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Binomial Coefficient – Harmonic Sum Identities Associated to Supercongruences
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Dermot McCarthy
Published/Copyright:
August 4, 2011
Abstract
We establish two binomial coefficient-generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of the author, they are used to establish modulo pk (k > 1) congruences between truncated generalized hypergeometric series, and a function which extends Greene's hypergeometric function over finite fields to the p-adic setting. A specialization of one of these congruences is used to prove an outstanding conjecture of Rodriguez-Villegas which relates a truncated generalized hypergeometric series to the p-th Fourier coefficient of a particular modular form.
Received: 2011-01-31
Accepted: 2011-05-03
Published Online: 2011-08-04
Published in Print: 2011-December
© de Gruyter 2011
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- Completely Multiplicative Automatic Functions
- The k-Periodic Fibonacci Sequence and an Extended Binet's Formula
- Robin's Theorem, Primes, and a New Elementary Reformulation of the Riemann Hypothesis
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- Euler's Pentagonal Number Theorem Implies the Jacobi Triple Product Identity
- On Directions Determined by Subsets of Vector Spaces over Finite Fields
- A Remark on a Paper of Luca and Walsh
- On the Tennis Ball Problem
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- Convolution and Reciprocity Formulas for Bernoulli Polynomials
- Counting Finite Languages by Total Word Length
Keywords for this article
Binomial Coefficient;
Harmonic Sum;
Identity;
Supercongruence;
Hypergeometric Function
Articles in the same Issue
- Completely Multiplicative Automatic Functions
- The k-Periodic Fibonacci Sequence and an Extended Binet's Formula
- Robin's Theorem, Primes, and a New Elementary Reformulation of the Riemann Hypothesis
- Reducing the Erdős–Moser Equation 1n + 2n + ⋯ + kn = (k + 1)n Modulo k and k2
- On Some Conjectures Concerning Stern's Sequence and Its Twist
- Number of Weighted Subsequence Sums with Weights in {1, –1}
- Binomial Coefficient – Harmonic Sum Identities Associated to Supercongruences
- Euler's Pentagonal Number Theorem Implies the Jacobi Triple Product Identity
- On Directions Determined by Subsets of Vector Spaces over Finite Fields
- A Remark on a Paper of Luca and Walsh
- On the Tennis Ball Problem
- On the Conditioned Binomial Coefficients
- Convolution and Reciprocity Formulas for Bernoulli Polynomials
- Counting Finite Languages by Total Word Length
