anything, less error-checking would be necessary).

Besides the obvious application to finding Taylor coefficients, there are combinatorial ones. For instance, whereas C(n,k) is the number of ways of choosing a k-element subset of an n-element set; we have that abs(C(-n,k)) is the number of ways of choosing a k-element multiset from an n-set; in other words, a k-element "subset" chosen from an n-element set, but allowing repeated choices.

[In reality the above can be reduced to the usual binomial coefficients via C(-n,k) = (-1)^k C(n+k-1,k), but it's hard to beat the beauty and simplicity of the formula abs(C(-n,k)). However, note that this only applies to negative integer values of n and not in general to other real values, much less complex values of n.]

Any hope that this modification, both useful and likely trivial to implement, could make its way to a future ROM? Even the extension of C(x,y) to all real x (even if not complex) would be useful, and both welcome and unique among all calculators.

Cheers,

Eduardo