Related papers: The q-WZ Method for Infinite Series
We use the Algortihm Z on partitions due to Zeilberger, in a variant form, to give a combinatorial proof of Ramanujan's $_1\psi_1$ summation formula.
In this paper we derive uniform asymptotic expansions for the partial sums of the exponential series. We indicate how this information will be used in a later publication to obtain full and explicitly computable asymptotic expansions with…
Two vectors in $\BZ^3$ are called \emph{twins} if they are orthogonal and have the same length. The paper describes twin pairs using cubic lattices, and counts the number of twin pairs with a given length. Integers $M$ with the property…
Surprisingly, apart from some special cases, simple asymptotic expansions for the associated Legendre functions $P_\nu ^\mu (z)$ and $Q_\nu ^\mu (z)$ for large degree $\nu$ or large order $\mu$ are not available in the literature. The main…
In this paper, by use of matrix inversions, we establish a general $q$-expansion formula of arbitrary formal power series $F(z)$ with respect to the base $$\left\{z^n\frac{(az:q)_n}{(bz:q)_n}\bigg|n=0,1,2\cdots\right\}.$$ Some concrete…
We revisit the q-deformed counterpart of the Zassenhaus formula, expressing the Jackson $q$-exponential of the sum of two non-$q$-commuting operators as an (in general) infinite product of $q$-exponential operators involving repeated…
We provide combinatorial proofs of some of the q-series identities considered by Andrews, Jimenez-Urroz and Ono [q-series identities and values of certain $L$-functions. Duke Math. J. 108 (2001), no. 3, 395--419].
Recently, Schlosser and Zhou proposed many conjectures on sign patterns of the coefficients appearing in the $q$-series expansions of the infinite Borwein product and other infinite products raised to a real power. In this paper, we will…
Parametric families in the centre ${\bf Z}({\bf C}[S_n])$ of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements.…
We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x^2+5y^2. Making use of Ramanujan's_1\psi_1 summation formula we establish a new Lambert series identity for…
In this study, using q-generalized bit cumulants (q is the nonextensivity parameter of the recently introduced Tsallis statistics), we investigate the asymmetric unimodal maps. The study of the q-generalized second cumulant of these maps…
The $q$-Whittaker function $W_\lambda(\mathbf{x};q)$ associated to a partition $\lambda$ is a $q$-analogue of the Schur function $s_\lambda(\mathbf{x})$, and is defined as the $t=0$ specialization of the Macdonald polynomial…
We provide several new $q$-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric…
We present a theorem on taking the repeated indefinite summation of a holomorphic function $\phi(z)$ in a vertical strip of $\mathbb{C}$ satisfying exponential bounds as the imaginary part grows. We arrive at this result using transforms…
Lucy Slater used Bailey's $_6\psi_6$ summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type. In the present paper we apply the same techniques to Chu's…
We give multidimensional generalizations of several transformation formulae for basic hypergeometric series of a specific type. Most of the upper parameters of the series differ multiplicatively from corresponding lower parameters by a…
We present algorithms to factorize weighted homogeneous elements in the first polynomial Weyl algebra and $q$-Weyl algebra, which are both viewed as a $\mathbb{Z}$-graded rings. We show, that factorization of homogeneous polynomials can be…
Product identities in two variables $x, q$ expand infinite products as infinite sums, which are linear combinations of theta functions; famous examples include Jacobi's triple product identity, Watson's quintuple identity, and Hirschhorn's…
We define the concept of an affinized projective variety and show how one can, in principle, obtain q-identities by different ways of computing the Hilbert series of such a variety. We carry out this program for projective varieties…
In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial n'th Weyl algebra, the polynomial n'th shift algebra, and ZZ^n-graded polynomials in the n'th q-Weyl algebra. The most…