On Series Expansions of Capparelli’s Infinite Product

Further Results on Vanishing Coefficients in Infinite Product Expansions

We extend results of Andrews and Bressoud on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if (qr−tk, qmk−(r−tk); q)∞ (qr, qmk−r; q)∞ =: ∞ ∑ n=0 cnq , for certain integers k, m s and t, where r = sm+t, then ckn−rs is always zero. Our theorems also partly give a simpler reformulation of results of Alladi and Gordon, but also...

Nonharmonic Gabor Expansions

We consider Gabor systems generated by a Gaussian function and prove certain classical results of Paley and Wiener on nonharmonic Fourier series of complex exponentials for the Gabor expansion‎. ‎In particular, we prove a version of Plancherel-Po ́lya theorem for entire functions with finite order of growth and use the Hadamard factorization theorem to study regularity‎, ‎exactness and deficienc...

On the Infinite Product Representation of Solution and Dual Equation of Sturm-Liouville Equation with Turning Point of Order 4m+1

Recurrences and explicit formulae for the expansion and connection coefficients in series of the product of two classical discrete orthogonal polynomials

Suppose that for an arbitrary function $f(x,y)$ of two discrete variables, we have the formal expansions. [f(x,y)=sumlimits_{m,n=0}^{infty }a_{m,n},P_{m}(x)P_{n}(y),] $$‎ ‎x^{m}P_{j}(x)=sumlimits_{n=0}^{2m}a_{m,,n}(j)P_{j+m-n}(x)‎,$$ ‎we find the coefficients $b_{i,j}^{(p,q,ell‎ ,‎,r)}$ in the expansion‎ $$‎ ‎x^{ell }y^{r},nabla _{x}^{p}nabla _{y}^{q},f(x,y)=x^{ell‎ ‎}y^{r}f^{(p,q)}(x,y) =sumli...

Generalized Moonshine Ii: Borcherds Products

The goal of this paper is to construct infinite dimensional Lie algebras using infinite product identities, and to use these Lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and Lie algebras. The Lie algebras that we construct conjecturally appear in an orbifold conformal field theory with symmetries given by the monster ...