ON THE MOMENTS OF A DISTRIBUTION DEFINED BY THE GAUSSIAN POLYNOMIALS

HELMUT PRODINGER

Di Bucchianico [1] has considered the probability distribution defined by the (probability) generating function

F_{m,n}(q) = \begin{matrix}\left[{m+n \atop m}\right] & q \\ ( & {m+n \atop m} & ) \end{matrix} ,

with the Gaussian polynomials

\left[{m + n \atop m}\right]_{q} = \frac{(1 − q)(1 − q^{2}) … (1 − q^{m+n})}{(1 − q)(1 − q^{2}) … (1 − q^{m}) · (1 − q)(1 − q^{2}) … (1 − q^{n})}.

He discussed several methods to compute the moments of this distribution.

In this short note, I want to draw the attention of the reader to another (potentially superior) method that is due to Panny [2]. This method first computes the cumulants and translates them into the moments by a standard formula. This yields explicit formulæ for all the moments.

Note that

F_{m,n}(q) = \frac{g_{1}(q) … g_{m+n}(q)}{g_{1}(q) … g_{m}(q) · g_{1}(q) … g_{n}(q)}

with

g_{k}(q) = \frac{1 − q^{k}}{k(1 − q)},

where g_{k}(q) is a probability generating function of a random variable X_{k}. The function log g_{k}(e^{it}) is the generating function of the cumulants^{1}. Panny has computed that as

log g_{k}(e^{it}) = \frac{(k − 1)it}{2} + ∑_{j≥1} \frac{B_{2j}}{2j} (k^{2j} − 1)\frac{(it)^{2j}}{(2j)!} , |t| < \frac{2π}{k} ,

where B_{i} denotes the ith Bernoulli number.

Reading off coefficients we find the cumulants of X_{k} as

κ_{1} = \frac{k − 1}{2},  κ_{2r} = \frac{(k^{2r} − 1)B_{2r}}{2r} ,  κ_{2r+1} = 0,  r = 1, 2, 3, ….

Since

log F_{m,n}(e^{it}) = ∑_{k=1}^{m+n} log g_{k}(e^{it}) − ∑_{k=1}^{m} log g_{k}(e^{it}) − ∑_{k=1}^{n} log g_{k}(e^{it}),

the cumulants κ_{r}^{m,n} are obtained by summing the κ_{r}’s.

Date: June 12, 2000.

1We write log x for the (natural) logarithm with base e.

So we get

κ_{1}^{m,n} = \frac{\binom{m+n}{2}}{2} − \frac{\binom{m}{2}}{2} − \frac{\binom{n}{2}}{2} = \frac{mn}{2},
κ_{2r}^{m,n} = \frac{B_{2r}}{2r(2r + 1)}(B_{2r+1}(m + n + 1) − B_{2r+1}(m + 1) − B_{2r+1}(n + 1)),
κ_{2r+1}^{m,n} = 0.

Here is a little list:

κ_{1}^{m,n} = \frac{mn}{2},
κ_{2}^{m,n} = \frac{mn(m + n + 1)}{12},

κ_{4}^{m,n} = −\frac{mn(m + n + 1)(m(m + 1) + mn + n(n + 1))}{120}.

The formula to compute the moments μ_{r} from this is

μ_{r} = ∑_{π_{1}+2π_{2}+···+rπ_{r}=r} (\frac{κ_{1}}{1!} )^{π_{1}}(\frac{κ_{2}}{2!} )^{π_{2}} … (\frac{κ_{r}}{r!} )^{π_{r}} \frac{r!}{π_{1}!π_{2}! … π_{r}!},

which is a finite sum since π_{k} ∈ {0, 1, …, r}. Hence

μ_{1} = \frac{mn}{2},
μ_{2} = κ_{1}^{2} + κ_{2} = \frac{mn(m + n + 3mn + 1)}{12},

μ_{3} = κ_{1}^{3} + 3κ_{1}κ_{2} + κ_{3} = \frac{m^{2}(m + 1)n^{2}(n + 1)}{8},  &c.

We finish by mentioning that, as in [2], one could also get asymptotic formulæ.

References

[1] A. Di Bucchianico. Combinatorics, computer algebra, and the Wilcoxon–Mann–Whitney test. Journal of Statistical Planning and Inference, 79:349–364, 1999.

[2] W. Panny. A note on the higher moments of the expected behavior of straight insertion sort. Information Processing Letters, 22:175–177, 1986.

Helmut Prodinger, The John Knopfmacher Centre for Applicable Analysis and Number Theory, Department of Mathematics, University of the Witwatersrand, P. O. Wits, 2050 Johannesburg, South Africa, email: helmut@gauss.cam.wits.ac.za,

homepage: http://www.wits.ac.za/helmut/index.htm