Related papers: Transformation Formulae and Applications for Doubl…
We announce a higher-dimensional generalization of the Bailey Transform, Bailey Lemma, and iterative ``Bailey chain'' concept in the setting of basic hypergeometric series very well-poised on unitary $A_{\ell}$ or symplectic $C_{\ell}$…
In this paper, we use two-variable Laurent polynomials attached to matrices to encode properties of compositions of sequences. The Lagrange identity in the ring of Laurent polynomials is then used to give a short and transparent proof of a…
Based on an interesting identity of Bat{\i}r we derive new identities for double sums involving famous number sequences. We also prove some double sum identities for binomial transform pairs.
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…
In the present paper, we establish a general transformation for $q$-series which contains L. Wang et al's transformation involved in Nahm series. As direct applications, some concrete new transformation formulas for the ${}_{r+1}\phi_r$…
Recently, the author and Yamamoto invented a new proof of the duality for multiple zeta values. The technique is applicable in other series identities. In this article, we exhibit such proofs for some series identities.
We consider special Lambert series as generating functions of divisor sums and determine their complete transseries expansion near rational roots of unity. Our methods also yield new insights into the Laurent expansions and modularity…
We summarize the known useful and interesting results and formulas we have discovered so far in this collaborative article summarizing results from two related articles by Merca and Schmidt arriving at related so-termed Lambert series…
We present two general finite extensions for each of the two Rogers-Ramanujan identities. Of these one can be derived directly from Watson's transformation formula by specialization or through Bailey's method, the second similar formula can…
We investigate Newton series for truncated multiple $L$-values and thereby obtain a class of relations for multiple $L$-values. In addition, we give a formulation and a proof of extended derivation relations for multiple $L$-values.
The double exponential formula was introduced for calculating definite integrals with singular point oscillation functions and Fourier-integrals. The double exponential transformation is not only useful for numerical computations but it is…
We consider certain Lambert series as generating functions of divisor sums twisted by Dirichlet characters and compute their exact resurgent transseries expansion near $q=1^-$. For special values of the parameters, these Lambert series are…
The paper contains a systematic theory of the one-dimensional Double Hecke algebra, including applications to the difference Fourier transform, Macdonald's polynomials, Gaussian sums at roots of unity, and Verlinde algebras. The main result…
We study multivariable (bilateral) basic hypergeometric series associated with (type $A$) Macdonald polynomials. We derive several transformation and summation properties for such series including analogues of Heine's ${}_2\phi_1$…
In terms of Sear's transformation formula for $_4\phi_3$-series, we give new proofs of a summation formula for ${_4\phi_3}$-series due to Andrews [2] and another summation formula for${_4\phi_3}$-series conjectured in the same paper.…
As a continuation of the authors and Wakatsuki's previous paper [5], we study relations among Dirichlet series whose coefficients are class numbers of binary cubic forms. We show that for any integral models of the space of binary cubic…
We show that the Bailey lattice can be extended to a bilateral version in just a few lines from the bilateral Bailey lemma, using a very simple lemma transforming bilateral Bailey pairs relative to $a$ into bilateral Bailey pairs relative…
We take matrix decompositions that are usually applied to matrices over the real numbers or complex numbers, and extend them to matrices over an algebra called the double numbers. In doing so, we unify some matrix decompositions: For…
Through symbolic methods, we state explicit formulae for Tchebychev, Gegenbauer, Meixner, Mittlag-Leffler, and Pidduck polynomials. This is done by underlining the crucial role played by the Abel identity in revisiting the Lagrange…
Employing the Lagrange inverting series, a solution of the transcendental equation $(x-a)(x-b)=le^{x}$, that can be considered a quadratic generalization of the equation defining Lambert $W$ function, has been found in terms of Bessel…