Congruences involving F-partition functions.

Sellers, James

Displaying similar documents to “Congruences involving F-partition functions.”

Congruences involving generalized Frobenius partitions.

Sellers, James (1993)

International Journal of Mathematics and Mathematical Sciences

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Congruences involving the Fermat quotient

Romeo Meštrović (2013)

Czechoslovak Mathematical Journal

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Let $p > 3$ be a prime, and let $q_{p} (2) = (2^{p - 1} - 1) / p$ be the Fermat quotient of $p$ to base $2$ . In this note we prove that $\sum_{k = 1}^{p - 1} \frac{1}{k \cdot 2^{k}} \equiv q_{p} (2) - \frac{p q_{p} {(2)}^{2}}{2} + \frac{p^{2} q_{p} {(2)}^{3}}{3} - \frac{7}{48} p^{2} B_{p - 3} (mod p^{3}),$ which is a generalization of a congruence due to Z. H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z. H. Sun, we show that $q_{p} {(2)}^{3} \equiv - 3 \sum_{k = 1}^{p - 1} \frac{2^{k}}{k^{3}} + \frac{7}{16} \sum_{k = 1}^{(p - 1) / 2} \frac{1}{k^{3}} (mod p),$ which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum $\sum_{k = 1}^{p - 1} 1 / (k^{2} \cdot 2^{k})$ modulo $p^{2}$ that also generalizes...

Congruences for Stirling numbers and Eulerian numbers

Hui-Qin Cao, Hao Pan (2008)

Acta Arithmetica

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Infinite families of divisibility properties modulo 4 for non-squashing partitions into distinct parts.

Hirschhorn, Michael D., Rødseth, Øystein J., Sellers, James A. (2009)

Integers

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Relation products of congruences and factor congruences

Jaromír Duda (1991)

Czechoslovak Mathematical Journal

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Kummer congruences for expressions involving generalized Bernoulli polynomials

Glenn J. Fox (2002)

Journal de théorie des nombres de Bordeaux

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We illustrate how a particular expression, involving the generalized Bernoulli polynomials, satisfies systems of congruence relations if and only if a similar expression, involving the generalized Bernoulli numbers, satisfies the same congruence relations. These congruence relations include the Kummer congruences, and recent extensions of the Kummer congruences provided by Gunaratne.

On congruence properties of the partition function.

Getz, Jayce (2000)

International Journal of Mathematics and Mathematical Sciences

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Congruences for hyper $m$ -ary overpartition functions.

Ma, Lin, Lu, Qing-Lin (2010)

Integers

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Exceptional congruences for powers of the partition function

Matthew Boylan (2004)

Acta Arithmetica

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Congruences in ordered pairs of partitions.

Hammond, Paul, Lewis, Richard (2004)

International Journal of Mathematics and Mathematical Sciences

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Some congruences for 3-component multipartitions

Tao Yan Zhao, Lily J. Jin, C. Gu (2016)

Open Mathematics

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Let p3(n) denote the number of 3-component multipartitions of n. Recently, using a 3-dissection formula for the generating function of p3(n), Baruah and Ojah proved that for n ≥ 0, p3(9n + 5) ≡ 0 (mod 33) and p3 (9n + 8) ≡ 0 (mod 34). In this paper, we prove several congruences modulo powers of 3 for p3(n) by using some theta function identities. For example, we prove that for n ≥ 0, p3 (243n + 233) ≡ p3 (729n + 638) ≡ 0 (mod 310).