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The modified q-Bessel functions and the q-Bessel-Macdonald functions of the first and second kind are introduced. Their definition is based on representations as power series. Recurrence relations, the q-Wronskians, asymptotic…
Using the theory of representations of the symmetric group, we propose an algorithm to compute the invariant ring of a permutation group. Our approach have the goal to reduce the amount of linear algebra computations and exploit a thinner…
Given a partition $\lambda$ of $n$, the {\it Schur functor} $\mathbb{S}_\lambda$ associates to any complex vector space $V$, a subspace $\mathbb{S}_\lambda(V)$ of $V^{\otimes n}$. Hermite's reciprocity law, in terms of the Schur functor,…
We formulate and prove the analogue of the Ramanujan Conjectures for modular forms of half-integral weight subject to some ramification restriction in the setting of a polynomial ring over a finite field. This is applied to give an…
The new representation formula for the spectral shift function due to F.Gesztesy and K.A.Makarov is considered. This formula is extended to the case of relatively trace class perturbations.
The cycle polynomial of a finite permutation group $G$ is the generating function for the number of elements of $G$ with a given number of cycles: \[F_G(x) = \sum_{g\in G}x^{c(g)},\] where $c(g)$ is the number of cycles of $g$ on $\Omega$.…
Symmetry analysis of Ramanujan's system of differential equations is performed by representing it as a third-order equation. A new system consisting of a second-order and a first-order equation is derived from Ramanujan's system. The Lie…
It was recently shown that $q\omega(q)$, where $\omega(q)$ is one of the third order mock theta functions, is the generating function of $p_{\omega}(n)$, the number of partitions of a positive integer $n$ such that all odd parts are less…
Ramanujan in his second notebook recorded total of seven $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are…
A new recursion in only one variable allows very simple verifications of Bressoud's polynomial identities, which lead to the Rogers-Ramanujan identities. This approach might be compared with an earlier approach due to Chapman. Applying the…
In solving $q$-difference equations, and in the definition of $q$-special functions, we encounter formal power series in which the $n$th coefficient is of size $q^{-\binom{n}{2}}$ with $q\in(0,1)$ fixed. To make sense of these formal…
Rousseau's simple proof of the quadratic reciprocity law, followed by the proof of its equivalence with Hilbert's product formula. The Hilbert symbol is explained in terms of the reciprocity isomorphism, and the places of Q are determined.
By using two known transformation formulas for basic hypergeometric series, we establish a direct extension of Bailey's $_6\psi_6$-series identity. Subsequently, it and Milne's identity are employed to drive multi-variable generalizations…
Recently, Rosengren utilized an integral method to prove a number of conjectural identities found by Kanade and Russell. Using this integral method, we give new proofs to some double sum identities of Rogers-Ramanujan type. These identities…
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 define an overpartition analogue of Gaussian polynomials (also known as $q$-binomial coefficients) as a generating function for the number of overpartitions fitting inside the $M \times N$ rectangle. We call these new polynomials over…
We build on a recent paper on Fourier expansions for the Riemann zeta function. We establish Fourier expansions for certain $L$-functions, and offer series representations involving the Whittaker function $W_{\gamma,\mu}(z)$ for the…
We introduce four q-analogs of the double Laplace transform and prove some of their main properties. Next we show how they can be used to solve some q-functional equations and partial q-differential equations.
We offer some further applications of some Bailey pairs related to some mock theta functions which were established in a recent study. We discuss and offer some double-sum $q$-series, with new relationships among mock theta functions. We…
Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third order mock theta functions $\omega(q)$ and $\nu(q)$. In this paper, we find several new exact generating functions for those partition…