Related papers: Two generalizations of Jacobi's derivative formula

We state and prove various new identities involving the Z_K parafermion characters (or level-K string functions) for the cases K=4, K=8, and K=16. These identities fall into three classes: identities in the first class are generalizations…

Since their introduction by Andrews, generalized Frobenius partitions have interested a number of authors, many of whom have worked out explicit formulas for their generating functions in specific cases. This has uncovered interesting…

By employing the classical tools from the theory of $q$-series and theta functions, new fascinating identities on different continued fractions can be achieved. In this article, we use the product expansion of Jacobi's theta function to…

We define and study a multidimensional process that generalizes the eigenvalues of matrix Jacobi processes on the one hand and whose stationary distribution is given by the beta Jacobi ensemble on the other hand.

Landen formulas, which connect Jacobi elliptic functions with different modulus parameters, were first obtained over two hundred years ago by making a suitable quadratic transformation of variables in elliptic integrals. We obtain and…

We extend A.B. Mingarelli's method for constructing generalized factorials. Our extension uses a pair of arithmetic functions $(x, y)$, where $x$ is superadditive. When $x$ is the identity function, our generalized factorial reduces to…

Poisson-Lie T-duality/plurality was recently generalized to Jacobi-Lie T-plurality formulated in terms of Double Field Theory and based on Leibniz algebras given by structure coefficients $f_{ab}{}^{c},f_{c}{}^{ab},$ and $Z_a,Z^a$. We…

Using a general $q$-series expansion, we derive some nontrivial $q$-formulas involving many infinite products. A multitude of Hecke--type series identities are derived. Some general formulas for sums of any number of squares are given. A…

The Mordell-Tornheim zeta function and the Herglotz-Zagier function $F(x)$ are two important functions in Mathematics. By generalizing a special case of the former, namely $\Theta(z, x)$, we show that the theories of these functions are…

We recast Byerly's formula for integrals of products of Legendre polynomials. Then we adopt the idea to the case of Jacobi polynomials. After that, we use the formula to derive an asymptotic formula for integrals of products of Jacobi…

For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by…

Using a probabilistic approach, we derive some interesting combinatorial identities involving gamma and beta functions. These results generalize certain well-known combinatorial identities involving binomial coefficients and special…

We prove new bounds for the Fourier coefficients of Jacobi forms using a method of Iwaniec. In view of the Fourier-Jacobi expansion of degree two Siegel modular forms, we can use these to obtain strong bounds on fundamental Fourier…

Jacobi's $\theta$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $\theta(z,\tau)$, for $z, \tau$ verifying certain conditions, with precision $P$ in $O(\mathcal{M}(P)…

We show how Andrews' generating functions for generalized Frobenius partitions can be understood within the theory of Eichler and Zagier as specific coefficients of certain Jacobi forms. This reformulation leads to a recursive process which…

George Andrews and Ae Ja Yee recently established beautiful results involving bivariate generalizations of the third order mock theta functions $\omega(q)$ and $\nu(q)$, thereby extending their earlier results with the second author.…

The formula of expanding the Abel variety theta function restricted to Abel subvariety into theta functions of this subvariety is obtained. With the help of this formula the solution of differential equations with Jacobi theta functions,…

In this paper we review and derive hyperbolic and trigonometric double summation addition theorems for Jacobi functions of the first and second kind. In connection with these addition theorems, we perform a full analysis of the relation…

We give a criterion to distinguish between a genus three Jacobian and its [-1] twist in terms of the product of the 36 even theta nulls. We also express the product of the 36 theta nulls in terms of the discriminant of a genus three curve.…

Generalized Jacobi polynomials are orthogonal polynomials related to a weight function which is smooth and positive on the whole interval of orthogonality up to a finite number of points, where algebraic singularities occur. The influence…