Related papers: Theta function identities and q series
In this note, we describe a general procedure to prove functional equations involving quasi-periodic functions. We give novel proofs for fundamental identities of Weierstrass sigma and Jacobi theta functions. Our method is based on the…
The mock theta conjectures are ten identities involving Ramanujan's fifth-order mock theta functions. The conjectures were proven by Hickerson in 1988 using q-series methods. Using methods from the theory of harmonic Maass forms,…
In this paper we generalize the famous Jacobi's triple product identity, considered as an identity for theta functions with characteristics and their derivatives, to higher genus/dimension. By applying the results and methods developed in…
Liu established an addition formula for the Jacobian theta function by using the theory of elliptic functions. From this addition formula he obtained the Ramanujan cubic theta function identity, Winquist's identity, a theta function…
The two Rogers-Ramanujan $q$-series \[ \sum_{n=0}^{\infty}\frac{q^{n(n+\sigma)}}{(1-q)\cdots (1-q^n)}, \] where $\sigma=0,1$, play many roles in mathematics and physics. By the Rogers-Ramanujan identities, they are essentially modular…
Partitions associated with mock theta functions have received a great deal of attention in the literature. Recently, Choi and Kim derived several partition identities from the third and sixth order mock theta functions. In addition, three…
Using new $q$-functions recently introduced by Hatayama et al. and by (two of) the authors, we obtain an A_2 version of the classical Bailey lemma. We apply our result, which is distinct from the A_2 Bailey lemma of Milne and Lilly, to…
We give new generalizations of some q-series identities of Dilcher and Prodinger related to divisor functions. Some interesting special cases are also deduced, including an identity related to overpartitions studied by Corteel and Lovejoy.
Applying the theory of elliptic functions we establish two Jacobi theta function identities. From these identities we confirm two q-trigonometric identities conjectured by Gosper. As an application, we give a new and simple proof of a…
In this article we study properties of Ramanujan's mock theta functions that can be expressed in Lerch sums. We mainly show that each Lerch sum is actually the integral of a Jacobian theta function (here we show that for $\vartheta_3(t,q)$…
The zeta function attached to a finite complex $X_\Gamma$ arising from the Bruhat-Tits building for $\PGL_3(F)$ was studied in \cite{KL}, where a closed form expression was obtained by a combinatorial argument. This identity can be…
Various new identities, recurrence relations, integral representations, connection and explicit formulas are established for the Bernoulli, Euler numbers and the values of Riemann's zeta function. To do this, we explore properties of some…
In 2003, Zudilin presented a $q$-analogue of Euler's identity for one of the variants of $q$-double zeta function. This article focuses on exploring identities related to another variant of $q$-double zeta function and its star variant.…
We obtain several expansions for $\zeta(s)$ involving a sequence of polynomials in $s$, denoted in this paper by $\alpha_k(s)$. These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities…
Studied is a generalization of Zagier's q-series identity. We introduce a generating function of L-functions at non-positive integers, which is regarded as a half-differential of the Andrews--Gordon q-series. When q is a root of unity, the…
In this paper, we explore the role that Liu's transformation formula can play in discovering Rogers-Ramanujan type identities. Specifically, we combine Liu's transformation formula with other $q$-series summations to derive a series of…
After obtaining some useful identities, we prove an additional functional relation for $q$ exponentials with reversed order of multiplication, as well as the well known direct one in a completely rigorous manner.
In 1991, the Borweins established a cubic analogue of Jacobi's identity for theta functions, which is used by B.C. Berndt, S. Bhargava, and F.G. Garvan in the development of Ramanujan's cubic theory of elliptic functions. In 2013, D.…
We show that all of Ramanujan's mock theta functions of order 3, Watson's three additional mock theta functions of order 3, the Rogers-Ramanujan q-series, and 6 mock theta functions of order 5 take on irrational values at the points q=\pm…
We prove the transformation laws of the four Jacobi theta functions using Gordon's proof for the transformation law of the Dedekind eta function.