Some new $q$-congruences for truncated basic hypergeometric series: even powers

Abstract

We provide several new $q$-congruences for truncated basic hypergeometric series with the base being an even power of $q$. Our results mainly concern congruences modulo the square or the cube of a cyclotomic polynomial and complement corresponding ones of an earlier paper containing $q$-congruences for truncated basic hypergeometric series with the base being an odd power of $q$. We also give a number of related conjectures including $q$-congruences modulo the fifth power of a cyclotomic polynomial and a congruence for a truncated ordinary hypergeometric series modulo the seventh power of a prime greater than 3.

13 pages, more background and references added

References in corpus (4)

• Some new $q$-congruences for truncated basic hypergeometric series: even powers
• Proof of a basic hypergeometric supercongruence modulo the fifth power of a cyclotomic polynomial
• Some new $q$-congruences for truncated basic hypergeometric series
• On some conjectural congruences

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• Divisibility results concerning truncated hypergeometric series
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• Proof of Sun's supercongruence involving Catalan numbers
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• A $p$-adic analogue of Chan and Verrill's formula for $1/\pi$
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