arXiv subjects
Publications and source records attributed to Jerome Malenfant.
The determinant of an $N \times N$ circulant matrix $M = {\rm CIRC}[x_0, x_1, ..., x_{N-1}$] can be expanded in the form det$ ~M= \sum C_{a_0 a_1 ...a_{N-1}} x_{a_0} x_{a_1}...x_{a_{N-1}}$. By using the generating function of a restricted, mod-$N$ partition function, we derive a formula for the coefficients in this expansion as finite sums over products of binomial coefficients with integer variables.
We generalize Ramanujan's expansions of the fractional-power Euler functions (q^{1/5})_{\infty} = [ J_1 - q^{1/5} + q^{2/5} J_2 ](q^5)_{\infty} and (q^{1/7})_{\infty} = [ J_1 + q^{1/7} J_2 - q^{2/7} + q^{5/7} J_3 ] (q^7)_{\infty} to (q^{1/N})_{\infty}, where N is a prime number greater than 3. We show that there are exactly (N+1)/2 non-zero J functions in the expansion of (q^{1/N})_{\infty}, that one of these functions has the form +-q^{X_0}, that all others have the form +-q^{X_k} times the ratio of two Ramanujan theta functions, and that the product of all the non-zero J's is +-q^Z, where Z and the X's denote non-negative integers.
We derive expressions for the partition function p(n), with n in the form 7k+a, as (k+1)-dimensional determinants.
We find general solutions to the generating-function equation sum c_q^{(X)} z^q = F(z)^X, where X is a complex number and F(z) is a convergent power series with |F(0)| >0. We then use these results to derive finite expressions containing only integers or simple fractions for partition functions and for Euler, Bernoulli, and Stirling numbers.
We derive a compact determinant formula for calculating and factorizing the hypersum polynomials S^{(L)}_k(N) \equiv \sum_{n_1=1}^N ...\sum_{n_{L+1}=1}^{n_{L}}(n_{L+1})^k expressed in the variable N(N+L+1)