Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler’s constant γ. The proof is by reduction to known irrationality criteria for γ involving a Beukers-type double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, γ, and logari...

We study the combinatorics of two classes of basic hypergeometric series. We first show that these series are the generating functions for certain overpartition pairs defined by frequency conditions on the parts. We then show that when specialized these series are also the generating functions for overpartition pairs with bounded successive ranks, overpartition pairs with conditions on their Du...

Ramanujan studied the analytic properties of many q-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious q-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have q-expansions resembling modular theta functions, is not well understood. Here we con...

We derive a formula for the n-row Macdonald polynomials with the coefficients presented both combinatorically and in terms of very-well-poised hypergeometric series.

Stable sampling formula using basic hypergeometric series for reconstructing analytic functions from exponentially spaced samples is considered. Criterion for selecting regularizing parameter and error estimates are obtained.

Let p be an odd prime. The purpose of this paper is to refine methods of Ahlgren and Ono [2] and Kilbourn [13] in order to prove a general mod p congruence for the Gaussian hypergeometric series n+1Fn(λ) where n is an odd positive integer. As a result, we extend three recent supercongruences. The first is a result of Ono and Ahlgren [2] on a supercongruence for Apéry numbers which was conjectur...

Abstract. 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 2φ1 transformation, the q-Pfaff-Kummer and Euler transformations, the q-Saalschütz summation formula, and Sear’s transformation for terminating, balanced 4φ3 series. For ...

We show that some q-series such as universal mock theta functions are linear sums of theta quotient and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are multiplied by suitable powers of q. And we prove that certain linear sums of q-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or com...

The author introduces overpartitions and particle seas as a generalization of partitions. Both new tools are used in bijective proofs of basic hypergeometric identities like the q-binomial theorem, Jacobi’s triple product, q-Gauß equality or even Ramanujan’s 1Ψ1 summation. 1. Partitions In 1969, G. E. Andrews was already looking for bijective proofs for some basic hypergeometric identities. The...

We study the perturbative power series expansions of the eigenvalues and eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d. The (small) expansion parameters are the entries of the two diagonals of length d-1 sandwiching the principal diagonal that gives the unperturbed spectrum. The solution is found explicitly in terms of multivariable (Horn-type) hypergeometric series in 3d...