相关论文: An esoteric identity with many parameters and othe…

In this paper, we use the effect of the $q$-differential and deformed $q$-exponential operators on basic hypergeometric series to find new $q$-identities from the $q$-Gauss sum, the $q$-Chu-Vandermonde's sum, and Jackson's transformation…

Using multiple q-integrals and a determinant evaluation, we establish a multivariable extension of Bailey's nonterminating 10-phi-9 transformation. From this result, we deduce new multivariable terminating 10-phi-9 transformations, 8-phi-7…

Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…

In this paper we provide some relationships between Catalan's constant and the $_3{\rm F}_2$ and $_4{\rm F}_3$ hypergeometric functions, deriving them from some parametric integrals. In particular, using the complete elliptic integral of…

We prove $q$-analogues of identities that are equivalent to the functional equation of the arithmetic-geometric mean. We also present $q$-analogues of $F(\sqrt{k},\frac{\pi}{2})$, the complete elliptical integral of the first kind, and its…

We give a method to embed the q-series in a (p,q)-series and derive the corresponding (p,q)-extensions of the known q-identities. The (p,q)-hypergeometric series, or twin-basic hypergeometric series (diferent from the usual bibasic…

We use elliptic Taylor series expansions and interpolation to deduce a number of summations for elliptic hypergeometric series. We extend to the well-poised elliptic case results that in the $q$-case have previously been obtained by Cooper…

We deduce several curious q-series expansions by applying inverse relations to certain identities for basic hypergeometric series. After rewriting some of these expansions in terms of q-integrals, we obtain, in the limit q -> 1, some…

We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…

We obtain new elliptic function identities, which are an elliptic analogue of Fukuhara's trigonometric identities. We show that the coefficients of Laurent expansions at $z=0$ of our elliptic identities give rise to some reciprocity laws…

Motivated by the telescoping proofs of two identities of Andrews and Warnaar, we find that infinite q-shifted factorials can be incorporated into the implementation of the q-Zeilberger algorithm in the approach of Chen, Hou and Mu to prove…

There are many identities for the hypergeometric series presented in the article "Special values of the hypergeometric series" by Ebisu. In this note, we obtain a new hypergeometric identity, which includes some of these identities as…

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…

We revisit the classical integrals introduced by Coxeter, not to recalculate their well-known exact values, but to use them as a tool to derive elliptic integral identities. By embedding Coxeter's first integral into a one-parameter family…

We prove a conjecture that arose in the context of a subspace enumeration problem over finite fields. We prove, more generally, a bibasic, double-sum identity, which extends a $q$-analogue of the (terminating) binomial theorem.

Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we…

We prove a duality relation for generalized basic hypergeometric functions. It forms a $q$-extension of a recent result of the second and the third named authors and generalizes both a $q$-hypergeometric identity due to the third named…

Elliptic 6j-symbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6j-symbols (or q-Racah polynomials) and Wilson's biorthogonal 10-W-9 functions. We…

We obtain eigenvalue equations satisfied by various elliptic modular graphs with five links where two of the vertices are unintegrated. Solving them leads to several non--trivial algebraic identities between these graphs.

We present several identities with a form of polynomials or rational functions that involve Pochhammer and q-Pochhammer symbols and q-binomials (i.e. Gauss polynomials). All these identities were obtained by some analytical methods based on…