Related papers: An Elliptic $BC_n$ Bailey Lemma, Multiple Rogers--…
Using Abel's five-term relation, we derive a new two-parameter series identity for the Rogers dilogarithm. By specializing this identity, we obtain dilogarithm series involving Lucas sequences. These results generalize certain series…
We present various identities involving the classical Bernoulli and Euler polynomials. Among others, we prove that $$ \sum_{k=0}^{[n/4]}(-1)^k {n\choose 4k}\frac{B_{n-4k}(z) }{2^{6k}} =\frac{1}{2^{n+1}}\sum_{k=0}^{n} (-1)^k…
In 2018, Stanton proved two types of generalisations of the celebrated Andrews--Gordon and Bressoud identities (in their $q$-series version): one with a similar shape to the original identities, and one involving binomial coefficients. In…
We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial…
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 prove a master theorem for hypergeometric functions of Karlsson-Minton type, stating that a very general multilateral U(n) Karlsson-Minton type hypergeometric series may be reduced to a finite sum. This identity contains the…
The potential of the $BC_1$ quantum elliptic model is a superposition of two Weierstrass functions with doubling of both periods (two coupling constants). The $BC_1$ elliptic model degenerates to $A_1$ elliptic model characterized by the…
We give elementary proofs of the univariate elliptic beta integral with bases $|q|, |p|<1$ and its multiparameter generalizations to integrals on the $A_n$ and $C_n$ root systems. We prove also some new unit circle multiple elliptic beta…
It is shown how many of the partial theta function identities in Ramanujan's lost notebook can be generalized to infinite families of such identities. Key in our construction is the Bailey lemma and a new generalization of the Jacobi triple…
Basil Gordon, in the sixties, and George Andrews, in the seventies, generalized the Rogers-Ramanujan identities to higher moduli. These identities arise in many areas of mathematics and mathematical physics. One of these areas is…
We obtain some new bifurcation criteria for solutions of general boundary value problems for nonlinear elliptic systems of partial differential equations. The results are of different nature from the ones that can be obtained via the…
We use the exterior product of double forms to reformulate celebrated classical results of linear algebra about matrices and bilinear forms namely the Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton identities and…
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 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…
Via the contour integral method, we establish a reduction formula from a double series to a single series with parameters, which not only implies Uncu and Zudilin's two results and Cao and Wang's two results, but also is related to…
We compare previously found finite-dimensional matrix and integral operator realizations of the Bailey lemma employing univariate elliptic hypergeometric functions. With the help of residue calculus we explicitly show how the integral…
We introduce the symmetric (respectively, non-symmetric) $\tau_{-\ell}-$hypergeometric functions associated with a root system of type $BC$ as joint eigenfunctions of a commutative algebra of differential (respectively,…
A pair of sequences $(\alpha_{n}(a,k,q),\beta_{n}(a,k,q))$ such that $\alpha_0(a,k,q)=1$ and \[ \beta_{n}(a,k,q) = \sum_{j=0}^{n} \frac{(k/a; q)_{n-j}(k; q)_{n+j}}{(q;q)_{n-j}(aq;q)_{n+j}}\alpha_{j}(a,k,q) \] is termed a \emph{WP-Bailey…
By virtue of Bailey's well-known bilateral 6\psi_6 summation formula and Watson's transformation formula,we extend the four-variable generalization of Ramanujan's reciprocity theorem due to Andrews to a five-variable one. Some relevant new…
The Askey--Wilson polynomials are the most general classical orthogonal polynomials that are known and the Nassrallah--Rahman integral is a very general extension of Euler's integral representation of the classical $_2F_1$ function. Based…