Related papers: On three third order mock theta functions and Heck…
In this paper we offer some new identities associated with mock theta functions and establish new Bailey pairs related to indefinite quadratic forms. We believe our proof is instructive use of changing base of Bailey pairs, and offers new…
We prove Rogers-Ramanujan type identities for the Nahm sums associated with the tadpole Cartan matrix of rank $3$. These identities reveal the modularity of these sums, and thereby we confirm a conjecture of Penn, Calinescu and the first…
We prove that the coefficients of certain mock theta functions possess no linear congruences modulo 3. We prove similar results for the moduli 2 and 3 for a wide class of weakly holomorphic modular forms and discuss applications. This…
In this paper, we will study various connections between double zeta values relative to $\mu_N$, Hecke operators $T_N$, and newforms of level $\Gamma_0(N)$ for $N=2,3$. Those various connections generalize the well-known of Baumard and…
Let $f_k$ be the $k$-th Fourier coefficient of a function $f$ in terms of the orthonormal Hermite, Laguerre or Jacobi polynomials. We give necessary and sufficient conditions on $f$ for the inequality $\sum_{k}|f_k|^2\theta^k<\infty$ to…
It is well known that Ramanujan conjectured congruences modulo powers of $5$, $7$ and and $11$ for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences…
We exhibit an infinite family of vector-valued mock theta functions indexed by positive integers coprime to $6$. These are built from specializations of Dyson's rank generating function and related functions studied by Watson, Gordon, and…
While examples of Ramanujan-type congruences are amply available via their relation to Hecke operators, it remains unclear which of them should be considered of combinatorial origin and which of them are mere artifacts of the connection…
A stream of new theta relations is obtained. They follow from the general Thomae formula, which is a new result giving expressions for theta derivatives (the zero values of the lowest non-vanishing derivatives of theta functions with…
We compute the Quot and finitized Coh zeta functions of the inert quadratic orders $\mathbb{F}_q[[T]]+T^{m}\mathbb{F}_{q^{2}}[[T]]$ for every $m\geq 1$ in terms of a $2m$-fold multisum, and then show this multisum equals an $m$-fold…
In this note we consider infinite series similar to the "strange" function $F(q)$ of Kontsevich studied by Zagier, Bryson-Ono-Pitman-Rhoades, Bringmann-Folsom-Rhoades, Rolen-Schneider, and others in connection to quantum modular forms. We…
Generalized $m$-gonal numbers are those $p_m(x)= [ (m - 2)x^2 - (m - 4)x ]/2 $ where $x$ and $m$ are integers with $m \geq 3$. If any nonnegative integer can be written in the form $ap_r(h)+bp_s(l)+cp_t(m)+dp_u(n)$, where $a,b,c,d$ are…
We show that identities involving trigonometric sums recently proved by Harshitha, Vasuki and Yathirajsharma, using Ramanujan's theory of theta functions, were either already in the literature or can be proved easily by adapting results…
We present a unified approach which gives completely elementary proofs of three weighted sum formulae for double zeta values. This approach also leads to new evaluations of sums relating to the harmonic numbers, the alternating double zeta…
In this paper, we initiate a generous amount of new-found general theorems for explicit evaluations of product of the theta functions $b_{m, n}$ using Kronecker's limit formula and other various novel explicit evaluations that were…
We provide new proofs to five of Ramanujan's intriguing identities on false theta functions without using the Rogers-Fine identity and Bailey transforms.
We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the Reshetikhin-Turaev invariants of closed 3-manifolds constructed from the quantum groups U_q sl(N) by…
We consider integral and series transformations, which are associated with Ramanujan's identities, involving various arithmetic functions and a ratio of products of Riemann's zeta functions of different arguments. Reciprocal inversion…
We prove three variations of recent results due to Andrews on congruences for $NT(m,k,n)$, the total number of parts in the partitions of $n$ with rank congruent to $m$ modulo $k$. We also conjecture new congruences and relations for…
In this paper, we establish some new inequalities for functions whose third derivatives in the absolute value are m-convex.